Download Rule 3 - elearnocean

48
PRACTICE BOOK ON QUICKER MATHS
5.
The quotient arising from the division of 24446 by a
certain number is 79 and the remainder is 35,what is the
.divisor?
a) 309
b)319
c)310
d) 379
6. A boy had to divide 49471 by 210. He made some mistake
in copying the divisor and obtained as his quotient 246
with a remainder 25. What mistake did he make?
a) He made no mistake
b) He put down 120 for 21 0
c) He put down lO2fo~ 210 d) He put dwn 201 for 21 0
7. Ina division sum the dividend is 57324 and quotient 123.
If the remainder is greater than the' quotient but less han
twice the quotient. Find the divisor.
a) 465
b) 475
c) 645
. d) 565
:. the least number to be added = 58.
Ex. 2: Find the least number of 3 digits, which is exactly
divisible by 14.
Solo: The least number of3 digits = 100 On
dividing 100 by 14, remainder = 2
To determine exactly divisible least number, the above
method will be applied.
:. The required number
= Dividend + (Divisor...: Remainder)
= 100+(14-2)= 112.
Exercise
1.
Answers
I.b
2.a·
3.b
4.c
5.a
6.d
7.a
2.
Rule 3
A number (Dividend) can be made completely divisible with the
help of either of the following methods:
Divisor) Dividend. (Quotient
3.
4.
,,
1"
Iii[
I
I.
Remainder
Method I: By subtracting remainder from dividend. For finding
the greatest n-digit number completely divisible by a divisor,
this rule is applicable.
5.
Illustrative Examples
6.
Ex. I: Find the greatest number of 3 digits, which is exactly
divisible by 35.
Solo: The greatest number of3 digit = 999 On
dividing 999 by 35, remainder =-19.
Now, applying the above method,
the required number = dividend - remainder = 999 19=980
Ex.2: Find the least number that must be subtracted from
87375, to get a number exactly divisible by 698.
Solo: On dividing 67375 by 698, the remainder is 125. By the
above method,
The least number to be subtracted is the remainder from
dividend.
:. the least number to be subtracted = 125.
Method II: By adding (divisor- remainder) to dividend. For
finding the least n-digit number completely divisible by a
divisor, this rule is applicable.
Illustrative Examples
Ex. I: What least number must be added to 49123 to get a
number exactly divisible by 263.
Solo: On dividing 49123 by 263, the remainder is 205.
By the above method,
The least number to be added to the dividend =
divisor - remainder =263-205=58.
7.
8.
9.
10.
11.
12.
13.
14.
What least number must be subtracted from 5731625, to
get a number exactly divisible by 3546?
a) 1189
b) 1829
c) 1289
d) 1982
Find the least number of 5 digits which is exactly divisibleby456.
a) 10456
b) 10424
c) 10032
d) 10023
Find the number which is nearest to 68624 and exactly
divisible by 587 ..
a) 68679
b)69156
c) 68569
d) 68689
Find the number nearest to 144759 and exactly divisible by
927.
a) 144906
b) 144612
c) 144169
d) 144621
Find the greatest number of 5-digits, which is exactly
divisible by 547.
a) 99456
b) 99554
c) 10545
d) 99545
What least number must be added to 9541-31, to get a
number exactly divisible by 548?
a) 63
b)563
c) 485
d)611
What least number be subtracted from 650 I to get a
number exactly divisible by 135?
a)21
b) 12
c)35
d)53
What least number be added to 5200 to get a number
exactly divisible by 180.
a) 160
b) 60
c) 20
d) 180
Findthe number which is nearest to 6555 and exactly
divisible by 21.
a) 6558
b) 6576
c) 6552
d) 6534
Find the number which is nearest to 8845 and exactly
divisible by 80.
a) 8890
b) 8810
c) 8800
d) 8880
What least number must be subtracted from 13601 to get a
number exactly divisible by 87.
a)39
b)29
c)27
d)33
What least number must be added to 1056 to get a number
exactly divisible by 23.
a)21
b)23
c)2
d)4
The largest number offour digits exactly divisible by 88 is
a) 9856
b) 9944
c) 9988
d) 9994
Find the greatest number of five digits exactly divisible by
279.
Number System
~
.
i
i
3-digitnumber= 156
a) 99882
b) 99720
c) 99782
d) 99982
15. Find the nearest integer to 56100 ••• hich is exactly divisible by456.
a) 56556
b) 56088
c) 56112
d) 56188
16. What is the nearest whole number to one million which is
divisible by 537 without remainder?
ta)999894
b) 999994
c)999~84
d) 999948 .
17. What least number must be added to 2716321 to make it .
exactly divisible by 3456?
a)3361
b)95
c) 105
d)3316
18:' What least number must be subtracted from 2716321 to
make it exactly divisible by 3456?
a)3361
b) 95
c) 85
d) 3613
19. Find'the least number of five digits which is exactly divisible by 654.
a) 10190
b) 10654
c) 10464
d) 10644
20. Which least number should be subtracted from 427396 so
that the remainder would be divisible by 15?
(BSRB Delhi PO, 2000)
a)6
b)1
c) 16
d)4
Answers
I.c
8.c
15.b
2.c
9.c
16.a
3.a
1O.d
17.b
4.b
11. b
18.a
5.b
12.c
19.c
6.c
13.b
20.b
7.a
14.a
Rule 4
Theorem: When two numbers, after being divided by a third
number, leave the same remainder, the difference of those two
numbers must be perfectly divisible by the third number.
IIInstrative Examples
I
r
Ex. I: 24j45 and 33334 are divided by a certain number of ,
three digits and the remainder is the same in both the cases. Find
the divisor and the remainder.
Soln: By the above theorem, the difference of 24345 and 33334
must be perfectly divisible by the divisor. We have the
difference = 33334 - 24345 = 8989 = 101 x 89 Thus, the
three-digit number is 101.
The remainder can be obtained by dividing one of the
numbers by 101. Ifwe divide 24345 by 101, the remainder is 4.
Ex. 2: 451 and 607 are divided by a number and we get the
same remainder in both the cases. Find all the possible
divisors (other than 1).
Soln: By the above theorem:
607 - 451 = 156 is perfectly divisible by those numbers
(divisors).
Now, 156=2x2x3 x 13
Thus, I-digit numbers = 2,3,2 x 2,2 x 3 = 2, 3, 4, 6
2-digit numbers = 12, 13,26,39,52,78
umber of foui digits and the remainder is the same
4 in both the
9
.. cases. Find the divisor.
a) 1423
1432
c) 1422
. d) 1433
2. 31 593and 23456 are divided by a certain number of
three digits and the remainder is the same in both the
cases. Find the remainder .
q)
E
x
~
rc
is
e
1.4
5
7
2
1
3
~~
~~
0~
~~
Answers
1. a
2. a
Rule 5 .'\..
To find the prOduCt of the two numbers w!ten the sum and the
difference of the two numbers are given.
Product of the numbers
(Sum + Difference )(Sum - Difference) 4
Illustrative Example
a
n
d
Ex.
The sum of two numbers is 14 and their difference is
10. Find the product of the two numbers.
Soln: Detail Method: Let the two numbers bex andy, then
x+y= 14andx-y= 10
3
4
3
3
7
3
a
r
e
d
i
v
i
d
e
d
b
y
a
c
e
r
t
a
i
n
n
Now, we have, (x+ yf = (x- yf +4xy
or, (14)2 = (lO)i +4xy.
:. xy = (14)2 -(lOf = 96 = 24
4
4
Quicker Method: Applying the above formula, we
have
Product
= (i4+10~14-:10)_24
Note: The numbers can also be found by the direct formula x =
Sum + Difference = 14 + 10 = 12
2
2
y = Sum-Difference = 14 -10
2
=2
2
Exercise
1.
2.
3.
4.
The sum of two numbers is 20 and their difference is 10.
Find the product of the two numbers.
a) 60
b) 100
c) 80
d) 75
The sum of two numbers is 49 and their difference is 3.
Find the product of the two numbers.
a) 598
b) 958
c)589
d) 859
The sum of two numbers is 38 and their difference is 4.
Find the produc~ of the two numbers.
a) 537
b) 375
c)357
d) 753
The sum of two numbers is 24 and their difference is 18 .•
50
PRACTICE BOOK ON QUICKER MATHS
Find the product of the two numbers.
a) 54
b) 63
c)36
d) 64
5 The sum of two numbers is 33 and their difference is 21.
Find the product of the two numbers.
.
a) 162
b) 126
c) 102
d)216 1
6. The difference of two numbers is .11 and "5 th of their
sum is 9. The numbers are:
[RRB Exam 1991
J
a)31,20
b) 30, 19
Answers
c) 29, 18
4.b
d)28,17
5.a
1. d
2. a
3. c
6. d; Hint: See Note.
Rule 6
Ex.
If one-fifth of one-third of one-half of number is 15,
find the number.
Soln: Detail Method: Let the number be x. Then we have,
.
(*) The required number = 1
s( T)( T )( T )
I.
~%
~~
~~
435
10. If "9 of 10 of g of a number is 45, what is the number?
[BSRBHyderabadPO 1999)
a)450
b) 540
c) 560
d)650
11.Two-thirds of three-fifths of one-eighth of a certain number
is 268.50. What is 30 per cent ofthat number? [NABARD
19991
a)1611.0
b) 716.0
c}1342.5
d) 596.60
124
12. Ifg of3 of5 of an umber is 12 then 30 per cent of the
number will be
~~
[SBI Bank PO 2001)
~64
~~
1. c
2. b
8.c
9.d
3.a
lO.b
4.c
Il.a
3.
4.
5.
6.
7.
8.
~~
5.a
12.c
6. b . 7.d
The sum of the digits of a two-digit number is S./fthe digits are
reversed, the number is decreased by N, then the numberisgivenby
s[s+
:]+~[S- :]
If one-third of one-sixth of two-third of number is 64, find
or
the number.
~ [
Decrease] ) [
Decrease]
a) 1278
b) 1782
c) 1728
d)3456
5 Sum of digits +
9' + Sum of digits -9
If one-tenth of one-fourth of one-fifth of number is 10, find
the number.
Illustrative Example
a) 200
b) 2000
c) 500
d) 1000
Ex.
The sum ofthe digits ofa two-digit number is 8. Ifthe
If three-fourth of two-third of two-fifth of one-half of
digits are reversed, the numbe{ is decreased by 54. Find
number is 60, find the number.
the number.
a) 600
b}400
c) 650
d) 575
Solo: Detail Method: Let the two-digit number be lOx + y.
If two-fifth of one-third of two-third of number is 16, find
Then, we have; x + y = 8 ... (1) and
thenmber.
10y+ x= 10x+y-54
~1~
~~O
~IW
~1~
. = 54 =. 6
If one-fifth of two-third of one-half of number is 30, find
or,x-y 9
.... (2)
the number.
From equations (1) and (2)
a)450
b) 900
c) 950
d) 400
Three-fourth of one-fifth ofanumber is 60. The number
8+6
x = -- = 7 and y = 1
is:
[Baok PO Exam, 1990]
2
a) 300
b) 400
c) 450
d) 1200
... The required number = 7 x 10+ 1 =71
Four-fifths of three-eighths of a number is 24. What is
Quicker Method:
250 per cent ofthat number?
[BSRB Mumbai, 1998J
The required number =
a) 100
b) 160
c) 120
d) 200
DeCrease] I [
Decrease]
Two-fifths ofthirty per cent of one-fourth ofa number is
15. What is 20 per cent of that number?
[
[BSRB Mumbai 1998J
5 Sumofdi gits+-- 9- +"2 Sumofdigits--- 9-
'2
2.
~~
Rule 7
=. 450
Note:(*) The resultant should be multiplied by the reverse of
each fraction.
Exercise
a) 90
b) 150
c) 100
d) 120
Two-fifths of one-fourth of five eighths of a number is 6.
What is 50 per cent ofthat number?
[BSRB Calcutta PO 19991
Answers
X(~X~)(~)=15
;. x = 15x5x3><2 =450
Direct Formula:
9.
-
7.
51
Number System
Detail Method: Let the number = x.
Then, x2 + x = 182
Exercise
or, x2 +x-182 = 0
1.
or, x2 + 14x-13x-182
The sum of the digits of a two-digit number is 12. lfthe digits
are reversed, the number is decreased by 18. Find the number.
~~
~~
~M
= 0 or,
x(x+14)-13(x+14)=0 or, (x 13 )(x + 14) = 0
~~
2.
The sum of the digits ofa two-digit-number is 9. If the digits
are reversed, the number is decreased by 63. Find the number.
a) 72
b)63
c) 54
d)81
3. The sum of the digits of a two-digit number is 10. 1f.the digits
are reversed, the number is decreased by 72. Find the number.
a)91
b) 82
c)73
d) 64
4. The sum ofthe digits ofa two-digit number is 13. lfthe digits
are reversed, the number is decreased by 45. Find the number.
a)85
b) 76
c)49
d)94
5. The sum of the digits of a two-digit number is 7. If the digits
are reversed, the number is decreased by 45. Find the number.
a) 52
b)43
c)61
d) 25
6. A certain number consists of two digits whose sum is 9.
Ifthe order of digits is reversed; the new number is 9 less than
the original number. The original number is
a)45
. b)36
. c)54
d) 63
7. In a two-digit number the digit in the unit's place is more than
the digit in the ten's place by 2. If the difference between the
number and the number obtained by interchanging the digits
is 18. What is the original number.
[8m Associates PO 1999J
c) 24
d) Data inadequate
a) 46
b) 68
Answers
6.c
1. a
2. d
3. a
4. d
5.c
·7. d; Hint: Let the no. be lOx + y
theny'=x+20r,y-x=2 .... (i)
(IOy+x)-(IOx+y)= 18
CPL 6 3 6 2 3 6
or, 9y - 9x = 18
or, y-x =2 ..... (ii)
From eqn (i) and (ii) we can't get any conclusion;
\\\ \\\\\\\\\\\\\\\\\\\\\\\\\
Rule 8
If the sum of a number and its square is x, then the number
~-I~J
[
is
given by Example
2
.
