Download Trades Math Practice Test and Review

Trades Math Practice Test
and Review
This material is intended as a review only. To help prepare for the assessment, the
following resources are also available:
1. online review material (free of charge) www.viu.ca/assessments
2. A 2.5 hour assessment preparation workshop
($25 per person) www.viu.ca/assessments
3. Up to one month of individualized instruction through the Adult Basic Education Learning Centre.
Must have a VIU student number. ($20 per person)
For more information contact:
Assessment Services: email [email protected]
Tel 250 740 6416 local 6276/2831
Nanaimo ABE Learning Centre: Call (250)740-6425
Cowichan Campus Learning Centre: Call (250) 746-3509
Powell River Campus Learning Centre: Call (604) 485-2878
Preparing for the Trades Math Assessment
The Trades Math Assessment will assess your math skills to ensure you are ready for your chosen program.
The best way to use this review package is to start with the practice test. Once you have completed and
scored the test, you will be able to see where you need to do more study. Go to the appropriate place(s) in
the study package and PRACTICE, PRACTICE, PRACTICE!
More practice material is available if you click on ‘Additional Online Resources’
The following skills will be covered in the 40-question test. You will need to score 70% or above to
successfully complete the assessment (minimum 80% for the Electrical Entry program)
**YOU CAN USE A CALCULATOR**











