Download Grade 6 Natural and Whole Numbers

ID : ae-6-Natural-and-Whole-Numbers [1]
Grade 6
Natural and Whole Numbers
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Answer t he quest ions
(1)
T wo tankers contain 784 litres and 224 litres of petrol respectively. What is the capacity of the
largest measuring container which can measure the petrol of either tanker exactly?
(2)
T he length, breadth and height of a room are 8 m 55 cm, 2 m 79 cm and 4 m 14 cm respectively.
What is the length of the longest rod which can measure the dimensions of the room exactly?
(3)
A rectangular courtyard with length 6 m 40 cm and breadth 3 m 50 cm is to be paved with square
stones of the same size. Find the least number of such stones required.
Choose correct answer(s) f rom given choice
(4) In whole numbers, the associative property is satisf ied with the f ollowing operations
(5)
a. Addition and Subtraction
b. Subtraction and Multiplication
c. Division and Multiplication
d. Addition and Multiplication
Jibril wrote an exam which had 30 questions. Each question he answers correctly gets him 10
marks. However, he loses 5 marks f or each question he answers incorrectly.
He attempted all the questions and got a total of 255 marks. How many questions did he answer
wrong?
a. 8
b. 5
c. 2
d. 3
(6) 42(5 + 20) = (42 x 5) + (42 x 20) is an example of
a. Commutative Property
b. Associative Property
c. Distributive property
d. Closure Property
(7) T he number 256 can be represented by a 16 x 16 square grid. Out of the f ollowing numbers,
which number can not be represented by such a square grid?
(8)
a. 470
b. 144
c. 729
d. 484
T he product of two digit number is 2378. If the product of their units digit is 8 and their tens
digits is 20, then the numbers are
a. 81, 54
b. 58, 41
c. 55, 41
d. 51, 48
(9) (79 + 6) + 15 = 15 + (6 + 79) is an example of
a. Associative Property
b. Distributive property
c. Commutative Property
d. Closure Property
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ID : ae-6-Natural-and-Whole-Numbers [2]
Fill in t he blanks
(10)
T he two numbers nearest to 50000 which are exactly divisible by 7, 9, 5, 2, 3 are
and
.
(11)
(12)
T he only even prime number is
.
T he total number of whole numbers that are there between 31 and 81 is
.
Check True/False
(13) T he sum of two even numbers is an even number.
T rue
False
(14) T here is a whole number which does not change to value of any whole number it is added to.
T rue
False
(15) Every whole number has its predecessor.
T rue
False
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ID : ae-6-Natural-and-Whole-Numbers [3]
Answers
(1)
112 litres
Step 1
T he container which can measure petrol of both tanks, should be such that its volume in
litres should f ully divide 784 and 224.
T heref ore, capacity of the largest measuring container which can measure the petrol of
either tanker exactly is the HCF of 784 and 224.
Step 2
Let us f ind all prime f actors of 784:
2 | 784
2 is a factor of 784
2 | 392
2 is a factor of 392
2 | 196
2 is a factor of 196
2 | 98
2 is a factor of 98
7 | 49
7 is a factor of 49
7 |7
7 is a factor of 7
|1
784 = 2 × 2 × 2 × 2 × 7 × 7
Step 3
Let us now f ind all prime f actors of 224:
2 | 224
2 is a factor of 224
2 | 112
2 is a factor of 112
2 | 56
2 is a factor of 56
2 | 28
2 is a factor of 28
2 | 14
2 is a factor of 14
7 |7
7 is a factor of 7
|1
224 = 2 × 2 × 2 × 2 × 2 × 7
Step 4
T he HCF of 784 and 224 is = 2 × 2 × 2 × 2 × 7
= 112
Step 5
T heref ore, the largest measuring container which can measure the petrol of either tanker
exactly will have a capacity of 112 liters.
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ID : ae-6-Natural-and-Whole-Numbers [4]
(2)
9 cm
Step 1
According to the question, the length, breadth and height of the room are 8 m 55 cm, 2 m
79 cm and 4 m 14 cm respectively.
Now you have to convert each dimension into same unit. Since you know that 1m = 100 cm,
the length, breadth and height of the room in centimeters is 855 cm, 279 cm and 414 cm
respectively.
Step 2
T he length of the rod which can be used to exactly measure all three dimensions of a room
should be a f actor of all three dimensions, that is, a f actor common to all three dimensions.
Since we have been asked to f ind the length of longest such rod, the length should be
equal to HCF of all three dimensions of the room.
Step 3
Let us now f ind the HCF of 855, 279 and 414.
Step 4
All prime f actors of 855:
3
| 855
3 is a factor of 855
3
| 285
3 is a factor of 285
5
| 95
5 is a factor of 95
19 | 19
19 is a factor of 19
|1
T heref ore,
855 = 3 × 3 × 5 × 19.
Step 5
All prime f actors of 279:
3
| 279
3 is a factor of 279
3
| 93
3 is a factor of 93
31 | 31
31 is a factor of 31
|1
T heref ore,
279 = 3 × 3 × 31.
Step 6
All prime f actors of 414:
2
| 414
2 is a factor of 414
3
| 207
3 is a factor of 207
3
| 69
3 is a factor of 69
23 | 23
23 is a factor of 23
|1
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ID : ae-6-Natural-and-Whole-Numbers [5]
T heref ore,
414 = 2 × 3 × 3 × 23.