Illustrative
Ex.:
If the sum of a number and its square is 182, what is
the number?
or, x = 13 (negative value is neglected).
Quicker Method: Applying the above rule, we have
the required answer
.J729 -1
27-1 =13 2
.J1+182x4 -1 2
2
Exercise
1.
Ifthe sum ofa number and its square is 240, what is the
number?
a) 15
b) 18
c)25
d) 22
2. If the sum of a number and its square is 306, what is the
number?
a)16
b)18
c) 17
d)19
.3. If the sum of a number and its square is 702, what is the number?
~~ ~n ~~ ~~
4 .. lfthe sum of a number and its square is 1560, what is the
number?
a)38
'b)37
c)36
d) 39
5. lfthe sum of a number and its square is 156, what is the
number?
a) 16
b) 14
c) 12
d) 13·
6. , If the sum of a number and its square is 210, what is the
number?
a) 12
b) 13
c) 14
d) 16
7. If the sum of a number and its square is 90, what is the
number?
a)7
b)8
c)9
d)8
8. Ifthe sum of a number and its square is 380, what is the
number?
a) 17
b) 18
c) 19
d)21
9. 'If the sum ofa number and its square is 342, what is the number?
a) 14
b)28
c) 18
d) 23
10. lfthe sum ofa numper and its square is 552, what is the
number?
a)21
b)22
c)23
d) 24
Answers
1. it
2. c
8. c
9. c
3.a
10.c
4.d
5.c
6.c
7.c
Rule 9
The sum of the digits of a two-digit number is S./fthe digits are
reversed, the number is increased by N, then the num-
PRACTICE BOOK ON QUICKER MATHS
52
ber is given by
Illustrative Example
5[ S ~ '~]+~[ S + ~] or
.
Ex.
Increase] 1 [.
[
5 Sum of digits -
Increase
+ 2 Sum of digits +
9
Illustrative Example
l
If40% ofa number is 360, what will be 15%0f15%of
that
numb(lr?
,
.
~
Soln: Detail Method: Let the number bex.Then we have
40%ofx=360
(
9j
The sum ofthe digits ofa two-digit number is 8. If the
digits are reversed, the number is increased by 54. Find the
number.
Soln: Detail Method:
Letthe two digit number be lOx + y
Then, we have, x + y = 8 ... (i) and
lOy+x= lOx+y+54
or,y-x=6 .... (ii)
From eqn (i) and (ii)
x= 1 and y = 7.
:. the required number = 1 x 10+7= 17
Quicker Method: Applying the above formula, we have
:.x=360x100 =900
40
Ex.:
Now, 15% ofx = ~x900 = 135 100
Again, 15%of135 =~x135=20.25
100
Quicker Method: Apply.ing the above rule, we have
,
the required answer =
Exercise
1.
.
.
[ +254J
ReqUIred
number = 5 88+91 [
9
54]
2.
= 10+7= 17
Exercise
1.
2.
3.
f
I
,
I
i
4.
1i'
'\ '
f
5.
Ii:
I
3.
The sum of the digits of a two-digit number is 7, If the digits
are reversed, the number is increased by 27. Find the number.
a)25
b)34
c) 16
d) None of these
The sum of the digits ofa two-digit number is 6. Ifthe digits
are reversed, the number is increased by 36. Find the number.
a) 24
b) 15
c)51
d) 42
The sum of the digits of a two-digit number is 9. If the digits
are reversed, the number is increased by 63. Find the number.
a) 27
b)36
c)45
d)18
The sum of the digits ofa two-digit number is 5. If the digits
are reversed, the number is increased by 27. Find the number.
a)23
b)32
c) 14
d)41
A number consists oftwo digits whose sum is 15. If9 is added
to the number, then the digits change their places. The number
is
a) 69
b) 78
c) 87
d)96
15x15x360
= 20.25. 40xlOO
----. -
4.
5.
If90% ofa number is 540, what will be 10% of5% ofthat
number .
a)30
b)3.5
c)3
d) 35
If35% ofa number is 385, what will be 5% of5% of that
number.
a) 11
b)5.5
<;:)2.5
d)2.75
If 17% of a number is 68, what will be 15% of25% ofthat
number.
a) 20
b}15
c)35
d)25
If 18% ofa number is 144, what will be 12% of25% of that
number.
a)8
b)12
c)16
d) 24
If39% of a number is 780, what will be 35% of 13% of that
number.
a)91
Answers
1. c
2. d
b)52
3.b
c)65
4.d
d)78
5.a
Rule 11
If the ratio of the sum and the difference aftwo numbers is
J
a+b\ a: b, then the ratio afthese two numbers is given by a _ b
(
Answers
1. a
Illustrative Example
2. b
Ex.
3.d
4.c
5.b
Rule 10
If x% of a number is n, then y% of z% of that number is
yzn ]
I
I,
\i
.,~
,
[
given by x x 100 .
The ratio of the sum and the difference of two numbers is 7 : 1. Find the ratio of those two numbers.
Soln: Detail Method: Let the two numbers be x and y. Then we
have
x+y='2x- y 1
=:::>x+y=7x-7y
x84
or 6x = 8y . - == - = - = 4 : 3
,
.. Y 6 3
•
•
Number System
5
3
. Quicker Method: Applying th.above rule, we have 7+1 8 4
the required ratio = -- = - = - = 4 : 3 7 -I 6
3
2.
Exercise
I.
Ratio of the sum and the difference of the two numbers is 5:
3. Find the ratio of those two numbers.
'a)4:1
b)3:2
c)3:1
d)2:1.
2. Ratio of the sum and the difference of the two numbers is 9 :
I. Find the ratio of those two numbers.
a)5:3
b)5:4
c)4:1
d)5:2
3. Ratio of the sum and the difference of the two numbers is 7 :
3. Find the ratio of those two numbers.
a)5:2
b)5:3
c)3:2
d)7:4'
4. Ratio of the sum and the difference ofthe two numbers /is 2 : 1.
Find the ratio oflthose two numbers.
a)l:2
b)3:2
c)4:3
d)3:1
5. Ratio of the sum and the difference of the two numbers is 13 :
3. Find the ratio ofthose two numbers.
a)5:8
b)8:3
c)8:5
d)8:7
3.
~7
4.
5.
6.
Answers
I. a
2. b
4.d
3.a
5.c
Rule 12
To find the difference of the two digits of a two-digit number,
when the difference between two-digit number and the number
obtained by interchanging the digits is given. Difference of two
digits
Diff. in original and interchanged number
9
Note: We cannot get the sum of two digits.
Illustrative Example
Ex. The difference between a two-digit number and the number
obtained by irit~rchanging the digits is 27. What are the
sum and the difference of the two digits of the number?
Soln: Detail Method: Let the number be lOx + y. Then we have
(lOx + y)-(10y+x)= 27
the sum of the two digits of the number?
a)2
b) 1
c)9 d) Can't be determined
The difference between two-digit number and the number
obtained by interchanging the digits is 36. What is the
difference ofthe two digits of the number?
~4
~3
02
~8
The difference between two-digit number and the number
obtained by interchanging the digits is 63. What is the
difference of the two digits of the number?
7.
~9
~6
08
The difference between two-digit number and the number
obtained by interchanging the digits is 9. What is the
difference of the two digits of the number?
~2
~5
03
~I
The difference between two-digit number and the number
obtained by interchanging the digits is 72. What is the
difference of the two digits of the number?
a)7
b)9
c)8
d) Can't be determined
The difference between two-digit number and the number
obtained by interchanging the digits is 45. What is the
difference of the two digits ofthe number?
a)6
b)5
c)8
d) Can't be determined
The difference between the digits of a two-digit number is
one-ninth of the difference between the original number and
the number obtained by interchanging the positions
the
digits. What definitely is the sum of the
digits of that number?
[BSRB Mumbai PO, 19981
a)5
b) 14
c) 12
d) Data inadequate
of
-
-
1
8. The sUm of the digits of a two-digit number is - of the
.
11
sum of the number and the number obtained by interchanging
the position of the digits. What is the difference between the digits of that number? _________ _
[Bank of Baroda PO, 19991a)3
b)2
c)6
d) Data inadequate
9. The difference between a two-digit number and the number
obtained by interchanging the position of the digits of that
number is 54. What is the sum ofthe digits of that
number?
[BSRB Calcutta PO, 1999)
a)6
b)9
c)15
d) Data inadequate
I
10. The sum of the digits ofa two-digit number is "5 of the
27
or,9(x-y)=27
:.x-y=9=3
Thus, the difference is 3, but we cannot get the sum of two
digits.
Quicker Method: Applying the above rule, we have
Required answer =
27
9
=3
Exercise
I.
The difference between a two-digit number and the number
obtained by interchanging the digits is 18. What is
difference between the number and the number obtained by
interchanging the positions of the digits. What definitely is the
difference between the digits of that number?
[BSRB Chennai PO, 2000J
a) 5
b) 9
c) 7
d) Data inadequate
Answers
. 1. d
2. a
7. d; Hint:
3.a
4.d
5.c
6.b
x- y = ~{(lOx+ y)-(10y+x)}= ~(9X-9Y)= x- y
r
,
.
"
54
PRACTICE BOOK ON QUICKER MATHS
8. d; Hint: Let, the two no. be xy, ie lOx + y then, x +y
= ~[(10x+ y)+(lOy + x )]= x+ y
11
Thus we see that the difference of x and y can't be determined.
, Hence, the answer is,data inadequate.
9. d; Hint: See note.
Let the two-digit no. be lOx + y
According to question,
(lOx +y)-(lOy+ x)= 54
9x-9y=54 :. x-y=6
10. a; Hint: Let the two-digit number be lOx + y
Exercise
1.
2.
3.
The average of 5 consecutive integers is 4. Find the
average of the squares of these integers.
a)22.5
b)45
c)l8
d) Can't be determined
The average of 15 consecutive integers is 15. Find the
average of the squares of these integers.
a)243.66approx
b) 300
c) 225.4 approx
d) 394.26 approx
The average of 9 con!>ecutive integers is 9. Find the
average of the squares of these integers.
a) 87
4.
b) 87%
c)88
d) 85+
Then, x + y = ~(IOX+ y-IOy-x)
The average of 7 consecutive integers is 6. Find the
average of the squares of these integers.
9
or, x+y= -(x-
a) 46~ 3
,
5
y)
b) 46L 3
or,4x-I4y=0::;.-=-
x
c)40
L
d) 47 3
7
y2
Using componendo & dividendo, we have,
x+y 7+2 9
x- y = 7-2 =5 iex-y=5K
Here, K has the only possible value, K = 1.
Because the difference of two single-digit numbers will
always be of a single digit.
'. II !
11
Rule 13
Ex.
I
The average of7 consecutive integers is 7. Find the
average of the squares of these integers.
Soln: Use the formula: [for odd number of consecutive integers)
Average of squares
[.
'I
II
I
i
~
I'
,
~
=
1
'x[nJ(nl+lX2nJ+l) n2h+lX2n2+1)]
No. of integers
6
6
Where, n] = Average + No. of integers -I 2
and n2 = Average _ No. of integers + 1 2
In the above case
7-1
n] =7+--=10 2
7 +1
n2 =7---=3 2
X
:. Average of squares = L[10XllX21 3(4 7)]
7
6
-6
= L[385 -14]= ~ = 53
7
7
5.
f the squares of these integers.
T
h
e
a
v
e
r
a
g
e
o
f
3
Rule 14
To find the least number which when divided by x], x2 and
x3 leaves the remainders G], G2' and G3respectively.
(XI -
G]) = (Xl - G2) = (X3 - G3)' We have an established method
that is given below.
Required least number = (LCM of Xl' x2 and x3) (XI -al) or
(X2 -a2) or (X3 -a3)
c
o
n
s
e
c
u
t
i
v
e
i
n
t
e
g
e
r
s
i
s
3
.
F
i
n
d
t
h
e
a
v
e
r
a
g
e
o
Illustrative Example
Ex.:
Find the least number which, when divided by 13, 15
and 19, leaves the remainder 2,4 and 8 respectively.
Soln: Applying the above rule, 13-2=
15~4= 19~ 18= II
, Now, LCMof13, 15, 19=3705
:. therequiredleastnumber=3705-I1 =3694
Note: Find the least number which, when divided by 13, 15 and
19, leaves the remainders 1, 2, 3 respectively. Can we
fmd the specific solution. No, because 13 - 1 '* 15-2,*
19-3
Exercise
1.
2.
Find the least number which when divided by 24, 32 and
36 leaves the remainders 19,27, and 31 respectively.
a) 288
b)283
c) 287
d) 285
Find the least number which when divided by 12,21 and
35 leaves the remainders 6,15, and 29 respectively.
5
5
Number System
I
3.
4.
5.
I
I
a)414
b)418
c) 420
d) 410
Find the least number which when'l!ivided by 48, 60 and 64
leaves the remainders38, 50, and 54 respectively.
a) 860
b) 960
c) 950
d) 850
Find the least number which when divided by 5, 6, 8, 9 and
12 leaves the remainders 3, 4, 6, 7 and 10 respectively.
a) 360
b)358·
c) 362
d}258
Find the least number which when divided by 9, 10 and 15
leaves the remainders 4, 5, and 10 respectively.
~W
~~
~~
~W
3.
4.
5.
Answers
1.b
Find also the common remainder ..
a)70,6
b)71,5
c) 75, Id)73,3 .