Addition, subtraction, multiplication and division (whole numbers and decimals)
Converting fractions to decimals (and vice versa)
Writing percentages as decimals
Tenths/hundredths/thousandths
Simple exponents (4² etc.)
Simple square roots
Percentages (adding and deducting tax)
Word problems using ratio/cross-multiplication
Converting imperial to metric, km - miles
Simple algebra (not for Automotive or Motor and Marine Technicians)
Geometry (area, volume, perimeter):
o Circles, triangles, cylinders, rectangles, squares
o **Please note formulae for circles will be given. Other formulas must be
memorized.
VIU Trades Math Assessment – Practice Test
**Round up to 2 decimal places**
1. Write 3,456,394 in words
__________________________________________________________________
2. 3.5 + 8.09= _______
3. 3.5 – 0.34= _______
4. 3.5 x 8.09= _______
5. 3.5 ÷ 8.09= _______
6. Write the following as a decimal : 25 hundredths
7. Write ⅓ as a decimal
_______
8. Write 1.35 as a faction
_______
9. Write 18% as a decimal _______
10. Calculate 7²
_______
_______
11. Calculate 2³
_______
12. What is the square root of 144?
_______
13. 5 is 20% of what amount?
_______
14. What is 7% of 138?
_______
15. John buys 3 t-shirts for $18.32 each, 1 pair of shorts for $29.99. How much is the
total with 6% tax? _______
16. If Max is paid $795.45 for a 35 hour work week. What is his hourly wage? _______
17. Use this table to convert the quantities/distances below
1 litre = .26 gallons
1 metre = 39.37 inches
1 kilogram = 2.2 pounds
1 kilometre = .62 mile
A.
B.
C.
D.
E.
F.
G.
H.
25 km =
15 miles=
93 litres=
5 gallons=
12 pounds (lbs)=
5 kg=
15”=
3’=
______miles
______km
______gallons
______litres
______kg
______pounds
______’ _____”
______”
**Question 18 (algebra) is not required for Automotive, HDCT, Welding or
Motorcycle and Marine Technician)
18. Find the value of x
a) 3x = 25 + 1 _______
b) x/6 = 72
_______
19. Find the perimeter and area of the following shapes:
A.
Length of one side = 8 ft.
Width of one side = 8 ft.
a) Perimeter (of one side of the cube) = __________
b) Area of entire cube = ___________
B.
16’
25’ 7”
18’ 7”
a) Perimeter of full shape = __________
b) Area of full shape = ___________
C.
Use the following information to answer this question
A
5”
a) Area of the circle = ___________
b) Radius of the circle =______________
c) Circumference of the circle = _____________
You have completed the practice test. Review your answers below:
Answers:
1. Three million, four hundred and fifty six thousand, three hundred and ninety-four
2. 11.59
3. 2.16
4. 28.315
5. 0.433
6. 0.25
7. 0.278
8. 1 35/100
9. 0.18
10. 49
15. $51.21
16. $22.73
17. a) 15.5 miles b) 24.1 km c) 24.1 gallons d) 19.2 L e) 5.4 kg f) 11 lb g) 1’ 3” h) 36”
11. 8
18. x = 12 19. X = 432 20. A. a) 32 ft
C. a) 78.5 sq.in b) 5 “ c) 31.4 “
12. 12
13. 25
14. 9.66
b) 384 square feet B. a) 125 ft 4“ (1504”) b) 7489’ (89869 sq.in.)
WHOLE NUMBERS REVIEW
Before studying fractions, you should be familiar with all operations on whole numbers.
This work will provide a quick review. Answers are at the end on p.4.
1. ADDITION
In addition, remember to begin with the right hand column, and work to the left.
e.g. 423
+ 134
557
Try these:
(a) 19
+2
(b) 345
+123
(c) 951
+111
Remember that when the total of any column is greater than 9, you must carry the left digit in that total to
the column to the left.
1
e.g. 437
+126
563
Here is some practice:
(d) 489
+ 96
(e)
2754
+ 1666
(f) 238
+ 777
(g)
2. SUBTRACTION
Remember to begin at the right hand column and move column by column to the left.
e.g. 768
– 15
753
Do these:
1069
+ 888
(a)
1349
– 36
(b)
864
– 333
(c) 3834
– 2222
Remember that you should check the accuracy of your answer by adding it to what you took away. If it is
correct, your result should be the number you began with. In the example above, 753 + 15 = 768, so we
can assume that it is correct.
Remember that when a digit in the bottom number is too big to subtract from the digit in the top number,
you borrow from the next column in the top number first.
3 1
e.g.
You can not
subtract 9 from
3, so you
borrow 1 ten
from the tens
column first
43
– 19
24
Try these:
(d) 753
– 409
(e)
70
– 18
1. Cross out the 4 in the tens place, and replace it with 3.
2. Borrow 10 from 4 to make 13 in the ones column.
3. Subtract 9 from 13
4. Subtract 1 from 3.
(f) 8104
– 1987
(g) 7512
– 943
(h)
1000
– 369
(i)
3084
– 2294
3. MULTIPLICATION
Take the time to learn your multiplication tables. That will save you a lot of time later on!
Multiply each digit in the top number by the bottom number.
Write the answer from right to left, starting in the ones column.
e.g.
12
41
x4
x 9
Remember that any number multiplied by 0 is 0.
48
369
Try these:
(a) 430
x 3
(b)
702
x 41
(c) 3011
x
7
When you multiply by a two or three digit number, be sure to begin your answer under the ones column.
Then continue on the line below the tens and then the hundreds. Always multiply from right to left.
Leave a space under the bottom digit that has already been multiplied OR add a zero in that space to keep
the columns lined up neatly and correctly.