Step 7
T he HCF of 855, 279 and 414 is = 3 × 3
= 9.
Step 8
Hence, the length of the longest rod which can measure the dimensions of the room
exactly is 9 cm.
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ID : ae-6-Natural-and-Whole-Numbers [6]
(4) d. Addition and Multiplication
Step 1
T he Associative Property states that if we are adding or multiplying three or more numbers,
it does not matter where we put the parenthesis. It is applicable f or addition and
multiplication.
For example, 3 + (5 + 7) = (3 + 5) + 7,
3 × (5 × 7) = (3 × 5) × 7.
Step 2
T heref ore, we can say that, in whole numbers the associative property is satisf ied with
Addition and Multiplication.
(5)
d. 3
Step 1
T otal number of questions in the exam = 30
Step 2
Since each correct answer gets 10 marks, the maximum possible marks in the exam = 30 ×
10 = 300 marks.
Step 3
Since Jibril attempted all the questions and got a total of 255 marks, the marks lost by him
= 300 - 255 = 45 marks.
Step 4
Since he loses 5 marks f or each wrong answer and each correct answer otherwise could
get him 10 marks, the marks lost f or each wrong answer = 10 + 5 = 15.
Step 5
T he number of questions Jibril answered wrong =
T otal marks lost
Marks lost f or each wrong answer
=
45
15
=3
Step 6
Hence, Jibril got 3 answers wrong.
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ID : ae-6-Natural-and-Whole-Numbers [7]
(6) c. Distributive property
Step 1
Distributive property of multiplication means multiplication distributes over addition.
Formally,
'a(b + c) = ab + ac'
Step 2
T hus, 42(5 + 20) = (42 x 5) + (42 x 20) is an example of distributive property.
(7) a. 470
Step 1
A number which is a perf ect square can be represented on a square grid and the number
which is not a perf ect square can not be represented on a square grid.
Step 2
A perf ect square is a natural number which is the square of some natural number. In other
words, only a natural number whose square root is a natural number is a perf ect square.
Step 3
Now, if we look at all the options caref ully, we notice that only the number 470 is not a
perf ect square. All other numbers are perf ect squares (squares of the numbers 16, 22 or
27).
Step 4
T heref ore, we can say that the number 470 can not be represented on a square grid.
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ID : ae-6-Natural-and-Whole-Numbers [8]
(8)
b. 58, 41
Step 1
Lets assume two digit numbers are,
10x1 + y1 and 10x2 + y2
Step 2
It is given that the product of their units digit is 8 and their tens digits is 20,
theref ore x1 × x2 = 20 -----(1)
y1 × y2 = 8 -----(2)
and the product of two digit number is 2378,
(10x1 + y1) (10x2 + y2) = 2378 -----(3)
Step 3
Now we notice that there are f our variables and only three equations are specif ied. We
know that f our variables cannot be uniqualy determined f rom three equations, theref ore
lets check which option satisf y above three equations.
Step 4
If you look at the option 58, 41, you will notice that the product of their units digit is 8, the
product of their tens digits is 20
and the product of two digit number i.e 58 × 41 = 2378
Step 5
Since 58, 41 satisf ies the all three equations, theref ore we can say that the numbers are
58, 41.
(9) a. Associative Property
Step 1
T he word "associative" comes f rom "associate" or "group". T he Associative Property is the
rule that ref ers to grouping. For addition, the rule is "(a + b) + c = a + (b + c)".
Step 2
T his means, "(79 + 6) + 15 = 15 + (6 + 79)" is an example of the Associative Property.
(10)
49770
50400
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ID : ae-6-Natural-and-Whole-Numbers [9]
(11)
2
Step 1
An even number is a number that is divisible by 2.
For example {2, 4, 6, 8, 10........} are even numbers.
Step 2
A prime number is a number that cannot be divided by a number other than 1 and itself .
For example {2, 3, 5, 7, 11, 13, 17........} are prime numbers.
Step 3
If you look at the all even and prime numbers, you will notice that only 2 is a number which is
even and prime.
(12)
49
Step 1
T he dif f erence between two numbers is same as the whole numbers between two numbers
plus it also includes one of the number.
Step 2
T hus, the total number of whole numbers that are there between 31 and 81 = 81 - (31 + 1)
= 81 - 32 = 49.
(13) T rue
Step 1
We know that every even number is divisible by 2.
Step 2
Consider two even numbers x and y.
Since they are even, they can be written as x=2a and y=2b respectively, f or some numbers
a and b.
T heref ore, the sum x+y=2a+2b=2(a+b).
It can be clearly seen that x+y is divisible by 2 and theref ore is even.
Step 3
We have seen that the sum of two even numbers is an even number.
T heref ore, the answer is true.
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ID : ae-6-Natural-and-Whole-Numbers [10]
(14) T rue
Step 1
Whole numbers are the numbers 0, 1, 2, 3 .....
If we add the whole number 0 to any other whole number, we get that whole number itself
as the sum. For example:
0+7=7
0+1=1
Step 2
So, the answer is true.
(15) False
Step 1
Whole numbers are the numbers 0, 1, 2, 3 ...
Negative numbers like -1, -2, -3 ... are not whole numbers.
T he predecessor of 0 is -1, which is a negative number.
T heref ore, every whole number does not have its predecessor.
Step 2
Answer is f alse.
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