The greatest number which when divides 99, 123 and 183
leaves the same remainder is
a) 12
b) 24
c) 18
d) 26
Find the greatest number which divides 77, 112 and 287 and
leaves.the same remainder in each case;
a)35
b) 25
c)45
d)I5
Find the greatest number which divides 95, 195 and 175 and
leaves the same remainder in each case.
a)5
b) 10
c)20
d)25
Answers
2. a
3.c
4.b
1. a
5.c
2. c
3.a
Rule 15
"To find the greatest number that will divide given numbers
say XI' X2'''' Xll so as to leave the same remainder in each . case,
wefind the HCF of the positive difference of numbers
4.a
Rule 16
The ratio between a two-digit number and the sum of the digits
of that number is a : b. If the digit in the unit's place is n more
than the digit in the ten's place, then the number
9a)
ie Ix) - x21, IX2 ~ x31, ... and so on.
5.c
.
is given by lIb _ 2a n and the digits in unit's place and
Illustrative Example
Ex.
. Find the greatest number that will divide 55, 127 and
175 so as to leave the same remainder in eacQ case.
Soln: Detail Method: Let x be the remainder, then the numbers (55
-x); (127 -x) and (175 -x) must be exactly divisible by the
required number.
Now, we know that if two numbers are divisible by a
certain number, then their difference is also divisible by
that number. Hence, the numbers
(27-x)-(55-x1
(I75-x)-(127-x)
and
(I75-x)-(55-x)
or, 72, 48 and 120 are also divisible by the required
number. HCF of72, 48 and 120 is 24.
Therefore, the required number is 24.
Quicker Method: If you don't want to go into the details of
the method, find the HCF of the positive differences of
numbers. It wiII serve your purpose . quickly. For example,
in the above case, positive difference of numbers are (127 55 = 72), (I 75 - 127 = 48) and (175-55 = 120).
HCF of72, 48 and 120 is 24
:. required number = 24.
Exercise
]. Find the greatest number which is such that when, 12288, 19139
and 28200 are divided by it, the remainders are al! the same.
a)221
b)212
c) 122
d)321
2. Find the greatest number which is such that when 76, 151 and
226 are divided by it, the remainders are all alike.
lOb - a )
.(d -b)
ten ~ place are n lIb _ 2a and n lIb _ 2a respectively.
Illustrative Example
Ex.:
The ratio between a two-digit number and the sum of
the digits of that number is 4 : 1. If the digit in the unit's
place is 3 more than the digit in the ten's place, what is the
number?
SoIn: Detail Method: Suppose
the two-digit number = lOx + y
lOx+
Then we have x+ y =
y 4
1
or,lOx+y=4x+4y
or,6x=3y or,2x=y
or, x = y-x = 3 (given) andy = 6 :. the
number is 36 .
Quicker Method: Applying the above rule, we have
Required number
=(9x4
llxl-2x4
}x 3 = 9 x 4 x 3 = 36
3
(
(
Exercise
1.
2.
The ratio between a two-digit number and the sum of the digits
ofthat number is 5 : 1. If the digit in the unit's place is I more
than the digit in the ten's place, what is the value of unit's place
digit of that number?
a)4
b)5
c)3
d)7
The ratio between a two-digit number and the sum of the digits
of that number is 2: 1.Ifthe digit in the unit's place
56
3.
4.
5.
PRACTICE BOOK ON QUICKER MATHS
is 7 more than the digit in the ten's place. What is the
va-Iue often's place digit of that number?
a)1
b)2
c)3
d) 64
The ratio between a two-digit number and the sum of the
digits of that number is 3 : 1. If the digit in the unit's place
is 5 more than the digit in the ten's place. What is the value
often's place digit of that number?
~4
~3
02
~I
The ratio between a two-digit number and the sum ofthe
digits of that number is 14: 5. If the digit in the unit's place
is 6 more than the digit in the ten's place. What is the sum
of the digits of that number?
a) 10
b)12'
c) 13
d)9
The ratio between a two-digit number and the sum ofthe
digits of that number is 4 : 1. If the digit in the unit's place
is 4 more than the digit in the ten's place. What is the sum
of the digits of that number?
a)9
b) 10
c) 15
d) 12
Answers
Lb
2.a
3.c
4.a
(xn + K) is divided byx-l. (i)
Remainder = 1 + K; when K < x-1
(ii) Remainder = (1 + Remainder obtained when K is divided
by x - 1);
when K>x-1.
Illustrative Example
Ex.:
Find the remainder when i
3
+ I is divided by 6.
Soln: Detail Method: See the following binomial expansion'
(x+ YY' =
x" + nC1x.II -1y+ nC'].x"-2y2 + nC3 x"-3y 3 + ... + IlC,,_lxyn-1 +
I. Find the remainder when (919 + 6} is divided by 8.
a)2
2.
b)3
b)3
c)?
d)5
Find the remainder when (523 + 3) is divided by 4.
~7
4.
d)7
Find the remainder when (7 + 8) is divided by 6.
a)2
3.
c)5
13
~4
03
Find the remainder when ~ i
a)19
b)7
50
~2
+ 8) is divided by II.
c)9
")8
. 5. Find the remainder when (25625 + 241) is divided by 24.
a) 23
b)2
c)1
d)Can'tbedetermined
3. b
4. c
5. b
Answers
I. d
2. b
Rule 18
5.d
Rule 17
To find the remainder when
Exercise
ytl
We find
that each of the terms except the last term
0" ) ~c~)Jltains x. It means each term except yn is
perfectlyoivisible by x.
Note: y" may be perfectly divisible by x but we cannot say
without knowing the values of x and y. FojJowingthe
same logic,
i 3 = (6 + I Y 3 has each term except ]13 exactly
3
divisible by 6. Thus, when i
is divided by 6 we
have the . remainder 113 = I and hence, when (713 + I) is
divided by 6 the remainder is I + I = 2.
Quicker i\1ethod: Applying the above rule, we have K
= I and x - I = 6
ie K <x-I.
Therefore, we apply rule (i)
:. required answer = I + I = 2.
To find the all possible numbers, when the product of two
numbers andth~ir HCF are given, we/ollow thefollowing
method.
Product
Step I: Find the value of (HCF
r.
Step II: Find the possible pairs of value got in step I.
Step III: Multiply the HCF with the pair of prime factors
obtained in step II.
Illustrative Example
Ex.:
The product of two numbers is 7168 and their HCF is
16. Find the numbers.
7168 =.28
Soln: Step I: (16)2
Stepn: (1,28),(2, 14),(4,7)
StepID: (I x 16,28 x 16)and(4 x 16,7 x 16)or(l6,448) and
(64, 112)
Note: (2, 14), which are not prime to each other should be
rejected.
Exercise
1.
The product of two numbers is 286 and their HCF is 12.
Find the sum of the numbers.
·a) 12
b) 24
c)36
d)48
2. Th'e product of two numbers is 3125 and their HCF is 25.
Find the sum of the numbers .
a) 75
b) 100
c) 125
d)50
3. The product of two numbers is 20 16 and their HCF is 12.
Find the number of all possible pairs of numbers.
a)1
b)2
c)3
d) Can't be determined
4. The product of two numbers is 338 and their HCF is 13.
Find the difference of the numbers.
a)13
-b)26
c)39
d)52
Number System
57
Answers
I. c
2. b
3.b
5.
4.a
Rule 19
A number on being divided by d, and d2 successively leaves the
6.
remainders r] and r2 respectively. If the number is divided by d l
x d2, then the remainder is given by
(d] xr2 H])'
a)12
b) 10
c)14
d)16
A number on being divided by 10 and 11 successively
leaves the remainders 5 and 7 respectively. Find the remainder when the same number is divided by 110.
a) 70
b)98
c) 74
d)75
A number on being divided by 3 and 7 successively leaves
the remainders 2 and 5 respectively. Find the sum ofdigits
of the remainder when the same number is divided by 21.
a)7
b) 17
c)S
d)6
Illustrative Example
Answers
Ex.
l. b
A number on being divided by 5 and 7 successively
leaves the remainders 2 and 4 respectively. Find the
remainder when the same number is divided by 5 x 7
=35.
Soln: Detail Method:
Ifffi
A
7 B· 2
.C4
In the above arrangement, A.is the number which,
when divided by 5, gives B as a quotient and leaves 2
as a remainder. Again, when B is divided by 7, itgives
C as a quotient and 4 as a remainder. For simplicity,
we may take
C = I.
:.B=7xl+4=11 andA=5xIl+2=57 Now, when 57 is
divided by 35, we get 22 as the remainder.
Quicker Method: The required remainder =
d] xr2 +r,
= the first remainder = 2
r2 = the second remainder = 4
.'. the required remainder = 5 x 4 + 2 = 22.
Exercise
I.
2.
3.
4.
A number on being divided by 12 and 15 successively
leaves the remainders 4 and 6 respectively. Find the remainder when the same number is divided by 180.
a)46
b) 76
c) 84
d) 18
A number on being divided by 5 and 7 successively leaves
the remainders 3 and 6 respectively. Find the remainder
when the same number is divided by 35.
a) 33
b) 23
c) 32
d) Can't be determined
A number on being divided by 8 and 9 successively leaves
the remainders 5 and 7 respectively. Find the remainder
when the same number is divided by 72.
a)61
b)S
c) 71
d)9
A number on being divided by 4 and 6 successively leaves
the remainders 2 and 3 respectively. Find the remainder
when the same number is divided by 24.
3:a 4.c
5.d
6.c
Rule 20
To find the number of zeros at the end of the product.
We know that zeros are produced only due to the following
reasons.
(i) If there is any zero at the end of (lny multiplicand.
(ii) If 5 or multiple qf 5 are multiplied by any even number . To
generalise the above two statements, we may say that
(2)"' (5t has n zeros ifm > nor m zeros ifm <no
Note: Always lesser value of the exponents of 5 and 2 will be .
the required answer. Thus, write the product in the form
bmx5 x ... )
n
I
Illustrative Example
Ex.:
Find the number of zeros at the end of the products.
12 x 18x 15 x40x25 x 16x55 x 105 Soln: 12 x 18
x 15 x40 x 25x 16x 55 x 105 = 12 x is x 16 x 40
x 15 x 25 x 55 x 105
Where, dl = the first divisor = 5
r]
2. a
, = (22x3~ (2x9)x (2f x (2 3 x 5)x (5 x 3)" (5)2 x (5 x II)x (5 x 21) = 2'0 x56
x ....
[Since numbers other than 2 and 5 are useless] Since
10> 6, there are 6 zerbsat the end of the product.
Note: This is the easiest way to count the number of zeros in the
chain of products. By this method, we can easily find
that the product of 1 x 2 x 3 x ... x 100 contains 24 zeros.
Exercise
1.
2.
3.
4.
Find the number of zeros at the end of the product 15
x 16 x IS x 25 x 35 x 24 x 20
a) 10
b) S
c) 5
d) Can't be determined
Find the number of zeros at the end of the product
52 x20x28 xlOxl6xl25
a) 15
b) 22
c)7
d) 8
Find the numberof zeros at the end of the product 50
x 625 x 15 x lOx 30
a) 10
b)9
c) 12
d)3
Find the number of zeros at the end of the product
PRACTICE BOOK ON QUICKER MATHS
58
150 x 250 x 625 x 125 x 75 x 20 x 16
a)9
b) 14
c)23
d)5
5. Find the number of zeros at the end of the product
70x80x 16x64x56x 13 x 18x3125
a) 16
b) 12
c)JO
d)25
Answers
1. c
2. c
3.d
To findthe number of different divisors.
Find tlte prime factors of tlte number and increase tlte index of
eaclt factor by 1. Tlte continued product of increased indices
will give tlte result including unity and tlte number itself.
Note: Also see Rule - 36.
Illustrative Examples
Ex.I: Find the number of different divisors of 50, besides unity and
the number itself.
Soln: If you solve this problem without knowing the rule, you will
take the numbers in succession and check the divisibility.
In dOIng so, y'ou may miss some numbers. It will a~o.take
more time.
Different divisors of 50 are: 1,2,5, 10,25,50
Ifwe exclude I and 50, the number of divisors will be 4.
2
By rule: 50 = 2x 5 x 5 = 2 x 5
... the number oftotal divisors = (I + I) x (2 + I)
=2x3=6
or, the number of divisors excluding 1 and 50 = 6 - 2 =4 Ex. 2:
Rule 22
Ex. 1 : How many numbers up to 100 are divisible by 6? Soln:
Divide 100 by 6. The quotient obtained is the required
'number of numbers.
100= lQ x 6+4
Thus, there are 16 numbers.
Ex. 2: How many numbers up to 200 are divisible by 4 and 3
toaether?
b
,Soln: LCM of 4 and 3 = 12
Now, divide 200 by 12 and the quotient obtained is the
required number of numbers.
200 = 16 x 12 + 8
Thus, there are 16 numbers.
Ex. 3: How many numbers between 100 and300 are divisibleby7?
Soln: Upto-l00, there are 14 numbers which are divisible by 7
(since 100= 14 x 7 + 2). Up to 300, there are 42 numbers
which are divisible by 7 (since 300=42 x 7 + 6) Hence,
there are 42 - 14 = 28 numbers .
3'-
Exercise
1.
Find the different divisors of 3 7800, excluding unity. Soln:
2.
37800=2x2x2x3x3x3 x5x5x7
3.
23
x
4.
33 x 52 x i
Total no. of divisors = (3 + 1)(3 + 1)(2 + 1) (I + 1)= 96 ...
the number of divisors excluding unity =96-1 = 95.
5.
Exercise
I.
Find the number of different divisors of307692.
a)96
b) 12
e)6
d)48
,2. Find the number of different divisors of 400, besides unity and
the number itself.
a) 15
b) 14
e) 13
d) 12
3. Find the number of divisors of999999, excluding unity.
~M
4.
6.
~~
~~
Find the number of different divisors of 13231.
~M
5.
~~
~4
~~
6.a
Illustrative Examples
Rule 21
1
5.c
To find the number of numbers divisible by a certain integer.
5.b
4.a
Answers
l.a
2.e
3.e
4. b;Hint: 13231=131 x 101,131 and 101 are primes
~5
Find the no. of different divisors of30030, besides unity and
the number itself.