Carrying in multiplication is like carrying in addition. Multiply first and then add the number
being carried. Line up the digits carefully under the correct column.
e.g.
23
x 22
46
46_
506
638
x 51
638
31900
32538
405
x 266
2430
24300
81000
107730
Now try these:
(d) 789
x 46
(e)
5106
x 203
When you multiply by 10, 100, 1000, etc,
add the same number of zeroes to the right of the number.
e.g. 35 x 10 = 350
41 x 100 = 4100
60 x 1000 = 60 000
Tricky!
4. DIVISION
You will need your multiplication tables here too! Learn them well!
Division is the opposite of multiplication. That helps you to check your division answer.
Pay careful attention to lining up digits, as you did in multiplying large numbers, so that you can keep your
working straight.
Here are the words: 168 (the dividend) divided by 7 (the divisor) is 24 (the quotient) OR 7 into 168 is 24.
e.g.
24
7 168
-14
28
-28
0
Step 1: Divide the divisor 7 into 16 = 2. Place the 2 above the 6.
Step 2: Multiply 7 x 2 = 14. Place the 14 under the 16.
Step 3: Subtract: 16 – 14 = 2.
Step 4: Bring down the next number to the right = 8
Step 5: The new number is 28.
Step 6: Divide 7 into 28 = 4. Multiply 4 by the divisor 7 and the answer is 28.
Step 7: Subtract 28 from 28 = 0. The division is complete.
Check:
In division, it is easy to check the answer by multiplying your answer
(the quotient)
the number
byanswer
(the divisor).
Division problems
do notbyalways
work you
out are
withdividing
an even
as in the last example.
24 x 7 = 168 so the answer is likely to be correct.
Sometimes there is an amount left over, which is called the remainder. It is placed on the top line with the
letter r for remainder and is part of the quotient
Try these:
569 r 3
(a) 4 1631
(b) 7 1046
(c) 9 3004
.e.g. 6 3417
– 30
41
– 36
57
– 54
3
When you check your answer for a division problem with a remainder, multiply your answer (the quotient)
by the number you are dividing by (the divisor) and add the remainder.
When you divide by a double or triple digit number, you need to use another different skill. You need to
estimate, which is a process of thoughtful guessing. This takes time and practice.
e.g.
223
28 6244
–56
64
– 56
84
– 84
0
Step 1: Think of about how many times 28 goes into 6244.
To do that, round 28 to 30 (28 ≈ 30) and divide that into 62.
The answer is approximately or close to 2.
Step 2: Place 2 above the last digit of 62.
Step 3: 2 x 28 (the divisor) = 56. Then 62 – 56 = 6.
Step 4: Bring down 4. Estimate how many times 28, ≈ 30,
divides into 64 ≈ 2. Place 2 on the top line.
Step 5: 2 x 28 = 56. 64 – 56 = 8.
Step 6: Bring down 4. Estimate how many times 28, ≈ 30, divides into 84 ≈ 3.
Sometimes your estimate is not quite right, but it should only be 1 more or 1 less than the correct number, so
the estimate gives a good starting point. Here is an example with a remainder:
e.g.
337 r 32
41 13849
– 123
154
–123
319
– 287
32
Try these:
(d)
57 3477
Round 41 ≈ 40. Divide 40 into the first part of the number.
40 goes into 138 about 3 times. Place 3 above the last digit of 138.
3 x 41 = 123. Subtract 123 from 138 = 15.
Bring down 4. Estimate how many times 41  40 goes into 154  3.
Multiply 3 x 41 = 123. 123 from 154 = 31.
Bring down 9. Estimate how many times 41  40
goes into 319  7. 7 x 41 = 287.
Step 7: 319 – 287 = 32. Divide 41 into 32. It cannot divide,
so 32 is the remainder.
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Step 6:
(e)
(f)
36 7601
67 13467
(g) 29 13796
ANSWERS
Addition:
Subtraction:
(a) 21
(a) 1313
(h) 631
Multiplication: (a) 1290
Division:
(a) 407 r 3
(b) 468
(b) 531
(i) 790
(b) 28782
(b) 149 r 3
(c) 1062
(c) 1612
(d) 585
(d) 344
(e) 4420
(e) 52
(f) 1015
(f) 6117
(g) 1957
(g) 6569
(c) 21077 (d) 36294 (e) 1036518
(c) 333 r 7 (d) 61
(e) 211 r 5 (f) 201 (g) 475 r 21
FRACTIONS REVIEW
A.
INTRODUCTION
1.
What is a fraction?
Fraction = numerator
denominator
A fraction consists of a numerator (part) on top of a
denominator (total) separated by a horizontal line.
For example, the fraction of the circle which is shaded is:
2 (parts shaded)
4 (total parts)
In the square on the right, the fraction shaded is 3 and
8
the fraction unshaded is 5
8
2.
Equivalent Fractions – Multiplying
The three circles on the right each have
equal parts shaded, yet are represented
by different but equal fractions. These
fractions, because they are equal, are
called equivalent fractions.
2
4
1
2
4
8
Any fraction can be changed into an equivalent fraction
by multiplying both the numerator and denominator by the same number
1 x2
2 x2
=
2
4
Similarly
5 x 2 = 10
9 x2
18
or
1 x4 4
=
2 x4 8
so 1  2  4
2 4 8
or
5 x 3 = 15
9 x 3 27
so 5 = 10 = 15
9 18 27
You can see from the above examples that each fraction has an infinite number of fractions that
are equivalent to it.
3.
Equivalent Fractions – Dividing (Reducing)
Equivalent fractions can also be created if both the
numerator and denominator can be divided by the
same number (a factor) evenly.
This process is called “reducing a fraction” by
dividing a common factor (a number which divides
into both the numerator and denominator evenly).
441
8 4 2
27  9  3
81 9 9
5 5 1
30 5 6
6
2 3
 