~M
~~
~~
~~
Find the no. of different divisors of 4452.
a) 24
b)32
e) 16
d)22
6.
7.
8.
How many numbers up to 150 are divisible by 9?
a) 16
b) 15
c) 10
d)6
How many numbers up to 200 are divisible by 7?
a)26
b) 22
e) 18
d)28
How many numbers up to 532 are divisible by 15?
a)25
b)26
e)36
d)35
How many numbers up to 300 are divisible by 5 and 7
together?
a)9
b)8
e) 10
d)7
How many numbers up to 450 are divisible by 4, 6 and 8
together?
'
a)19
b)18
e)17
d)16
How many numbers between 50 and 150 are divisible by 8?
a) 24
b) 12
e)18
d)8
How many numbers between 100 and 200 are divisible by 2
and 8 together?
a) 12
b) 13
e)9
d) 16
How many numbers between 100 and 300 are divisible by 9?
a)11
b)13
e)19
d) 22
Answers
l.a 2.d 3.d 4.b 5.b 6.b 7.b 8.d
1
5
9
Number System
Rule 23
.The number which when multiplied by x is increased by y is
Y)
(
Illustrative Example
5.
Answers
Multiplier -1 .
1. a
Ex.
Find the number which when multiplied by 16 is increased by225.
Soln: Detail Method: Let that number be x. Then
Ex.:
Exercise
..
= 225 = 225 = IS
the requ~~d number . 16 -lIS
I.
Find the sum of first 50 odd numbers.
a) 6250
b) 2500
c) 2520
d) 2450
Find the value of
(1 +3 + 5 + ... + 80thoddnumber)-(1 +3 +5 +7+ ... +
30th odd number)
a) 5500
b) 6100
c) 5400
d) 7300
Find the value of
35 + 37 + ... + 25th odd number.
a) 356
b) 336
c) 363
d)365
Find the value of I + 3 + 5 + ... + 199
a) 40000 b) 10000
c)39601 d)Can'tbedetermined
Find the value of 15 + 17 + ... + 51
a) 627
b) 676
c) 725
d) None ofthese
2.
Fmd the r umber which when~ultiplied by 36 is increased by
a)30
b)28
c)32
d)35
rind t;lt: number which when multiplied by 9 is increased by
128.
a) 12
b) IS
_£) 16
d) 18
Find the number which when multiplied by 17 is increased
by256.
a) 12
b) 14
c)18
d) 16
Find the number which when multipliedby IS is increased by
378.
a) 26
b) 16
c)27
d}28
Find the number which when multiplied by 26 is increased
by625.
a) 26
b)25
c) 24
d)27
Answers
1. a
2. c
3.
4.
5.
6. ~
3.d
5.b
3.
d) 1992
2. a
tn = nth term of the series a =
n(n+l)
Find the value of] + 2 + 3 + ... + 105.
Exercise
2.
c) 1990
tn =a+(n-I)d
105(105 + I)
Soln: Reuired sum = ---- = 5565 2
I.
b) 1989
3. b; Hint: We have the following formula,
first term of the series n =
number of numbers
d = common difference
2
Illustrative Example
Ex.:
is equal to
AnsWers
Rule 24
Theorem: Sum of all the firstn natural numbers = --;
1+3+5+ ... +3983
1992
a) 1988
1. b
4.c
Find the value of I + 3 + 5 + ... + 20th odd number .
Soln: 202 =400.
1050~
5.
5.b
Illustrative Example
Exercise
4.
4.a
2
Quicker Method: Applying the above rule, we have
3.
3.a
Theorem: Sum of first n odd numbers = n • .
:. x = 225 = 15
.
15
2.
2. b
Rule 25
16x-x = 225
I.
Find the value of I + 2 + 3 + ... + 62.
a) 1953
b) 1395
c) 1593
d) 1359
Find the value of
(1 + 2 + 3 + 4 + ... + 80) - (1 + 2 + 3 + ... + 60)
a) 1830
b) 1410
c)1140
d) 1380
(Increased Value)
given by x _lor
"
4.
Find the sum of first 45 natural numbers.
a) 1035
b) 1235
c) 1135
d) 1305
Find the sum of natural numbers between 20 and 100.
a) 4480
b) 4840
c) 4800
d) 4850
Find the value ofl +2+3+ .... +210.
a)22155
b)21255
c)22515
d) 22255
(
For the case of odd number
a= l,d=2
:.tn =1+(n-I)2=2n~1
We apply this formula for solving this question. First we
calculate 1+ 3 + 5 + ... + 33 and then 1+2 + 3 + ... +25th odd
number. For getting required answer, we subtract first from
second.
How do we calculate first ie (I + 3 + 5 + ... + 33)? We
have,
PRACTICE BOOK ON QUICKER MATHS
60 ..
33=2h-l Eseeforrnu!a)
:. n= 17
:. 1 +3 +5 + ... +33 =;01 +3+5 + ... + 17th odd number.
Exercise
= {17f =289
4. b ..
5. a
6. d
Rule 26
Theorem: Sum of first n even numbers = n (n + 1)
a) 5255
b) 5525
Find the value of2 +4 + 6 + 8 + ... + 1 00 (or 50th even
number)
Soln: 50 x (50 + 1) = 2550
Note: We have the following formula,
t" = a+(n-1)d
3.
5.
5.
Find the vahie of 1 + 2 + ... + (30th natural number?
a) 9454
2
b) 9544
2
2
b) 440
2
2
4.d
5.b
= n(I1~+IX2n+l)
6
)= 385, then the value of
c)1155
d)(385x385)
2.a
3.d
4.a
- -~----'-'----
,
2J
0
Rule 28
Theorem: Sum of cubes of first n natural numbers
Illustrative Example
Find the value of 13 + 23 + ... + 63 .
. [6X(6+1)l2 =(21)2 =441
S I
on.
..
2
.,
J
Exercise
2.
102 _ 1O(10+1X2xl0+1)
6
10x11x21 =----=385 6
6.a
{Ix 2)2 +(2x2? +(2x3)2 + ... +(2xl0)2
Find the value of 13 +23 + ... +123.
a) 6804
Find the value of 12 +22 +32 + ... +102
5.c
2
·=2l1"+2"+3~+ .... +10
=4 x 385= 1540
1.
Illustrative Example
+ + J + ... +
d) 660
Z
b) 1540
',
2
Ex..:
Rule 27
Solo:
c)550
+ ... +lO
7. b; Hint: 22+'42+ ... +20
2 22 ,,2
1
d)9555
Answers
Theorem: Sumofsquarcs offirst n natural numbers
Ex..:
2
If (1 +2 +3
l.b
t"
3.a
c) 9455
2
(1 +2 +3 + .... +10 )-(1+2+3+ ... +10) is equal to
a) 770
Answers
2. a
. d)4898
2
. (22 +42 +62 + ... +202) is
Find the value of2 + 4 + 6 + .... + lOOth even number.
a) 10000
b) 10100
c) 11000
d) 10101
Find the value of26 + 28 + ... + 28th even number.
a)656
b) 665
c) 566
d)565
Find the value 01'2 +4 + 6 + .... + 1002.
a)251502
b)250512
c)215502
d)255102
Find the value of68 + 70 + ... + 180
a) 7608
b) 7680
c) 6078
d) 7068
Find the value of2 + 4 + 6 ... + 56th even number.
a)3912
b)3192
c)3219
d)3129
I. b
c)4901
2
=
4.
d) 1496
2
b) 4900
Exercise
3.
2
Find the value of 2 + 3 + ... + 24 .
7.
= 2+2n-2 = 2n
2.
d) 38205
2
c) 1469
2
a)330
tn = 2+(n-l)2
l.
2
b) 1649
a) 4899
n = no. of numbers
d = common difference .
For the case of even numbers
2
Find the value of 1 +2 +3 + ... +16 .
4.
where, ~" = nth term
d) 5252
2
2
6.
a = first tertil
2
Find the value of 25 +26 + .... +50 .
a) 38025
b)30825
c) 38250
a) 1946
Ex.:
c )5552
2
2.
Illustrative Example
ro, n ,= 2
Find the value c.f 12 + 22 + ... + 252.
1.
3
d) 6408
3
b) 18495
c) 18497
3
3
3
d) 14895
Find the value of 8 + 9 + ... + 15 .
a) 16316
4.
c) 6048
3
Find the value of 2 + 3 + ... + 16 .
a) 18496
3.
b) 6084
b) 13661
3
c) 16361
d) 13616
3
Find the value of 1 +2 + ... +(10th natural numberY
a) 3025
b) 3205
c) 3052
d) 3250
Number System
5.
6.
61
Find the value of 23 + 33 + 43 + ... + 93.
a) 2024
b) 2025
c) 2225
Find the value of 33 + 43 + ... + 113.
a) 4356
b)4348
c) 4347
2.
d) 2205
3.
d) 4374
4.
Answers
1. b
2. b
4.a
3.d
5.a
5.
6.c
From I to 31, how many are the odd numbers?
a)15
b) 16
c)14
d)17
From I to 51, fmd the number of even and odd numbers.
a) 26, 25
b) 25, 26
c)24,25
d)25,24
From 51 to 91, find the number of even and odd numbers.
a)20,21
b)21,20
c)21,22
d) 19,20
From 51 to 90, find the number of even and· odd numbers.
Rule 29
n
50
For exampIe,from 1 to 50, there are - = 25 odd numbers 2
)0
amI - = 75 even numbers. 2
Exercise
1.
2.
3.
4.
5.
6.
In the first 62 counting numbers, fmd the number of even
numbers.
a)30
b)31
c)32
d) 34
From I to 78, how many are the odd numbers?
a)20
b)38
c)39
d) 40
From I t028, find the number of even numbers.
a) 14
b) 13
c)12
d) 15
From I to 100 find the number of even and the number of
odd numbers.
a) 50, 50
b)51,50
-.;)50,51
d) 49, 50
From 1 to 80 how many are the even numbers?
a)41
b) 42
c)39
d) 40
From 50 90, find the number of odd and even numbers.
a)20,21
b)20,20
c)21,22
d) 19,20
to'
. c)20,21
d) 19,20
Answers 1.
3.b
a 2. b
4.a
5.a
Rule 31
The difference between the squares of two consecutive
numbers is always an odd number and the difference
between the squares of two consecutive numbers is the sum
of the two consecutive numbers.
For example, 16 and 25 are squares of 4 and 5 respectively
(two consecutive numbers).
:. Differimce = 25 - 16 = 9 (an odd number)
and 52 - 42 (Difference) =5 + 4 = 9
2
2
Reasoning: a _b
= (a-bXa+b)=a+b [: a-b = I]
Exercise
a) 24
1.
2.
3.
b) 12
c) 18
d)8
c)8
d) 10
2
Find the value of 6 - 52 .
a)ll
b)9
2
2
Find the value of 35 -34 .
a) 59
b)69
c)70
Find the value of
d)71
102 _92 +82 _72 +62 _52 +42 _32 +22 _12 a) 50 b)
Answers
I. b
b)21,20
a) 20, 20
n
In the first n counting numbers, there are - odd and 2
. 2
even numbers provided n, the number of numbers, is even.
2. c
4.a
3.a
5.d
6.a
4.
292 +352 +332 +312 -342 -322 -302 -282. a) 250 b)
252 c)352 . d) 342
Rule 30
In the firstncounting numbers, if n, the number of num1
bers, is odd, then there are 2(n + 1) odd numbers and
65 c)45 d)55
Find the value of
5.
I
Find the value of 652 -642
a)129
b) 128
2(n -I) even numb~rs.
AnsWers
51+1
For example, from I to 51 there are -- = 26 odd numbers 2
I. a
2. b
c) 120
d) 125
5.a
3.d
Rule 32
51-1
and -- = 25 even numbers. 2
Exercise
1.
In the first 61 counting numbers, find the number of even
numbers.
a)30
b)31
c)32
d)29
To find the number in the unit-place for odd numbers. When
there is an odd digit in tire unit place (except 5), '1lultipIy
the number by itself· until you get 1 in the unit ?lace.
(. .. 1r=(. ..
1) (. .. 3ft' =
(. .. 1) (. .. 7/"
= (. .. 1)
~I
9.
62
(. .. 9/" = (. .. 1) ,
where n = 1, 2, 3, ....
Iilustrative Examples
Ex. 1 : What is the number in the unit place in (729)59? Soln:
When 729 is multiplied twice, the number in the unit
place is I. In other. words, if729 is multiplied an even
number oftimes, the number in the unit place will be
I. Thus, the number in the unit place in (729
(729 )59 = (729
i
8
Y8 is I. ~'.
x (729) = ( ... I)x (729) = 9 in the
unit place
Ex. 2: Find the number in the unit place in
(6~ (623)38 and (623)39
.
Solo: W~~~ 623 is multiplied twice, the number in the unit place
is 9. When it is multiplied 4 times, th~ humber in the unit
place is L Thus we say that if 623 is multiplied 4n number
oftimes, the number in the unit place will be 1:80,'
(623 )36 = (623 )4x9 == I in'the unit place
(623)38 = (623)4x9 x (623)l = ( .. :I)x(. .. 9)= 9 in the
~3
~6
~9
What is the number in theunit place in (333
Y4 ?
~I
~9
~6
02
P
R
A
C
T
I
C
E
B
O
O
K
O
N
Q
U
I
C
K
E
R
unit place.
(623 )39 = (623 )4x9 x (623)3 = (. .. ])x (...7) = 7 in the
unit place.
Exercise
J.
What is the number in the unit place in (659)56?
a) I
b) 9
c) 6
d) None of these ~
2.
What is the number in the unit place in (329
f
~I
~4
3.
~7
What is the number in the unit place in (147)48?
~7
4.
~9
?
~6
~9
What is the number in the unit place in (87
~I
~7
~9
6.
to ?