10 2 5
4.
Simplifying a Fraction (Reducing to its Lowest Terms)
It is usual to reduce a fraction until it can’t be reduced
any further.
A simplified fraction has no common factors which will
divide into both numerator and denominator.
Notice that, since 27 and 81 have a common factor of 9,
we find that 3 is an equivalent fraction.
9
But this fraction has a factor of 3 common to both
numerator and denominator.
So, we must reduce this fraction again. It is difficult to
see, but if we had known that 27 was a factor (divides
into both parts of the fraction evenly), we could have
arrived at the answer in one step
e.g.
8 8  1
24 8 3
45  15  3
60 15 4
27  9  3
81 9 9
331
9 3 3
27  27  1
81 27 3
5.
EXERCISE 1: Introduction to Fractions
a) Find the missing part of these equivalent fractions
1) 2 =
6
3
2)
3 =
12
4
3)
5 =
8
40
x2
Example:
3 =
10
5
x2
4)
1 =
32
16
5)
2 =
45
15
7)
7 =
100
10
8)
3 =
44
4
6.)
7 =
27
9
Since 5 x 2 = 10,
multiply the numerator
by 2, also.
So,
3 = 6
5
10
b) Find the missing part of these equivalent fractions.
÷5
1) 8 =
4
16
2)
24 =
27
9
Example:
5 =
2
10
÷5
3)
6 =
10
5
4)
25 =
7
35
5)
20 =
6
30
6)
90 =
50
100
Since 10 ÷ 5 = 2 divide
the numerator by 5, also.
So,
c) Simplify the following fractions (reduce to lowest terms).
1) 9
12
5) 20
25
8
12
6) 14
21
2)
6
8
7) 8
16
3)
4) 15
20
8) 24
36
5 = 1
2
10
9)
B.
66
99
10) 18
30
TYPES OF FRACTIONS
1. Common Fractions
A common fraction is one in which the numerator is less than the denominator
(or a fraction which is less than the number 1). A common fraction can also be called a proper
fraction.
1,
2
e.g.
3,
4
88 , 8
93 15
are all common fractions.
2. Fractions that are Whole Numbers
Some fractions, when reduced, are really whole numbers (1, 2, 3, 4… etc).
Whole numbers occur if the denominator divides into the numerator evenly.
8 is the same as 8 ÷ 4 = 2 or 2
4
4
4
1
30 is the same as 30 ÷ 5 = 6 or 6
5
5
5
1
e.g.
So, the fraction 30 is really the whole number 6.
5
Notice that a whole number can always be written as a fraction with a denominator of 1.
e.g.
3.
10 = 10
1
Mixed Numbers
A mixed number is a combination of a whole number and a common fraction.
e.g.
23
5
(two and three-fifths)
27 2 (twenty-seven and two-ninths)
9
3
9
= 9 1 (always reduce fractions)
6
2
4. Improper Fractions
An improper fraction is one in which the numerator is larger
than the denominator.
From the circles on the right, we see that 1 3 (mixed number)
4
is the same as 7 (improper fraction).
4
An improper fraction, like 7 , can be changed to a mixed
4
number by
dividing the denominator into the numerator and
expressing the remainder (3) as the numerator.
16 = 3 1
5
5
e.g.
29 = 3 5
8
8
1
7
4
3
4
1
= 4 7
4
3
14 = 4 2
3
3
7
4
=
= 13
4
A mixed number can be changed to an improper fraction by
changing
the whole number to a fraction with the same
denominator as the common fraction.
2 3 = 10 and 3
5
5
5
13
=
5
10 1
9
=
=
90
9
91
9
and
1
9
A simple way to do this is to multiply the whole number by the denominator,
and then add the numerator.
e.g.
4x95
36  5
45 =
=
= 41
9
9
9
9
10 x 7  2
70  2
10 2 =
=
= 72
7
7
7
7
5. Simplifying fractions
All types of fractions must always be simplified (reduced to lowest terms).
e.g.
6 = 2,
9
3
2 5 = 21,
25
5
27 = 3 = 1 1
18
2
2
Note that many fractions can not be reduced since they have no common factors.
e.g.
17 , 4 , 18
21 9 37
6. EXERCISE 2 : Types of Fractions
a) Which of the following are common fractions (C), whole numbers (W), mixed numbers (M)
or improper fractions (I)?
1) 2
3
2) 3 4
5
3) 7
5
4) 8
8
5) 24
2
6) 5 8
19
7) 2 3
3
8) 25
24
9) 24
25
10) 12
12
4) 12
5
5) 100
99
6) 25
2
3) 8 2
3
4) 11 1
5
5) 9 4
5
6) 4 3
4
3) 2 12
18
4) 5 27
54
5) 25
15
6) 90
12
b) Change the following to mixed numbers:
1) 7
5
2) 18
11
3) 70
61
c) Change the following to improper fractions:
1) 2 1
5
2) 6 3
8
d) Simplify the following fractions:
1) 28
40
2) 80
10
C.
COMPARING FRACTIONS
In the diagram on the right, it is easy to see that 7 is larger
8
than 3 (since 7 is larger than 3).
8
However, it is not as easy to tell that 7 is larger than 5 .
8
6
In order to compare fractions, we must have the same (common)
7
8
=
3
8
denominators. This process is called
“Finding the Least Common Denominator” and is usually
abbreviated as finding the LCD or LCM (lowest common multiple).
Which is larger:
7 or 5 ?
8
6
In order to compare these fractions, we must change both fractions
to equivalent fractions with a common denominator.
To do this, take the largest denominator (8) and examine multiples
of it, until the other denominator (6) divides into it.
Notice that, when we multiply 8 x 3, we get 24, which 6 divides
into.
Now change the fractions to 24th s.
When we change these fractions to equivalent fractions with an
LCD of 24, we can easily see
that 7 is larger than 5 since 21 is greater than 20 .
24
24
8
6
8x1 = 8
(6 doesn’t divide into 8)
8 x 2 = 16
(6 doesn’t divide into 16)
8 x 3 = 24 LCD
x3
x4
x3
x4
7 = 21
5 = 20
24
24
8
6
Which is larger:
4 or 5 ?
9
12
Examine multiples of the larger denominator (12) until the smaller
denominator divides into it. This tells us that the LCD is 36.
Now, we change each fraction to equivalent fractions with the LCD
of 36.
x4
4 = 16
9
36
x4
x3
5 = 15
12
36
x3
12 x 1 = 12
12 x 2 = 24
12 x 3 = 36 (LCD)
LCD is 36.
So, 4 is larger than 5
9
12
__________________________________________________________________________
Which is larger:
4 or 13 or 11 ?
5
12
15
Find the LCD by examining multiples of 15. Notice that, when we
multiply 15 x 4, we find that 60 is the number that all denominators
divide into.
x12
4 = 48
60
5
x12
x4
13 = 52
60
15
x4
So, 11 is the largest fraction.
12
15
15
15
15
x
x
x
x
1
2
3
4
=
=
=
=
x5
11 = 55
60
12
x5
15
30
45
60 (LCD)
Which is larger:
7 or 13 ?
9
18
Notice that one denominator (9) divides into the other denominator (18). This means that
the LCD = 18 and we only have to change one fraction  7  to an equivalent fraction.
9
x2
7 = 14
9
18
x2
So, 7 is larger than 13
9
18
______________________________________________________
______________________________________________________
1. EXERCISE 3: Comparing Fractions
Which is the largest fraction? (find LCD first)
1)
7 or 6
13
13
2) 1 9 or 18
10
10
3) 4 or 9
5
10
4)
3 or 5
13
12
5) 5 or 4
8
7
6) 1 or 6 or 7
11
2
12
8) 4 or 5 or 3
8
9
12
9) 1 or 3
4
16
7) 2 or 11
3
15
D.
ADDING FRACTIONS
There are four main operations that we can do with numbers: addition ( +
multiplication ( x ), and division ( ÷ ).
In order to add or subtract, fractions must have common denominators.
This is not required for multiplication or division.
), subtraction ( –
1. Adding with Common Denominators
To add fractions, if the denominators are the same, we simply add
the numerators and keep the same denominators.
e.g.
Add
1
+ 2 = 3
4
4
4
1 and 5
12
12
Since the denominators are common, simply add the numerators.
Notice that we must reduce the answer, if possible.
1 + 5 = 6
12 12
12
= 1
2
2. Adding When One Denominator is a Multiple of the Other
Add
5
2
and
9
27
Notice that the denominators are not common. Also notice that 27 is
a multiple of 9 (since 9 x 3 = 27). This means that the LCD = 27
(see the last example in “Comparing Fractions”).
2 + 5 = 6 + 5
27
27
27
9
11
=
27
x3
2 = 6
27
9
x3
),
3. Adding Any Fraction
Add
7 and 13
12
15
We must find a common denominator by examining multiples of the
largest denominator. We find that the LCD = 60.
Add 1
7 + 13 = 35 + 52
60
60
12 15
87
=
60
= 1 9
20
5 and 2 3
6
8
When adding mixed numbers, add the whole numbers and the
fractions separately. Find common denominators and add.
+
If an improper fraction occurs in the answer, change it to a common
fraction by doing the following.
1 5 = 1 20
24
6
2 3 = 2 9
8
24
total equals 3 29
24
29
5
3
= 3+1
24
24
5
= 4
24
4. The Language of Addition
1 + 2 CAN BE WORDED
2 3
1 plus 2
2
3
1 and 2
2
3
addition of 1 and 2
2
3
total of 1 and 2
2
3
1 combined with 2
2
3
sum of 1 and 2
2
3
1 more than (or greater than) 2
2
3
Note: All of these can be worded with the fractions in reverse order:
e.g. 2 plus 1 is the same as 1 plus 2
3
2
2
3
5.
EXERCISE 4: Adding Fractions
a) Add the following:
1)
1
2
+
5
5
2)
4
3
+
5
5
3)
4
2
+
9
9
4)
3
3
3
+
+
4
4
4
b) Find the sum of:
1)
2
1
+
3
9
2)
1
3
+
2
8
3)
1
5
+
4
16
4)
2
4
+
3
15
c) Add the following:
1
1
+ 4
2
4
2) 9
2
1
+ 3
3
6
1
4
+ 4
2
5
4) 2
3
1
+ 6
4
2
5) 4
1
5
+ 6
3
6
6) 6
1
3
+ 8
3
4
7) 7
2
4
+
3
5
8) 8
2
1
3
+ 6 + 1
3
4
8
1)
3
3) 8
d) Evaluate the following:
1)
2
3
and
3
7
2) total of
3)
1
1
plus
2
5
4)
5)
5
3
combined with
12
8
5
3
and
6
8
3
5
greater than
2
7
6) sum of
1
3
and
6
14
E.
SUBTRACTING FRACTIONS
1. Common Fractions
As in addition, we must have common denominators
in order to subtract. Find the LCD; change the fractions to
equivalent fraction with the LCD as the denominator. Then
subtract the numerators, but keep the same denominator.
5 3
8 8
=
2 3
3 8
=
2
1
or
8
4
16
9
24 24
7
=
24
2. Mixed Numbers
When subtracting whole numbers, subtract the whole numbers, and
then subtract the fractions separately.
3
5
3
2
-1 = 2
9
9
9
However, if the common fraction we are subtracting is smaller than
the other common fraction, we must borrow the number “1” from
the large whole number.
i.e.
4
2
7
2
9
=3+
+ , or 3
7
7
7
7
3
2
from 6 , first change the common fractions to
3
4
8
9
equivalent fractions with the LCD. Since
is smaller than
,
12
12
borrow from 6.
4
To subtract 1
6
8
12
8
20
= 5
+
= 5
12 12
12
12
6
2
5
9
5
-2 = 3 -2
7
7
7
7
4
= 1
7
2
3
8
9
-1 = 6
-1
3
12
4
12
20
9
= 5
-1
12
12
11
= 4
12
The Language of Subtraction
5 2
CAN BE WORDED
6 3
5
2
minus
6
3
2
minus)
3
(NOT
2 subtracted from 5
3
6
2
5
from
3
6
2
5
less than
3
6
5
2
decreased by or lowered by
6
3
the difference of
5
2
and
6
3
NOTE: Unlike addition, we can not reword the above with the fractions in reverse order:
1 2
2 1
i.e.
is NOT the same as
2 3
3 2
======================================================================
3. EXERCISE 5: Subtracting Fractions
a) Subtract the following:
1)
9
1
12
8
2)
14
1
15
6
4)
7
2
9
3
5) 9
7)
3
5
4
8
8)
3)
2
1
- 6
3
6
5 3
6 8
6) 4
1
1
-1
2
4
3) 5
5
6
- 4
7
7
11
2
12
3
b) Subtract the following:
1)
6
4) 4
1
2
- 2
3
3
3
4
- 1
2) 13
11
12
5) 16
1
3
- 5
4
4
2
3
- 5
3
6) 9
4
1
3
- 4
6
8
c) Find the following:
2
1
decreased by
is what?
7
21
1) What is
5
3
minus
?
8
16
2)
3) What is
4
7
less than ?
9
9
4) What is
1
9
from
?
6
24
F.
MULTIPLYING FRACTIONS
1. Common Fractions
When multiplying fractions, a common denominator is not needed.
Simply multiply the numerators and multiply the denominators
separately.
Sometimes, we can reduce the fractions before multiplying.
2 5
25
10
x
=
=
3 9
3 9
27
1
3 5
3
3 5
x
=
=
5 7
7
5 7
1
Any common factor in either numerator can cancel with the same
factor in the denominator. Multiply after cancelling (reducing).
1
4 3
43
1
x
=
=
9 8
98
6
3
Note that any whole number (16) has the number “1” understood in
its denominator.
1
2
2
5
5  16
x 16 =
= 10
8
8 1
1
If more than two fractions are multiplied, the same principles apply.
2
3
1
2  3 1
x
x
=
=
9
20 4
9  20  4
1
120
2. Mixed Numbers
Mixed numbers must be changed to improper fractions before
3
multiplying. Remember that a mixed number (like 2 ) can be
4
changed to an improper fraction by multiplying the whole number
(2) by the denominator (4) and then adding the numerator.
See page 6 for instructions on changing a mixed number to an
improper fraction.
2
5
3
8
11 52
x4
=
x
4
11
4
11
= 13
4
3
49 17
x 2 =
x
9
7
9
7
119
9
2
or 13
9
=
3. The Language of Multiplication
1 2
x
CAN BE WORDED
2 3
1
2
multiplied by
2
3
1
2
by
2
3
1
2
of
2
3
the product of
1
2
and
2
3
NOTE: When multiplying, it doesn’t matter which fraction is first.
1
2
2 1
i.e.
x
is the same as
x
2
3
3
2
=======================================================================
4. EXERCISE 6: Multiplying Fractions
a) Multiply (cancel first, when possible):
1)
2
3
x
3
4
2)
12
8
x
9
16
3)
5
9
x
3 15
4)
2
15
x
5
21
5)
5
48
x
8
125
6)
3 16
9
x
x
4
27 16
7)
5
7
16
x
x 90 x 20 x
3
4
14
8) 2
10)
18 57
6
8 2
x
x
x x
19
4
32 9 3
2 15
x
3 21
4
4
x 3
5
7
9) 8
12
1
x 2
25
37
12) 2
11) 1
5
x 6
12
b) Find the following:
1) What is
4)
4
5
of 2
1
7
2
of 45?
3
is what number?
2)
41
5
27
by 14 is what number? 3) What is times
?
9
10
7
5)
1
62
of
is what number?
2
63
G.
DIVIDING FRACTIONS
To divide fractions, we invert (take the reciprocal of) the fraction
that we are dividing by, then cancel (reduce), and then multiply.
Taking the reciprocal of a fraction involves “flipping” the fraction
so that the numerator and denominator switch places.
Note that a whole number is really a fraction (e.g. 4 =
4
).
1
2
3
reciprocal
3
2
8
19
reciprocal
19
8
4 reciprocal
4
1
1. Common Fractions
Simply invert (take the reciprocal of) the fractions that we are
8
dividing by ( ). Then cancel and multiply.
9
Note: you can only cancel after the division is changed to a
multiplication.
5 8
5 9 45
 =  =
7 9
7 8 56
9
3
9 32
=
=6