~3
~9
b)7
c)9
d)3
What is the number in the unit place in (6231)928?
~l
8.
~3
i
s
t
h
e
What is the number in the unit place in (5427)641 ?
a)]
7.
~7
10. W
h
a
t
~I
5 .. What is the number in the unit place in (127)127?
~I
M
A
T
H
S
~8
03
~4
What is the numbe~ in the unit place in (543)12 ?
n
u
m
b
e
r
••
in the unit place in (4673Y21 ?
a) I
b)6
c)3
d)9
II. What is the number, in the unit place in (5483)843 ?
~I
~7
~9
12. What is the number in the unit place 111
(1243YC x (I 547YoO ?
a) 1
b)2
c)3
13. What
is the number
in
~3
d)9
the
unit place
61
Y x
(12349l39 ?
(24533
a) 7
b) I
14. What . is the number in the
c)9
d)3
unit place
in
59
(157)157 x(159Y ?
a) 3
b}9
c)6
15. What is the number in the unit place
d) I
11
1
38
(751Y51 X (263)l71 x(137Y x(339~39?
a)7
b}9
c) I
d)6
Answers
1. a
2. b
3. d; Hint: When 7 is multiplied 4 times, the number in the unit place
is"l. ie if? is multiplied 4n number oftimes,the number in the
unit place will be I.
:. (147)48 = (147)4 x12 = I in the unit place.
4. c; Hint: (87)90 = (87)4x22 x 87 x87
=(. .. I)x( ... 9)=9
5.c
6.b
7.a
8.b
9.d
10. a
Il.b
12.a;Hint: (1243Y6 = (I 243)4x19 =(. .. 1) intheunitplace.
(1547)100 = (I 547)4x25 =(. .. 1) in the unit place. 13.
a; Hint: (24533
Y61 = (24533 )4 190 x (24533) .
X
= (. .. I)x( .. .3)=(. .. 3) in the unit place
(12349)839 = (12349)2X419 x(12349)= (. .. IX ... 9)= (. .. 9) in
the unit place.
14.a
15. a; Hint: (751YS! =(. .. 1) in the unit place
(263 f71 = (263
t67
x (263)3
= ( ... 1) x ( ... 7) f' ( ... 7) in the unit place
.
I
(137)138 = (I 37)4X34 X (lp7)2 = (. .. I)x(. .. 9) = (. .. 9) in the
unit place
(339)339 = (339YX169 x(339)= (. .. I)x( ... 9)= ( ... 9) in the unit
place.
6
3
Number System
3.
... required answer == ( .•. 1X .. 7X .. 9X···9)
== ( ••• 7)
Find the number in the unit place in (1602)602
in the unit
a)2
4.
place.
b)4.
a)2
5.
b)4
~6
~8
7.
~6
a)6
Ex. 1 : Find the number in the unit place in (122
9.
~2
~2
is 6. Therefore,
X
(J22)20 = (J22)4 S =( ... 6)0=6 in the unit place
(J22?2 = (J22)4X S x (122Y = ( ... 6)x( . ..4) = 4 in the unit
c)8
d)2
~6
08
Find the number in the unit place in (9S8 y 17
and (122f .
Soln: ( . 2) x (. .. 2) = .. .4
( .. 2) x ( .. : 2) x ( ..... 2) = 8
( . 2) x ( ... 2) x ( .... 2) x ( ... 2) == ...... 6
We know that( .... 6) x ( ......6) == .......6
Thus, when{ 122) is multiplied 4'n times, the last digit
~2
Find the number in the unit place ih (958 Y 16 .
8.
yo, (122 Y2
~4
08
b)4
~4
Illustrative Examples
02
Find the number in the unit place in (216?\6.
(. .. 6)" = ( 6)
( ... 8)4n =( ... 6); where n == 1,2,3, ...
d) 8
Find the number in the unit place in (5924)429.
~4
( .. .4yn =( .. 6)
c)6
Find the number in the unit place in (194)64 .
6.
( ... 2 )4n == ( •.• 6)
d)6
Find the number in the unit place in (1392)9\ .
Rule 33
To find the number in the unit placefor even numbers. When
there is an evell digit in the unit pletce, multiply the number
by itself until you get 6 in the unit place.
c)8
10.
06
~4
~8
Find the number in the unit place in (958
y 18.
~4
~2
06
~8
~2
~4
06
~8
11.Find the number in the unitplacein (958 t9
.
J2. Find the number in the unit place in
61
(1532Y62 x(34S4Y x (1236Y62 x{53 1 8)243 .
a)2 b)4 c)6d) 8
place
13. Find the number in the unit place in
(122)23 = (J22)4XS x (122Y ~ (. .. 6)x ( ... 8) = 8 in the
unit place.
2
Ex. 2: Find the number in the unit place in (98)40, (98t
{4152Yl X (3268)67 ;«S913)83 X (6217yo3 .
a)4
Answers
and (98)43 .
4.d
c)6
5.a
10.a
J1.a
12.a
d)8
6:a
7.a
l3.c
1. a
2. a
Rule 34
8.d
9.d
Ifthereis 1, 5 or 6 in the unit place oft/te given II u11lb er, then
after (tny times of its J1tultiplicatioll, it will have the
Soln: (98)4 = (. .. 6)
... (98)4n =( ... 6)
10
Thus, (98)40 = (98t
b)2
3.b
= (. .. 6)= 6 in the unit place
same digit in the unit place ie
X
( .. It = (
(98)42 =(98)4 IOx(98? =(. .. 6)><(. . .4)=4 intheunit place
(98t 3 =(98)4XIOx(98Y = ( ... 6)x(. .. 2) = 2 in the unit
...
1) ( ... S}' =
place
( ..
Exercise
1. Find the number in the ur,it place in (542 to.
a)6
2.
b)2
c)3
Illustrative Example
Ex.:
d)9
Find the number in the unit place in (1542)541.
a)2
b)4
c)6
.5) (. .. 6}'
d)8
=Find
(. ..the
6) number in the unit place in
(621?40, (62Sfs, (636Y6
Soln: From the above rule,
(621tO = ( ... 1 to = 1 in the unit place
t!
I
~
:I",
64
PRACTICE BOOK ON QUICKER MATHS
Now, apply the above rule,
Number of divisors = (7 + 1) (1 + 1)(2 + 1) = 84
(625)125 = ( .. .5)125 = 5 in the unit place (636)36
Exercise
= ( ... 6)36 = 6 in the unit place' Exercise
1.
2.
Find the number in the unit place in (1845
~5
~3
~9
~9
~6
r .
1.
Find the no. of divisors of225.
r56 •
2.
'Find the no. of divisors of63504.
a) 25
b)32
c)75
d) 56
Find the no. of divisors of 17640, besides unity and itself.
a) 12
b) 60
c) 72
d) 70
Find the no. of divisors of25200, excluding unity.
a) 90
b)89
c}88
d) 86
Find the no. of divisors of234.
a) 12
b)6
c)2
d)8
Find the no. of divisors of9000.
a)36
b)48
c) 54
d) 18
Find the no. of divisors of20570, besides unity and itself.
~1
Find the number in ,the unit place in (99026
~3
3.
5
\ ~1
Find the number in the unit place in
(441)441 X (495)126 X (1536)236 .
~1
4.
~5
~6
~4
3.
4.
5.
~O
6.
Find the number in the unit place in (321)321 x (325 )326 .
~1
~5
~6
~8
Answers
La
2.c
7.
~9
a) 24
8.
3.d 4.b
~8
b) 22
~6
c}21
d) 18
Find the no. of divisors of 1 0000, excluding itself.
a) 24
' b)25
c}16
d) 32
Answers
Rule 35
Ex.:
What is the number in the unit place when 781, 325,
497 and 243 are multiplied together?
Soln: Multiply all the numbers in the unit place, ie 1 x 5 x 7 x 3, the
result is a nU\TIber in which 5 is in the unit ,place.
l.b
2.c
7. b
8.a
3.d
Find the number in the unit place in 962 x <1,- 5 x 454 x 959.
~2
2.
~4
4.
~6
~8
Find the number in the ~nit place in 954 x 9625 x 43216 x
15437 x 12343.
~O
3.
6.b
~l
~1
~6
Find the number in the unit place in 14532 x 14531 x 243 x
245 x 247 x 249.
~3
~6
~4
~O
Find the number in the unit place in 1431 x 5343 x 9645 x
1489.
~3
~6
~O
~5
Let N = aP bq c" .... , then the sum of the divisors of a number'
P 1
q
a + -1 b + 1 -1 C"+1_1
--'-, x---x---x ...
a-I
b-l
c-l
'Note: This includes unity and the number itself as divisors.
Illustrative Example
Ex.:
Find the sum ofthe divisors of a number 8064.
Soln: Factorize 8064 into its prime factors.
8064= 27 x31 x72
Now, apply the above rule
Answers
27+1-1 31+ 1-1 72+ 1-1
1. a
--,-x--x---
2. a
3.d
2-1 3~1 7-1
256 -1 9 -1 343-1
4.d
=-'-x--x--
1
Rule 36
1.
(p+ 1) (q+ 1) (r+ 1) ...
2.
Find the no. of divisors of 8064.
Soln: 8064= 27 x31 x72
6
Exercise
Note: This includes unity and the number itself as divisors.
Illustrative Example
2
=255 x4 x 57=58140.
If N is a composite number and N = aP bq c" ...
Where a, b, c, ... are different prime numbers and p, q, rare
positive integers. Then the number of divisors are
Ex.:
5.a
Rule 37
Exercise
1.
4.b
3.
4.
Find the sum of the divisors of a number 225.
a)430
b) 403
c) 503
d) 303
Find the sum ofthe divisors of a number 63504.
a) 213870
b)231807
c)213807
d)213708
Find the sum of the divisors of a number 17640.
a) 66960
b) 66690
c) 96660
d) 69660
Find the sum of the divisors ofa number 180.
a) 465
b) 546
c)564
d) 654
"
,
65
Number System
5.
Find the sum of the divisors ofa number 120.
a) 360
b) 420
c)480
~)630
6. Find the sum of the divisors ofa number 64.
a) 128
b) 127
c) 63
d) 130
7. Find the sum of the divisors of a number 3 125.
a) 3906
b) 3609
c) 3096
d) 3069
8. Find the number and the sum of the divisors of the number2460 excluding one and itself.
a) 24, 7056 b) 42, 7056 c) 24,4594 d) 24,4595
Answers
J.b
2.c
3.b
4.b
5.a
6.b
7.a
8. d; Hint: Sumofthe divisors excluding 1 and itself= 7056. :.
sum of the divisors including I 'and itself =7056-(2460+
1)=4595.
Rule 38
es
/'
If theplac: of last two digits of a three-digit number are
interchanged, a new number greater than the originalltumbel' by N is obtained, then the difference betWeen the last
two digits of that numbefl is given by (~) or (Dif.fe~'ence
i;
two values).
Illustrative Example
Ex.:
If the places of last two digits of a three digit number
are interchanged, a new number greater than the original number by 54 is obtained. What is the difference
between the last two digits of that number?
[NABARD 1999]
Soln: Detail Method:
Let the three-digit number be 100x + 10 y + z .
interchanged, a new number greater than the original
number by 27 is obtained. What is the diffeience between
the last two digits of that number?
a)l
b)2
<::)3
d)4
4. If the places oflast two-digits bf a three-digit number are
interchanged, a new number greater than the original
number by 36 is obtained. What is the difference between
the last two digits of that number?
~l
~2
~3
~4
S. If the places oflast t\vocdigits of a tlu:ee-digit number are
interchanged, a new number greater than the original
number by 45 is obtained. What is the difference between
the last two· digits of that number?
a)3
b)4
c)5
d)6
6. Ifthe places oflast two-digits of a three-digit liumber are
interchanged, a new nuniber greater than the original
number by 63 is obtained. What is the difference between
the last two digits of that number?
~7
~S
06
~8
7. If the places oflast two-digits of a three-digitnumber are
interchanged, a new numb~r gi:eater than the original
number by 72 is obtained. What is the difference between
the last two digits of that number?
~7
~5
~4
~8
8. If the places of last two-digitI! of a threecdigit number are
interchanged, a new number greater than the original
number by 8 I is obtained. What is the difference between
the last two digits of that number?
a)7
b}8
c)9
d) 1
Answers
3.c
1. b 2. a 7.
d, 8. c
Acconling to the question,
(IOOx + IOz + y )-(lOOx + lOy + z)= 54 or,
4.d
5.c
6.a
Rule 39
Exercise
A Itumber is divided by a certain number NI and gives a
remainder 'R'. If the same number is divided by altother
number N z, then the new remainder is obtained by the
following method.
"Divide R by N z and the remainder obtained ilt this division
will be the new remainder". (Note: Here Nj > N 2 and NI is
divisible Nz.)
1.
Illustrative Exampl~
9z-9y=54 orz-y=6
QuiCker Method: Applying the above rule, we have 54
the required answer = 9" = 6 .
2.
3.
If the places oflast two-digits of a three-digit number are
ii1terchanged, a new number greater than the original
number by 18 is obtained. What is the difference between
the last two digits of that number?
a) I
b) 2
. c) 3
d) 4
If the places ofJast two-digits of a three-digit number are
interchanged, a new number greater than the original
number by 9 is obtained. What is the difference between
the last two digits of that number?
~l
~3
~4
~6
If the places of last two-digits of a three-digit number are
Ex.:
A number when divided by 899 gives a remainder 63.
What remainder will be obtained by dividing the same
number by 29.
So]n: Detail Method:
Number == Divisor x Quotient + Remainder = 899 x
Quotient + 63
= 29 x 3 I x Quotient + 2 x 29 + 5
Therefore, the remainder obtained by dividing the
number by 29 is clearly 5.
66
PRACTICE BOOK ON QUICKER MATHS
Quicker Method: Applying the above rule, we have, 63 +
29 i.e. 29) 63 (2
58
5
:. required answer = 5
Exercise
1.