16 32 16 3
2. Mixed Numbers
As in multiplication, mixed numbers must be changed to improper
fractions.
2
1
5 15
1 2 = 
3
7
3 7
5 7
= 
3 15
7
=
9
3. The Language of Division
1
2
CAN BE WORDED

2
3
1
2
divided by
2
3
2
3
into
1
2
divide
1
2
by
2
3
NOTE: In multiplication, the order of the fractions was not important.
1
2
2 1
i.e.
x
is the same as
x
2
3
3
2
In division, this is not the case. The order of the fractions is important.
Consider the following:
1 2
1 3
3
 =  =
2 3
2 2
4
but
2 1
2 2
4
1
 =  = =1
3 2
3 1
3
3
======================================================================
4. EXERCISE 7: Dividing Fractions
a) Divide:
1)
3 9

5 15
2)
3 12

7 5
3) 6 
4)
7
 14
12
5)
13 39

15 40
6) 3
7)
27
1
8
30
3
7
2
8) 1  1
9
3
10) 2
2
3
3 11

10 15
2
1
9) 4  3
3
2
3
9
3
11
22
b) Find the following:
1)
4
2
divided by
5
5
2)
2
4
divided by
5
5
3)
2
2
by
3
3
5)
4
4
into 1
9
5
6) Divide
4) Divide 1
9
12
into
4
25
1
2
by
4
5
FRACTION REVIEW:
Decide which operation ( +, -, x,  ) by the wording in the question. Then find the answer.
3
of 40?
5
1
2
3) How much is
from ?
2
3
4
2
5)
and
equals….?
5
3
9
7) What is
of 300?
10
3
8
9)
from equals….?
4
9
5
21
11) How much is
of
?
7
50
3
2
13)
of
equals….?
4
3
1) What is
2) How much is 4
4) How much is
2
1
from 6 ?
3
5
3
of 21?
7
2
divided by 14 is what number?
3
2
8) What is
into 12?
5
2
3
10) What is 1 by 3
3
8?
2
1
4
12) Find the total of
and
and ?
3
6
9
2
3
14) What is
greater than 4 ?
5
5
6)
ANSWER KEY – FRACTION REVIEW
EXERCISE 1: Introduction to Fractions (Page 3)
a)
1)
4
6
2)
9
12
3)
25
40
4)
2
32
5)
6
45
6)
21
27
b)
1)
2
4
2)
8
9
3)
3
5
4)
5
7
5)
4
6
6)
45
50
c)
1)
3
4
2)
2
3
3)
3
4
4)
3
4
5)
4
5
6)
2
3
7)
70
100
8)
33
44
7)
1
2
8)
2
3
9)
2
3
10)
3
5
EXERCISE 2: Types of Fractions (Page 6)
a)
1) C
2) M
b)
1) 1
c)
1)
11
5
2)
d)
1)
7
10
2) 8
2
5
2) 1
3) I
4) W
7
9
2
3) 1
4) 2
61
5
11
51
8
3)
26
3
3) 2
2
3
4)
5) W
5) 1
6) M
8) I
9) C
10) W
1
1
6) 12
99
2
56
5
5)
49
5
1
2
5)
5
2
or 1
3
3
4) 5
7) W
6)
19
4
6)
15
1
or 7
2
2
6)
7
12
EXERCISE 3: Comparing Fractions (Page 9)
1)
7
13
2) 1
9
10
3)
9
10
4)
5
12
5)
5
8
7)
11
15
8)
4
9
9)
1
4
EXERCISE 4: Adding Fractions (Page 12)
a)
1)
3
5
2)
7
2
2
or 1
3)
5
3
5
4)
9
1
or 2
4
4
b)
1)
7
9
2)
7
8
4)
14
15
c)
1) 7
3
4
d)
1) 1
2
5
2) 1
21
24
2) 12
3)
5
6
9
16
3) 13
3)
3
10
7
10
4) 9
1
4
5) 11
4) 2
3
14
5)
1
6
19
24
6) 15
6)
1
7
7
7) 8
8) 16
15
24
12
8
21
EXERCISE 5: Subtracting Fractions (Page 14)
5
8
a)
1)
b)
1) 3
c)
1)
2)
2
3
7
16
23
30
3)
11
24
4)
1
2
3)
6
7
4) 2
5
21
3)
1
3
4)
2) 7
2)
1
9
5) 3
5
6
1
2
5) 10
11
12
6) 3
1
4
6) 4
19
24
7)
1
8
8)
5
24
EXERCISE 6: Multiplying Fractions (Page 16)
a)
b)
1
2
2)
2
3
8) 10
9)
130
4
or 6
21
21
1) 30
2) 82
1)
3) 1
3)
4)
2
7
5)
10) 1
3
1
or 1
2
2
4)
1
2
6
25
11) 3
6)
1
4
12)
12
5
or 1
7
7
5)
7) 6000
29
1
or 14
2
2
31
63
EXERCISE 7: Dividing Fractions (Page 18)
a)
1) 1
8) 1
b)
1
15
1) 2
2)
5
28
3) 9
1
3
10)
9) 1
2)
1
2
3)
4)
1
24
5)
8
9
6) 4
1
2
7)
27
250
2
3
16
75
4) 2
1
2
1
20
6)
5
8
7
15
6)
1
21
7) 270
1
2
14) 5
5) 4
Fraction Review (Page 19)
a)
1) 24
2) 1
8) 30
9)
8
15
5
36
3)
1
6
10) 5
4) 9
5
8
11)
5) 1
3
10
12) 1
5
18
13)
1
4
DECIMAL REVIEW
A. INTRODUCTION TO THE DECIMAL SYSTEM
The Decimal System is another way of expressing a part of a whole number. A decimal is simply
a fraction with a denominator of 10, 100, 1 000 or 10 000 etc. The number of decimal places refers to
how many zeros will be in the denominator. Note that the number 5.62 is read as five point six two.
The first decimal place refers to tenths
The second decimal place refers to hundredths
The third decimal place refers to thousandths
3
10
31
2.31  2
100
319
2.319  2
1000
2.3  2
Similarly, six decimal places would be a fraction with a denominator of 1 000 000 (millionths). The
most common usage of decimals is in our monetary system where 100 cents (2 decimal places) make
up one dollar. For example, $2.41 is really two dollars and forty-one hundredths ( 41 ) of a dollar.
100
Examples:
1.
Change 2.30 to a fraction
Notice that 2.30 is the same as 2.3
In fact, 2.30 = 2.300 = 2.3000 etc.
2.30  2
2.
Change 0.791 to a fraction
Notice that 0.791 = .791
The zero in front of the decimal place is not
needed.
0.791 
3.
Change .003 to a fraction
Notice that the zeros in this example are
important.
.003 
4.
Simplify 0.0024000
Notice that zero at the end or zero as a whole
number (to the left of the decimal) is not
needed.
0.0024000  .0024
30
3
2
100
10
791
1000
3
1000
B. OPERATIONS WITH DECIMALS
1.
Addition:
Add $2.50 and $1.35
Place numbers in columns, so that the decimal places
are in line. Place the decimal point in the same line
for the answer.
Now, add as if adding whole numbers.
Add 1.3928 and 12.43 and .412
Add zeros to fill in columns. This will not change
the value of the number (see examples on previous
page.)
2.
1.3928
+ 12.4300
+ 0.4120
14.2348
Subtraction:
Subtract $1.30 from $5.45
Set up columns as in addition, and subtract as if
subtracting whole numbers.
Calculate 5 minus .2982
Fill in columns with zeros, as in addition.
Note the difference in wording in these two examples.
3.
$2.50
+ $1.35
$3.85
$5.45
- $1.30
$4.15
5.0000
- 0.2982
4.7018
Multiplication:
Multiply 2.12 by 4.2
At first, ignore the decimals, and multiply as if
calculating 212 x 42.
Now, add up the decimal places in both numbers and
your answer will have that total number of decimal
places.
Product of 0.0941 and .02
Multiply. Add enough zeros to show the correct
number of decimal places
2.12 (2 places)
x 4.2 (1 place)
424
8480
8.904 (3 places)
.0941
x .02
.001882
(4 places)
(2 places)
(total = 6)
Multiply
.5624 by 1000
Notice .5624 x 1000 = 562.4, so multiplying x
1000 (0 decimal places) is the same as moving
the decimal place 3 places (since 1000 has 3
zeros) to the right.
Similarly,
.58 x 10 = 5.8 5.636 x 10 000 = 56 360
.58 x .1 = .058
.58 x .001 = .00058
Division:
Divide 2.322 by .12
The divisor (.12) must be changed to a whole
number (12) by moving the decimal point 2
places to the right, in both numbers. In the
answer, place the decimal point directly above the
numbers. Note that zeros must be added to
complete the division.
.12
.
2.322
19.35
12 232.20
12
112
108
42
36
60
60
0
.5624
x 1000
562.4000
(4 places)
(0 places)
(total = 4)
or 562.4
Divide .003 into 51
Firstly, notice the difference in wording in these
two division questions. Zero must be added to the
number 51 in order to move the decimal
3 places to the right.
.003 51
17000.
3 51000.
3
21
21
0
Divide 254.25 by 1000
Notice 254.25 ÷ 1000 = .25425. So, dividing by
1000 is the same as moving the decimal 3
places to the left (since 1000 has 3 zeros).
0.25425
1000 254.250
200 0
54 25
50 00
4 250
4 000
2500
2000
5000
5000
0
Similarly, 32.52 ÷ 10 = 3.252
2.6 ÷ 10 000 = .00026
EXERCISE 1: Decimal Operations
a)
b)
c)
d)
e)
Do not use a calculator.
Change to fractions: (remember to reduce)
1) 5.8
2) 27.3400
3) 30.02
4)
0.590
Addition:
1) 2.49 + .32
2)
0.042 plus .00982
4)
5)
2.76 more than 8.4590
Subtraction:
1) 2.036 from 4.478
2)
12.258 from 13
4)
5)
19.6 decreased by 5.349
1743.2 + 2.984 + 12.35
0.1002 minus 0.05
Multiplication:
1) .21 by .04
2)
.42 x .218
4)
Product of .009 and 2.003
5)
.25 of 288
6)
9.4325 by 1000
7)
9.4325 by .001
3)
5)
3.075
7.342 and 2 and 7.65
3) 670.1 minus 589.213
3)
.75 times 132.786
Division:
1) 248 divided by 0.8
2)
15.47 divided by .091
3)
40.4 into 828.2
4)
0.0338 divided by 1.30
5)
.0025 into 1.875
6)
923.56 divided by 1000
7)
923.56 divided by .01
C. CONVERSION AND ROUNDING OFF
1.
Converting Decimals to Fractions:
When changing decimals to fractions, simply
create a fraction with 10, 100, 1000 etc. in the
denominator. The number of zeros in the
denominator is the same as the number of
decimal places.
2.
913
1000
25
1
5.25  5
5
100
4
0.913 
Converting Fractions to Decimals:
When changing a fraction to a decimal,
simply divide the denominator into the
numerator. Every time a fraction is changed
to a decimal, the division will either stop (as
4
in
= 0.8) or the division will go on forever
5

2
by repeating (as in
= .6 ). Notice that a
3
repeating decimal is shown by a dot over the
number (if only one number repeats) or as a
bar (if more than one number
repeats, as in 2 = .18 )
4
= 4 ÷ 5 = 0.8
5

2
= 2 ÷ 3 = 0.666…. = .6
3
2
= 2 ÷ 11 = .1818…. = .18
11
11
3.
Repeating Decimals
Following is a list of some repeating decimals:


1
1
1
= .3
= .16
 .142857
3
6
7

1
 .11 or .1
9

5
 .55 or .5
9
4.

2
 .22 or .2
9

7
 .77 or .7
9
2 
= .6
3

5
= .83
6

4
 .44 or .4
9

8
 .88 or .8
9
Rounding off:
If we want to divide $2.00 into 3 equal parts,
we would want our answer to be to the nearest
cent (or nearest hundredth). Since our answer
is closer to 67 cents than 66 cents, we would
round off our answer to $0.67.
When rounding to the nearest thousandth, we
want our answer to have 3 decimal places.
If the fourth decimal place has a 5 or greater,
round up. If less than a 5, do not round up.
0.666….
3 2.000
2.6549 to nearest thousandth = 2.655
2.6549 to nearest hundredth = 2.65
3.95 to nearest tenth = 4.0
EXERCISE 2: Conversion and Rounding Off
a) Change to Fractions: (Remember to reduce whenever possible)
1.
2.591
2.
25.030
3.
50.0250
4.
0.8
b) Change to Decimals: (Do not round off)
1.
6.
7
8
5
12
2.
7.
3
11
5
6
3.
8.
4
9
2
3
4.
9.
3
5
7
2
4
7
2
10.
9
5.
c) Round Off:
1.
2.864 to nearest hundredth
2.
35.9649 to nearest thousandth
3.
931.85 to nearest tenth
4.
2.091 to nearest tenth
5.
11.898 to nearest hundredth
6.
12.92 to nearest whole number
7.
11.74235 to nearest thousandth
8.
5
to nearest thousandth
9
9.
7
to nearest hundredth
12
ANSWERS
EXERCISE 1: Decimal Operations
4
17
2) 27
a) 1) 5
5
50
3) 30
1
50
4)
59
100
5) 3
3
40
b) 1) 2.81
2) .05182
3) 16.992
4) 1758.534
5) 11.219
c) 1) 2.442
2) .742
3) 80.887
4) .0502
5) 14.251
d) 1) .0084
2) .09156
3) 99.5895
4) .018027
5) 72
3) 20.5
4) .026
5) 750
6) 9432.5
e) 1) 310
7) .0094325
2) 170
6) .92356
7) 92356
EXERCISE 2: Conversion and Rounding Off
591
3
1
2) 25
3) 50
a) 1) 2
1000
100
40
b) 1) .875
c)
2) .27