2.
3.
5.
7.
8.
9.
5.d
4.c
6.a
Suppose, the larger divisor is N J ' and the smaller divi-
~7
Where, N J = K N z and K = any integer> I.
~6
08'
~J
09
~9
Now, when the number is divided by N z ' then remainder is Rz
(say) and when the
~2
05
2N 2 +R 2 = R)
In the given question,
357 _ 21
~8
a)4
b) 18
c)9
K=--
Nz = 17and KN2 =357
d) 6
17
Here, K> 1 an integer. Now,
we can apply the remainder
rule.
2N2 +R2 = RJ or, 2
x 17 + 5 = R,
~4
10. A number when divided by 1404 gives a remainder 93.
What remainder would be obtained by dividing the same
number by 39?
a)4
b) 13
c)19
d)15
II. A number when divided by 17, leaves a remainder 5.
What remainder would be obtained by dividing the same
number by 357? a)39
c)21
b)29
d)38
same number is divided by
N\(=KNz),retnainderis R, (say).
Then, by the remainder rule, we have the following formula,
A number when divided by 1491 gives a remainder 83.
What remainder would be obtained by dividing the same
number by 21 ?
a)21
b)2
c)20
d) 18
A number when divided by 1092 gives a remainder 60.
What remainder would be obtained by dividing the same
number by 28?
A number when divided by 1156 gives a remainder 73.
What remainder would be obtained by di" ,Jing the same
numberby34?
a)5
b) 17
c) 13
d)4
A number when divided by 1836 gives a remainder 79.
What remainder would be obtained by dividing the same
number by 36?
a)7
b)9
c) 19
d) 14
A number when divided by 1207 gives a remainder 85.
What remainder would be obtained by dividing the same
number by 17?
a)7
b)2
c)O
d)6
A number when divided by 2470 gives a remainder 80.
What remainder would be obtained by dividing the same
numher by 38?
7.a
sor is Nz .
A number when divided by 609 gives a remainder 65.
What remainder would be obtained by dividing the same
number by 29?
a)6
b)5
c)6
d)7
A number when divided by 738 gives a remairider 92.
What remainder would be obtained by dividing the same
number by 18?
~6
6.
3.a
8:c
9.a
10.d
11. a; Hint: Here we apply "Remainder Rule".
This rule is applicable when the same number (dividend) is
divided by two different divisors which are multiples of each
other.
A number when divided by 221 gives a remainder 43, what
remainder will be obtained by dividing the same number by
17?
~2
4.
Answers
I.d
2.d
:. RJ =39
Hence, the required remainder = 39.
Note: All the other questions can also be solved by this rule.
Rule 40
If the sum of two numbers is x and their difference is y, then the
difference of their squares is xy.
Illustrative Example
Ex.:
The sum oftwo numbers is 75 and their difference is
20. Find the difference of their squares.
Soln: Detail Method: Let the numbers bex andy.
According to the question,
x+y= 75 .... (i) and
x-y=20 ... (ii)
Now, multiplying eqn (i) and (ii), we get
2
x
-l
= Difference of the squares of numbers =75
x20= 1500
Quicker Method: Applying the above rule, we have,
required answer = 75 x 20 = 1500
Exercise
I.
The sum of two numbers is 100 and their difference is 37.
The difference of their squares is.
[Clerk's Grade Exam, 1991]
i
I
-.II
8.
Number System
6
7
a)37
b) 100
c) 63
d)3700 .
The sum of two numbers is 50 and their ditzerence is 6.
The difference of their squares is
a) 400
b)500
c)350
d) 300
3. The sum of two numbers is 75 and their difference is 9.
The difference of their squares is
a) 685
b) 625
c) 675
d) 775
4. The sum of two numbers is 160 and their difference is
39.
The difference of their squares is
a) 6420
b) 4620
c) 8420
d) 6240
5. The sum of two numbers is 175 and their difference is
75.
The difference of their squares is
a) 13025
b) 13125
c) 13215
d) 13152
3.c
5.b
4.d
2.
Answers
I. d
Answers
1. a
If the two consecutive numbers arex andy, then the difference of their squares is given by x + y.
Illustrative Example
Ex.:· Two consecutive numbers are 8 and 9. Find the difference
of their squares.
Soln: Detail Method:
Required answer = 92 - 82 = 81- 64 = 17
Quicker Method: Applying the above rule, we have the
required answer = 8 + 9 = 17
Exercise
2.
3.
4.
5.
or, x2 +1+2x-x2 =37 or,
2x = 37 - 1 = 36 :.x=18
and x+I=19
:. numbers are 18, and 19
Quicker Method: Applying the above rule, we have
the required an.swer =
37-1
37+1
-- and -- = 18 and 19
22·
Two consecutive numbers are 17 and 18. Find the difference of their squares.
a)36
b)25
c)35
d)34
Two consecutive numbers are 75 and 76. Find.the difference of their squares.
a) 141
b)151
c)l3l
d)115
Two consecutive numbers are 79 and 80. Find the differ- .
ence of their squares.
a) 159
b) 169
c) 149
d)158
Two consecutive numbers are 15 and 16. Find the difference of their squares.
a)31
b)32
c)30
d)21
Two consecutive numbers are 26 and 27. Find the difference of their squares.
a)53
b)52
c) 43
d) 63
Answers
I.c
2.b
3.a
4.a
2.
3.
4.
5.
The difference between the squares of two consecutive
numbers is 39. Find the numbers.
a) 19,20
b)20,21
c) 18, 19
d) 17, 18
The difference between the squares oEtwo consecutive
numbers is 27. Find the numbers.
a)14,15
b)13,14
c)15,16
d)16,7
The difference between the squares of two consecutive
numbers is 35. Find the numbers.
a) 14, 15
b) 15, 16
c) 17,18
d) 18, 19
The difference between the squares of two consecutive
numbers is 59. Find the numbers.
a)29,30
b)30,31
c)28,29
d)27,28
The difference between the squares of two consecutive
numbers is 77. Find the numbers.
a)3839
b)39,40
c)40,41
d)37,38
5.a
Rule 43
.
If the sum of two numbers is x and sum of their squres is y,
then the
Exercise
I.
5.a
4.a
3.c
Rule 42
1.
2. d
2. b
(i) product of numbers is given by
(X2 - YJ
l-2-
and·
X-~2y-x2]
[
(ii) tlte numbers are.
2
[x+~]
and
2
Illustrative Example
The sum of two numbers is 13 and the sum of their
squares is 85. Find the numbers.
Soln:- Detail Method: Let the numbers be x and y.
According to the question,
Ex.:
x+y= 13 .... (i) and x2 +.l = 85 .... (ii)
Now, from eqn (i) and eqn (ii), we have
(x+ y)2 = 169
68
PRACTICE BOOK ON QUICKER MATHS
2
Soln: Detail Method: Let the numbers be x and y.
According to the question,
J
or, x + y + 2xy = 169
or, 2xy= 169-85 = 84
:. xy;= 42 [xy = product of two numbers]
Again,
(x-
x2 + y2 = 90 ..... (i) and
(x- yf = 46 .... (ii)
yf = (x+y)2 -4xy
From eqn (ii)
=169-4x42=1 :.
x-y= 1 .... (iii)
From eqn (i) and eqn (iii) we have,
x= 7 andy=6
:. Numbers are 7 and 6
Quicker Method: Applying the above rule, we have,
'(x-yY ~ 46
2
or, x + y2 - 2xy = 46
or, 90 - 2xy = 46 [Puttingthe value of eqn (i)]
90-46
or, ,xy= ---=22
2
.
n-.J170-169
reqUired answers = ----- and 2
:. product of two numbers = 22
Quicker Method: Applying the above rule, we have 90-46
the required answer = --, - = 22 2
13+.J170-169 =6and7 2
Exercise
1.
The sum of two numbers is 15 and sum of their squares
is 113. The numbers are:
[CDS Exam, 1991]
a)4,11
b) 5, 10
c)6,9
d)7,8
2. The sum of two numbers is 25 and sum of their squares is
313. The numbers are:
a)12,13
b)20,25
c) 9, 16
d)21,4
3. The sum of two numbers is 26 and sum of their squares is
340. The numbers are:
a) 12, 14
b)11,15
c) 9, 17
d)8,18
4. The sum of two numbers is 30 and sum oftheir squares is
458. The numbers are:
a)14,16
b) 12; 18
c) 13, 17
d)II,15
5. The sum of two numbers is 14 and sum of their square.§ is
100. The numbers are:
a) 6, 8
b)5,9
c)4,10
d)3,11
6. The sum of two numbers is 13 and sum of their squares 89.
Find the product of the two numbers.
a) 40
b)36
c)22
d)30
7. ,The sum of two numbers is 32 and sum of their squares 514.
Findthe product of the two numbers.
a)510
b)225
c}255
d) 355
Answers
1. d
2: a
Exercise
1.
2.
3.
4.
5.
The sum of squares of two numbers is 80 and the square of
their difference is 36. The product of the two numbers
is
[Clerks' Grade Exam, t 9911
a) 22
b) 44
c)58
d)116
The sum of squares of two numbers is 40 and the square of
their difference is 20. The product ofthe two numbers is
a) 10
b)20
c)15
d)16
The sum of squares of two numbers is 95 and the square of
their difference is 37. The product of the two numbers is
a) 18
b) 19
c)29
d) 27
The sum of squares of two numbers is 94 and the square
oftheir difference is 24. The product ofthe two numbers is
a)36
b)40
c)30
d)35
The sum of squares oftV;'o numbers is 87 and the square of
their difference is 25. The product of the two. numbers is
b)35
a)31
Answers
3. a
4. c
5. a
6. a
7. c
1. a
2. a
3.c
c)32
4.c
d)30
5.a
Rule 44
Rule 45
If the sum of squares of two numbers is x and the square of
their difference is y, then the product of the tlllO numbers is
If the product of two !lumbers is x ami the sum of their squa}'es
is y, then (i) the sum of the two /lumbers is given by
(x;y).
~ Y + 2x ami (if) tlte difference is ~ y -
Illustrative Example
Illustrative Example
Ex.:
2; .
The sum of squares of two numbers is 90 and the
square of their difference is 46. The product of the two
numbers is
Ex.:
The product of two numbers is 143. The sum of their
squares is 290. Find the sum of the two numbers and also
find the difference of the two numbers.
Soln: Detail Method: Let the numbers be x and y.
.. ...- ..
69
Number System
Rule 46
According to the question, xy=
2
143 and x +
l
The denominator of a rational number is 'D' more than its
numerator. lfthe numerator is increased by x and the denominator
is decreased by y,we obtain p, then the rational
= 290 Now,
2
(x+yf =x +y2+2xy
=290+2 x 143 =576
X-P(D-y)]
or,x+y= ...)576 =24
numberis given by . x + (yP - D) .
.'. Sum of the numbers = 24
Again,
Illustrative Example
(x - y
[
Ex.:
y = x + y2 - 2xy
2
=290-286=4
or,x-y=2
... difference of the numbers = 2
Quicker Method: Applying the above rule, we have the
sum ofthe numbers
The denominator ofa rational number is 3 more than
its numerator. If the numerator is increased by 7 and the
denominator is decreased by 2, we obtain 2. The
rational number is ______ _
Soln: Detail Method: Let the numerator be x and the denominator
= x + 3.
According to the question,
x+7 x+3-2
= ...)290+2xI43 =...)576 =24 and the
difference of the numbers
2
or, x + 7 = 2x + 2 ... x = 5
= ...)290-2xI43 =14 =2
... Numerator = 5 and the denominator = 5 + 3 = 8
Exercise
1.The product of two numbers is 12.0. The sum of their squares is
289. The sum of the two numbers is __ '_. [Clerks' Grade Exam,
1991]
a) 20
b}23
c) 169
d)33
2. The product of two numbers is 48. The sum of their squares
is 100. The sum ofthe two numbers is __ .
~14
~12
~18
~24
3. The product of two numbers is 168. The sum of their squares
is 340. The sum of the two numbers· is __ .
a)36
b)24
c)26
d) 28
4. The product of two numbers is 36. The sum of their squares
is 97. The sum of the two numbers is __ .
a) 13
. b)12
c}15
d}l1
5. The product of two numbers is 35. The sum of their squares
is 74. The sum of the two numbers is __ .
a) 13
b) 12
c) 14
d) 17
6. The productof two numbers is 120. The sum of their squares
is 289. The difference of the two numbers is
a)7
b)9
c)8
d) 23
7. The product of two numbers is 180. The sum of their squares
is 369. The,difference of the two numbers is
8.
b)2
d)
15
c)4
Answers
2.a
~. b
3.c
4.a
5.b
6.a
5
Quicker Method: Applying the above rule, we have
.
7-2(3-2)_ 5
ReqUIred answer = 7 + (2 x 2 - 3) '8
Exercise
1.
The numerator of a rational number is 4 less than its
denominator. If the numerator is increased by 8 and the
denominator is decreased by 2, we obtain 3. Find the rational
number.
3
b) 7"
2.
3.
1
c) "5
5
d) "9
The denominator of a rational number is 6 more than its
numerator. If the numerator is increased by 9 and the
denominator is decreased by 5, we obtain 5. Find the rational
number.
1
2 b) .
8
a) 7
"
The denominator of a rational number is 3 more than its
numerator. If the numerator is increased by 6 and the
denominator is decreased by 2, we obtain 2. Find the rational
number.
a)3
b) 27
c)5
d) 17
The product of two numbers is 224. The sum of their squares
is 452. The difference of the two numbers is
-a)30
l.b
7.a
... rational number = '8
1
4.
5
4
d) 7"
b) '8
a) '
3
The denominator of a rational number is 8 more than its
numerator. If the numerator is increased by 7 and the
denominator is decreased by 8, we obtain 8. Find the
70
PRACTICE BOOK ON QUICKER MATHS
rational number.
123
a)
5.
6.
"9
b)
10
c)
IT
The denominator of a rational number is 2 more than its
numerator. If the numerator is increased by 9 and the
denominator is decreased by 5, we obtain 7. Find the rational
number.
5
7
9
3
a)- b) c} d)7
9
II
5
The denominator of a fraction is 2 more than thrice its
numerator. If the numerator as well as denominator is
1
increased by one, the fraction becomes "3' What was
the original fraction.
[SBI PO, 1999]
435
a)
5
13
b)
11
c)
13
Answers
I.c
2.a
d)
IT
x 1 y
:. x:y= 1:2
2
Quicker Method: Applying the above ruie, we have
150-100
1
.
the required ratio =
100
- =]:2
:2
Note: In case tHe total ie (A + B) becomes P% of the number
100 )
(
A, the r'atio between A and B is given py P -1 00 .
Exercise
1. When
a number
added
another
totalis the
becomes
333~isper
centto
ofthe
first number
number.the
What
3
ratio between the first and the second number?
a) 3 : 7 b) 7: 4 c) 7: 3 d) Data inadequate When a number is
2. added to another number the total
becomes 333 ~ per cent afthe second number. What is
3
- ..
the ratio between the first and the second number?
3.d . 4.~
5.a
ISBI PO 20001
6. b; Hint: This type of question may be solved by hit and trial
a)3:7
b)7:4
c)1:3
d)4:7
method.
When
a
number
is
added
to
another
number
the
total becomes
3
First divide the question in different parts. Then start from the
. 250 per cent ofthe second number. What is the
answer-choices one-by-one. The choice, which' satisfies all
ratio between the first and the second number? a)3:2
the parts of the given question, will be required answer. For
b)2:3 c)4:3 d)3:4
example, in the above question we have two parts.
4. When a number is added to another number the total becomes
(I) The denominator of a fraction is 2 more than thrice its
175 per cent of the first number. What is the ratio between the
numerator.
first and the second number?
(II) If the num.erator as well as denominator is increased ~ 5. by 1,
a)4:3 b)3:4 c)5:3 d)3:5
the fractIOn becomes 1/3.
When a number is added to another number the total becomes
. Both parts will be satisfied by the answer choice (b), hence
275 per cent of the first number. What is the
(b) is the required answer.
ratio between the first and the second number? a)4:7
b}7:4 c)3:8 d)8:3
6. When a number is added to another number the total becomes
Rule 47
125 per cent of the second number. What is the ratio between
When a lIumber 'A' is added to another number 'B' alltl the total ie
the first and the second number?
(A + B) becomes P% of the lIumber B, thell the ratio
a) 1:4 b)4:1 c)1:2 d)2:1
7 When a number is added to another number the total becomes
between A and B is given by
375 per cent of the second number. What is the
.
ratio
between the first and the second number? a)4:11
Illustrative Example
b)11:4 c)4:7 d)7:4
Ex.:
When a number is added to another number the total
When a number is added to another number the total becomes
becomes 150 per cent of the second number. What is the
8 375 per cent of the first number. What is the
.
ratio between the first and the second number?
ratio between the first and the second num ber?
Solo: Detail Method:
a)4:11 b}II:4 c)4:7 d)7:4
Let the numbers be x and y.'
When a number is added to another number the total becomes
According to the question, 150
9.
225 per cent of the first numbel. What is the
x+y= 150%ofy= lOOy
ratio between the first and the second number? a)5:4
b)4:5 c)3:4 d)4:3
]0. When a number is added to another number the total becomes
3
I
or x + y = --'-y or x = - y
225 per cent of the second number. What is the
,
2
'
2
.
(P-I00).
wo
7
1
Number System
Answers
ratio between the first and the second number? a)3:4
b)4:3 c)5:4 "p)4:5
4.d
3.b
1. a 2. a
5.c
Rule 49
Answers
I.a
2.c
3.a
8.a
9.b
10.c
4.a
5.a
6.a
7.b
Rule 48
The sum of three consecutive even or odd immbers is Pless
Two different numbers when divided by the Sllme divisor,
leaves remainders x and y respectively, and when their sum is
divided by the same divisor, remainder is z, then the divisor is
given by (x + Y - z). Or,
Divisor = (sum of remainders) - (Remainder when sum is
divided)
a
or more than b ofQ. Then the middle number is given by
Illustrative Example
Ex:
Two different numbers when divided by the same divisor, left remainders II and 21 respectively, and when their
sum was divided by the same divisor,remainder was 4.
What is the divisor?
Soln: Applying the above rule, we have the required answer= II +21-4=28
Note: +ve and -ve sign indicate more and less respectively
Exercise
Illustrative Example
Ex:
The sum of three consecutive even numbers is 15
less than three-fourth of60. What is
. the middle number?
Soln: DetailMethod:
Let the middile number be x
According to the question, 60x3
'
x-2+x+x+2= ---15
4
'
or, 3x= 30 :.x= 10
... required answer = 10
Quicker Method: Since we have less type of question, the above formula will be like
. Q(~)-p
Middle number = ----=
3
60x~-15
4
3
10.
Exercise
1.
The sum of three consecutive even numbers is 14 less
than one-fourth of 176. What is the middle number. [BSRB
Mumbai PO, 19981
a)IO
b)8
c)6
d)4
2. The sum of three consecutive odd numbers is 15 more than
one fourth of 120. What is the middle number?
a)15
b)13
c) 17
d)21
3. The sUm of three consecutive even numbers is 24 less than
one-sixth of324. What is the middle number?
a) 12
b) 10
c) 14
d)20
, 4. The sum of three consecutive even numbers is 8 less than
two-third of 66. What is the middle number?
a) 10
b)18
c)16
d)12
5. The sum of three consecutive odd numbers is 25 more than
two-fifth of 65. What is the middle number?
a)]5
b) 19
c) 17
d)21
-
,I. Two different numbers when divided by the same divisor, left remainders 10 and 15 respectively, and when their
sum was divided by the same divisor, remainder was 3. What
is the divisor?
a) 22
b)25
c)23
d)21
2. Two differ.ent numbers when divided by the same divisor, left
remainders 5 and 7 respectively, and when their sum was
divided by 'the same divisor, remainder was 2. What is the
divisor?
a)11
b)12
c) 10
d)9
3. Two different numbers when divided by the same divisor, left
remainders 13 and 23 respectively, and when their sum was
divided by the same divisor, remainder was 5. What is the
divisor?
a) 32
b)36
<:)30
d)31
4. Two different numbers when divided by the same divisor, left
remainders 12 and 21 respectively, and when their sum was
divided by the same divisor, remainder was 4. What is the
divisor?
a)28
b) 27
c)31
d)29
5. Two different numbers when divided by the same divisor, left
remainders 15 and 17 respectively, and when their sum was
divided by the same divisor, remainder was 8. What is the
divisor?
~~
~~
Answers
3.d
1. a 2. c
0TI
4.d
~~
5.a
Rule 50
If the product of two numbers is x and the sum o.lthese two
.
[
--]j ;
y+ y -4x
numbers isy, then the 1lumbers a~egive1l by ~----
2
,
72
PRACTICE BOOK ON QUICKER MATHS
an"
[y-~~'
Rille 51
-4x ).
If the product of two numbes is x and the difference between
these two numbers is y, then the numbers are
Illustrative Example
The product of two numbers is 192 and the sum of
these two numbers is 28. What is the smaller ofthese two
numbers?
[BSRB Calcutta PO 1999]
Soln: Detail Method:
Let the numbers be x and y.
:. xy= 192,x+y=28
..... (i)
~y2+4X+Y]
2
Ex:
,'. (x_y)2=(x+y)2_4xy
Illustrative Example
. The product of two numbers is 192 and the difference
of these two numbers is 4. What is the greater of these two
numbers?
Soln: Detail Method:
Let the numbers is x and y.
xy= 192 andx-y=4 ...• (i}
(x+y? = (x-
yf +4xy
= (4? +4x]92=784
x+y=28 .... (ii)
, Solving eqn (i) and eqn (ii) we have x=
]6andy= 12
:. Greater number = 16
Quicker Method: Applying the above rule, we have
req.uired
answer
28+4
~y2 +4x + y =
2
_ 32 = 16 2
2
2
Note:
~/+4x+y ~y2+4x-y
--'->-----
2
Exercise
I
I
:
1.. The product of two numbers is ] 54 and the sum of these two
numbers is 25. Find the difference between the numbers.
a)3
b)4
c)5
d)8
2. The product of two numbers is 252 and the sum of these two
numbers is 33. Find the greater number.
a)2]
b)]2
c)]3
d)23
3. The product of two numbers is 255 and the sum of these two
numbers is 32. Find the smaller number.
f;
I
,I
11
a)]7
4.
~
I
I
~
5.
b)]6
c)]5
d)]3
The product of two numbers is ] 68 and the sum ofthese two
numbers is 26. Find the smaller number.
a)12
b)]4
c) 16
d) 18
The product of two numbers is 486 and the sum of these two
numbers is 45. Find the smaller number.
a) 12
b) 18
c)26
d) 34
Answers
I. a 2. a
.fi84 +4 _ 28+4 222
16 and 28-·h82
-4xl92 _ 28-4 = 24 = 12.
2
:. smaller number = 12.
.
Ex:
..
28+~282 -4x192
the reqUIred numbers = ---2
=
2
[
, =784-768 = 16
:. x - y = 4
.... (ii)
Combining eqn (i) and eqn (ii)
x= 16,andy= 12
:.smallernumber= 12.
Quicker Method: Applying the above rule, we have
28+ .,}784 - 768 2
[~y2+4X-Y]
and
2
Exericse
1.
The product of two numbers. is 22] and the difference of these
two numbers is 4. Find the smaller number.
a) 13
b)14
c)16
d)17
2. The product oftwo numbers is ] 98 and the difference of these
two numbers is 7. Find the greater number.
a) 18
b) 15
c) 13
d) I]
3. . The product of two numbers is ] 80 and the difference of these
two numbers is 3. Find the sum of the numbers.
a)26
b)25
c)28
d) 27
4. The product of two numbers is 594 and the difference of these
tw'J numbers is 5. Find the sum of the numbers.
a)46
b)39
c)40
d)49
5. The product of two numbers is 468 and the difference of these
two numbers is 8. Find the sum oHhe numbers.
a) 42
b)44
c)48
d)34
Answers
3.c
4.a
5.b
I. a 2. a
3.d
4.d
S.b
Number System
73
Miscellaneous
J.
a) 125
d)25
••
If a fi'action's numerator is increased by 1 and the de2
"3' But when the numerator is increased by 5 and the
denominator is increased by 1 then the fraction becomes
~ .What is the value of the original fraction?
4
[Bank of Baroda PO, 1999]
3
5
5
~-
6
~-
~~
7
8
7
7
2. If the numerator of a fraction is increased by 2 and denominator is increased by 3, the fraction becomes 7/9; and if
numerator as well as denominator are" decreased by 1 the
fraction becomes 4/5. What is the original fi'action? [SBI
Associates PO, 19991
13
a) 1
6
-5
9
c) 6'
b)I
l
17
d) 21
3.
e) None of these
If the numerator of a fraction is increased by 2 and the
denominator is increased by 1, the fraction becomes "8
5
and if the numerator ofthe same fraction is increased by 3 and
the denominator is increased by I the fraction
3
. becomes -. Whatis the original fraction? 4
[Guwahati PO Exam, 1999)
2
a) Data inadequate b)
3
d)7
4.
5.
4
"7
c)
"7
e) None of these
In a two-digit number, the digit at unit place is I more than
twice of the digit at tens place. If the digit at unit and tens
place be interchanged, then the difference between the new
number and original number is less than I to that of original
number. What is the original
number?
[BSRB Hydcrabad PO, 1999]
a) 52
b) 73
c)25
d)49
e)37
I
c) 40
e) None of these
6.
riominator is increased by 2 then the fraction becomes
~-
b)70
5
"5 ofa number is equal to "8 of the second number.lf35
is added to the first number then it becomes 4 times of second
number. What is the value of the second number? [BSRB
Hyderabad PO, 19991
The ratio of two numbers is 3 : 2. If! 0 and the sum of the two
numbers are added to their product, square of sixteen is
obtained. What could be the smaller number?
[NABARD,19991
a) 14
b) 12
c) 16
d) 18
e) None of these
7. The numbers x, y, z are such that xy = 96050 and xz="· 95625
and y is greater than z by one. Find out the numberz.
[NABARD,1999j
a) 425
b) 220
c) 525
d) 226
e) 225
8 If the sum of one-half, one-third and one-fourth of a number
exceeds the number itself by 4, what could be
.
the number?
[NABARD,19991
a) 24
b)36
c) 72
d) 84
e) None of these
When
any
number
is
divided by 12 then dividend bel
9.
comes - of the other number. By how much per cent is 4
first number greater than the second number?
[BSRB Chennai PO, 2000)
a)200
b) 150
c)300
d) Data inadequate
e) None of these
10. A number gets reduced to its one-third when 48 is substracted
from it. What is two-third of that number?
[BSRB Bhopal PO, 2000J
c)36
a) 24
b) 72
d)46
e) None of these
II. The sum of three consecutive numbers is given. What is he
difference between first and third number?
IBSRB Bhopal PO, 20001
a)-One
2) Three
c) Either one or three e)
d) Two
None of these
12. If the two digits of the age of Mr Manoj are reversed
thcn the new age so obtained is the age of his wife. -
I
11
of the sum of their ages is equal to the difference between their ages. IfMr Manoj is elder than his wife then find
the difference between their ages.
[BSRB Bangalore PO, 20001
a) Cannot be detennined
b) 10 years
c) 8 years
d) 7 years
e) 9 years
13. A number is greater than the square of 44 but smaller than the
square of 45. If one part of the number is the square of 6 and
the number is a multiple of 5, then find
the number.
[BSRB Bangalore PO, 20001
a) 1940
b)2080
c) 1980
d) Cannot be determined
e) None of these
]4. Ifa number is decreased by 4 and divided by 6 the result
I
I
!
i
f
74
PRACTICE BOOK ON QUICKER MATHS
i~ 9. What would be the result in is subtracted from the
nllmber and then it is divided by 5?
[BSRB Delhi PO, 2000]
2
I'
2
a)9- b) 10c) 11.55
5
d) 11
e) None of these
]5. A two-digit number is seven times the sum of its digits.
Ifeach digit is increased by 2, the number thus obtained is 4
more than six times the sum of its digits. Find the
number.
[BSRB Patna PO, 2001]
a)42
b)24
c)48
d) Data inadequate
e) None of these
]6. The digit in the units place of a number is equal to the digit in
the tens place of half ofthat number and the digit in the tens
place of that number is less than the digit in units place of half
of the number by 1. ]fthe sum of the digits of the number is
seven, then what is the number?
ISBI BankPO, 2001]
, a)52
b) 16
c)34
d) Data inadequate
e) None of these
] 7. A fraction becomes 4 when 1 is added to both the numerator
and denominator, and it becomes 7 when I is subtraced from
both the numerator and denominator.
The numerator of the given fraction is:
a)2
b)3
c)7
d) ]5
(NDAExam 1990)
18. If I is added to the denominator of a fraction, the fraction
becomes (1/2). If I is added to the numerator, the fraction
becomes 1. The fraction is:
4
a) -:
:;
5
b) "9
2
10
d)-
c) 3'
11
[-CDS Exam 1991
I
19. The sum of two numbers is twice their difference. If one 6fthe
numbers is 10, the other number is:
.., 1
1
.., 1
a).J3
c)300r - - d)300r.Jb)30
3
3
3
[RRB Exam 1991]
,
l
'
4
20.
5 of a certain number is 64. Half ofthat number is:
a) 32
b)40
c)80
d) 16
IBSRB Exam 1991]
I
1
21. - ofa number subtracted from :;- of the number gives
4
.)
12. The number is:
c) 72
d)63
a)]44
b)]20
IHotel Management, 19911
22. If one fifth of a number decreased by 5 is 5, then the number
is:
c)60
d) 75
a)25
b)50
[Clerks' Grade Exam 19911
23. I I times a number gives 132. The number is
a)]]
b)]2
c) 13.2
d)Noneofthese
[Clerks' Grade Exam 1991/
2
24. A number is25 more than its
5th. The number is:
[Clerk.s' Grade E~am, 19911
125
a) -3"
125
b)7
c)60
d) 80
25. 24 is divided into two paJ1S such that 7 times the first part
added to 5 times the second part makes ] 46. The
first part is:
b) 13
c) 16
d) 17
a) 11
[RRB Exam 19911
4
26.
2
5 of a number exceeds its '3 by 8. The number is:
a)30
b) 60
c) 90
d)Noneofthese
IRRB Exam 19891
27. The difference between squares of two numbers is 256000 and
sum of the numbers is 1000. The numbers are:
a) 628, 372
b) 600, 400
c) 640, 630
d) None of these
[GICAAO Exam, 19881
28. Three numbers are in the ratio 3 : 4: 5. The sum of the largest
and the sm~lIlest equals the sum of the third and 52. The
smallest number is:
a) 20
b)27
c)39
d) 52
[Accountants' Exam 19861
29. A positive number when decreased by 4, is equal to 2 I times
the reciprocal of the number. The number is:
a)3
b)5
c)7
-(~I)~
--[NDAExam 19871
30. The sum on Immb(;}FS'r~n58. If the ratio between first and
second be 2 : 3 and that between second and third be 5
: 3, then the second number is:
d)48
a)30
b)20
c)58
[SSC Exam 1986/
31. Two numbers are such that the ratio between them is 3 : 5; but
if each is il)creased by 10, the ratio between them becomes 5:
7. The numbers are:
a)3,5
b)7,9
c) 13,22
d)]5,25
[RRB Exam. 19891
32. Divide 50 into twq parts so that the sum of their recipro
cals is (1/12):
a)20,30
b) 24, 26
c)28,22
d)36,14 [RRB
Exam 19881
33. The sum of seven numbers is 235. The average of the first
three is 23 and that of the last three is 42. The fourth number
is:
c) 69
d) ]95
a) 40
b)126
[Clerks' Grade Exam. 1991/ 34.
How many figures (digits) are required to number a book
9.
Number System
7
5
cont~jning 200 pages?
a) 200
b) 600
c) 492
2
35. In a question, divisor is 3" of the dividend and 2 times
the remainder. rfthe remainder is 5, find the dividend.
a) 15
b)25
c) 18
---:;:J
= "8 or, 8x •. 5y = -II
Y
.
x
]. c; Let the fraction be - then, y .
.... (i)
5
Also,.we have y + I ="4
or,4x+20=5y+5
or,4x=5y-]5
From (i) and (ii), we get
2y+1 5y-]5
Y = 2x+ ] .... (i) imd(lOy+x)-(IOx+y)=
lOx + y-]
or, 9y - 9x = ] Ox + Y - ]
or, ] 9x - 8y = ]
Puttingthe value of (i) in equation (ii) we get,
] 9x - 8(2x + ]) = ]
or, ] 9x - ] 6x - 8 = ]
or,3x= 9 or, x= 3
So, y= 2 x 3 + ] =7
... original number = ] a x 3 + 7 = 37
]
5
. -=- (I')
5 · -1=-11
C
.
2y+] 2x7+I
or, 7y=49
..... y=7and x=--=---=53
3
x 5 ...
y =7
.. 11 8 ....
25
or, -11 +35 = 411 8
x+2
7
tively. Then y + 3 = 9
x-I 4
.... (ii)
Solving (i) and (ii), we get x = 5, Y = 6
Reqd fraction = 5/6
x 3. d; Let the original fraction be - . y
(But -ve value cannot be accepted)
So, x = 6. Hence, smaller number = 2x = ]2
7. e; xy=96050 ..(i) and xz= 95625 .... (ii)
andy-z=] ... (iii)
.: ............. ............................ y 96050 3842
.
Dlvldmg (I) by (ll), we get -; = 95625 = 3825 .... (IV)
8. e; Let the number be x.
. (~+~+~)x=(6+4+3)x=~x
or, 9(x + 2) = 7(y + 3) or; 9x- 7y = 3
I
6
Combining (iii) and (iv), we get z = 225
2. c; Let the numerator and denominator be x and y respec-
=
1 25
-41
, or
4· ,
Y.=J = 5" or, Sx - 4y
. fraction = - 7
...
4. e; Let the original number be ] Ox + Y
.... {ii)
--. =-- or 8y+4= ]5y-45
required original fraction =.
3
x = 3 and y = T
... 1l=40
6. b; Let the two numbers be 3x and 2x.
According to the question,
]0 + (3x+2x)+(3x x 2x)=(]6)2
or, 6x2 + 5x - 246 = 0 or, 6x2 +4]x - 36x -246 =0 or,
6x(6x +4 I) - 6(6x +4]) = 0
or, (6x +4])(x -6) =0
... x=6
y+2 = 3" or, 3x +3 =2y+4
x+5
.... (ii)
solving eqn (i) and (ii), we get
1+35=411
2
or, 3x = 2y + 1
....
(i)
x+3 3
Again, y +] == 4' or, 4x - 3y = -9
. '5 .8
Answers
3
Then
d) 24
[SSe 94)
36. A number when divided by 5 leaves a remainder 3. What is
the remainder when the square ofthe same number is
divided by 5?
a)9
b)3
c) ]
d)4
[MBA 19901
37.Assuming that A, Band C are different single-digit numerical
values other than what is already used in the following equation,
what number C definitely cannot be? 8A2+3B5+C4- ]27]
b) 9
c) Either 7 or 9
e) None of these
a)7
d) 6
x+]
x+2 5
d) 372 .
'" [MBA 1980]
.... (i)
.. 2 3 4
]2
According to the question,
12
]3
-x-x
= 4 . x=48
12
..
9. d; Here neither the remainder nor the dividend nor the second
number is given, so can't be determined.
] O. e; Let the number be x.
x
2
then ,x--=48
3 . -x=48
.. 3
76
PRACTICE BOOK ON QUICKER MATHS
11. d; Let the three consecutive numbers be x,
x + I and x + 2 respectively.
:. Diff. between first and third numbers
=x+2-x=2
12. e; Let the age ofMr Manoj be (lOx + y)yrs. :. His
wife's age = (1 Oy + x) years
1
Solving (i) and (ii), we get x = ·15, Y = 3.
x
18. c; Let the required fraction =
xI
:. y+l ='2 ~ 2x-y=]
.. ~
x+1
And, ,-- = 1 ~ x - y= -1
Y
Solving (i}and (ii), we getx = 2,y= 3.
Then, (lOx+y+ lOy+x) - = 10x+y -lOy-x" II
x 5 or,
x + y = 9x - 9y or, 8x = I Oy or,
y = '4
:. x = 5 and y = 4 (because any other multiple of5 will make
x 6ftwo digits)
:~ Diff=IOx +y -lOy-x =9x-9y=9(x -y) =9(5 4)= 9 yrs
13.c; Let the number be x.
2
44 <x<452 ~ 1936<x<2025
.... (i)
From equation (i), the required number will be any numberbetweenl1936 and 2025 .
Since one part of the number is the square of6 means one
factor is 36.
:. LCMof36and5= 180
:. Number will be multiple ofl80 ie 180x II = 1980 the only
value which satisfies the equation (i)
14. d;Let the number be x
:. The fraction is '3
lO+x =2(x-1O) ~ x=30.
4
20 b·-xx=64 - x=--=80
·, 5
64x5
---
4
1
I
. -xx= -x80 = 40
.. 2
2
1)
1
1
( 21 a' -xx--xx =12 - -x=12~x=144
·,3
4
--- 12'
.
2 '(~OfX)~5= 5~~=10=>x=50
·,5
5
.
23. b; l1x= 132 ~ x= 12
2
3x
' 125
2 ·.,x--x=25~-=25~x=-'
5
'5
3.
~ 25. b; Let these parts be x and (24 - x). Then,
7x+5(24-x)= 146 ~ x= 13 .
So the first part is 13.
26. b;Let the number be x. Then,
4
2
2
-x--x =8~-x =8~ x =60
5
3
15
27. a; Let the numbers be x and y. Then,
x2 '- y2 = 256000 and x + y = 1000.
2
,
x _ y2 256000
x - y = -- = -- = 256
..
x+ y
1000
Solving x +y= 1000, x-y =256, we get x =628,y= 372. 28.
c; Let the numbers be 3x, 4x and 5x.
5x+3x=Ax+52 ~ x= 13 .
:. smallest number = 3x = 39.
29. c; Let the number be x. Then,
x+i
Then, y-:;:-T = 4 ~ x - 4y = 3
x-I
2
.
19. b;Let the other number = x
x-4
x-3 58-3
. --=9 -x=58Again --=--=11
6
--.,5
5
15. a; Let the two-digit number be lOx + y.
IOx+y=7(x+y) ~ x=2y .... (i) 10(x
+2)+y+2 = 6(x + y+4)+4
or, lOx+y+22=6x+6y+28 ~ 4x-5y=6 .. : .. (ii)
Solving equations (i) and (ii), We get x = 4 and y = 2
16. a; Let 1/2 of the no. = lOx + y and
the no. = 10V + W
From the given conditions, W = x and V = Y -I
Thus the no, = lO(y - I) + x
.... (A)
:. 2(10x+y)= lO(y-I)+x ~.8y-19x= 10
V+W,\=7 ~ y-l +x=;7
:. x+y=8
....
Solving equations (i) and (ii), we get
(i)
x = 2 and y = 6
:. From equation (A),
....
Number = 1O(y-I)+x=52
(ii)
x 17 d' Let the
required
. , fraction be.- .
Y
And -- = 7 ~ x - 7y = -6 , y-]
y.
4
21
24
2<'
x- =-~x
- x- l=U~X=
x
7
:. required number = 7
------_._._ .. _~ _. - --...
77 .
Number System
Number oftwo digit pages from 10 to 99 == 90 Number
of three digit pages froni 100 to 200 == 101 :. total
number of required figures =
9 x I + 90 x 2 + 10 1 x 3 = 492
35. a; According to the question
30. a; Let the numbers be x, y, z. Then,
x_2y_5
;-3'-;-3
3
..
.,
=> 3x=2yand5z=3y.
3
33
9
. y=_x,z=-y=-x-:-x=-x
.,
2·
5
52
10
39·
~ 34x=680
-. x=20
., x+-x+-,--x=98
2
10
-;"
2
Divisor = "3 x dividend and 2 x remainder
-;'"
3 2'(3 ",30)
So, second number = '2 x == .
x 20
31. d; Let the numbers be 3x and 5x. .
36. d; The number is of the form (5x + 3), where x is an integer
3x+IO =~=>x=5
., 5x+ 10 7
25x2 +30x+9 25x2 30x 5+4
---- --+-+--
Hence, the numbers are 15,25.
32. a; Let the numbers be x and (50 - x). Then,
I I I 50-x+x
=> x(50.~x)
;+
2
or - x dividend = 2 x 5 , 3
2x5x3
.. Dividend = --2-· - = 15.
50-x = 12
I
12
=> x2-50x+600=0 =:>x=30or20. 33. a;
(23 x 3 +x +42 x 3)=235 => x =40:
:. fourth number = 40.
34. c; Number of one digit pages from I to 9 = 9
5
555
:. the remainder is 4.
37. e; SinceA+ B +C= 16
(Possible values of A, Band C are 0, 6, 7 & 9).
Also kt: B, B;f: C, A ;f: C .
IfC = 6, A + B should be 10, which is not possible. IfC = 9,
A+ B should be 7, which is also not possible. IfC = 0, A + B
should be 16 which is also not possible.