3) . 4
4)
4
5
4) .6
5) .571428




6) .41 6
7) .8 3
8) . 6
9) 3.5
10) . 2
1) 2.86
2) 35.965
3) 931.9
4) 2.1
5) 11.90
6) 13
7) 11.742
8) .556
9) .58
PERCENT REVIEW
The use of percentage is another way of expressing numbers (usually fractions) in such a way as to
make comparisons between them more obvious. For instance, if you get 28 out of 40 in test A and 37
out of 50 in Test B, it may not be clear whether you have improved or not. The use of percentage will
allow this comparison, because a percent is part of 100. (i.e. a percent is a fraction with a denominator
of 100).
A.
CHANGING % TO FRACTIONS / DECIMALS
A percent means a part of 100. For example, if you get 95% on a test, your mark was 95 out of
100. A percent can be changed to a fraction or decimal by simply dividing the percentage
number by 100.
1. Changing % to Fractions
Divide by 100.
(i.e. put the % number over 100 and reduce if necessary)
50%
=
50
100
=
1
2
If a decimal appears in the fraction, multiply
the fraction by 10, 100, 1000 etc. to produce
an equivalent fraction without decimals.
24%
=
93%
24
100
26.3%
=
=
5.55%
=
=
=
93
100
6
25
=
26.3
100
263
1000
5.55
100
x
10
10
100
x 100
555
= 111
10000
2000
2. Changing % to Decimals
Simply divide by 100. (i.e. move the decimal
50%
= .50
9.23%
= .0923
4%
= .04
148%
= 1.48
point 2 places to the left)
or
.5
.
CHANGING FRACTIONS / DECIMALS TO PERCENTS
1. Changing Fractions to Percent
If you get 17 out of 20 on a test, it is convenient to change this mark to a percentage.
This means changing
17
17
?
to an equivalent fraction with 100 as denominator (i.e.
).

20
20 100
To change fractions to %, simply multiply the fraction by 100%.
17 17
1700
x 100% =
= 85%

20 20
20
1 1
 x 100% = 50%
2 2
2 2
 x 100% = 66. 6 %
3 3
1
19 19
x 100% = 47 % or 47.5%

40 40
2
Note: The mathematical wording for changing a fraction (
17
) to a percent would normally be:
20
17 is what % (out) of 20?
or
What % is 17 (out) of 20?
The word “out” is usually not included.
e.g. 19 is what % of 75?
What % is 7 of 5?
1
19 19
x 100% = 25 %

75 75
3
7 7
 x 100% = 140%
5 5
2. Changing Decimals to Percents
To change decimals to percents, simply
multiply by 100% (i.e. move the decimal
point 2 places to the right.)
C.
USING PERCENTS
.29
= .29
x
100%
=
29%
.156
= .156
x
100%
=
15.6%
1.3
= 1.3
x
100%
=
130%
When percents are used in calculations, they are first converted to either fractions or decimals.
Usually it is more convenient to change % to decimals.
1. Multiplying With Percents
If a test mark was 50% and it was out
of 40 total marks, what was the test
test score?
40 ( total marks )
50%
=
50%
x
40
50%
=
score?
So
=
.5
x
40 = 20
20 marks
40
50% (out) of 40 is what number?
What number is 50% (out) of 40?
\
To find the test score, or the part, we multiply the % by the total.
e.g. 85% of 25 is what number?
85%
x 25
=
.85 x 25 = 21.25
What number is 30% of 45.37?
30%
x
=
.3 x 45.37 = 13.611
45.37
2. Dividing with Percents
If a test mark was 50% and you
50%
20 marks
total ?
=
received a score of 20 marks,
what was the test out of?
20 ÷ 50%
or 20
1
2
÷
= 20
÷
.5
=
40
=
x
2
1
=
40
20
50% (out) of what number is 20?
So, 50%
20 is 50% (out) of what number?
=
20
40
total marks
To find the total marks, we divide by the %.
e.g. 40% of what number is 25?
25
÷ 40%
=
25
÷
.40
=
62.5
18 is 75% of what number?
18
÷ 75%
=
18
÷
.75
=
24
D.
SUMMARY AND EXERCISE
1. Three types of Percent Problems
In summary, there are three things
that we can do with percent. We will
use the example on the right side
of the page to summarize.
1.
Finding % or what % of 40 is 20?
2.
Finding the Part or 50% of 40 is what
number?
3.
Finding the total or 50% of what
number is 20?
50%
=
20
40
=
50%
x
20
÷
20 (part )
40 total
20
40
x
100%
=
50%
40
=
.5
x
40 = 20
50%
=
20
÷
.5 = 40
d)
45.3%
===============================================
2. EXERCISE: PERCENT PROBLEMS
1.
Change to Fractions
a)
97%
b)
82%
c)
150%
e)
9.25%
f)
40%
g)
5
9.37%
c)
2%
d)
243.9%
c)
18
75
d)
1
12
g)
0.865
h)
2.37
2.
1
%
2
Change to Decimals
a)
42%
e)
0.95%
3.
b)
Change to %
a)
e)
i)
19
20
5
9
.0092
b)
f)
j)
2
3
38
40
7
4
4.
Finding %
a)
What % of 72 is 18?
c)
What % of 30 is 18.5?
5.
b)
16 is what % of 80?
b)
What number is 16.5% of 30.2?
b)
18 is 55% of what number?
Finding the Part
a)
40% of 18 is what number?
c)
65% of 15 is what?
6.
Finding the Total
a)
40% of what number is 12?
c)
120 is 150% of what number?
7.
Percent Problems Combined
a)
What % of 25 is 5?
b)
70% of 15 is what number?
c)
85 is 20% of what number?
d)
90 is what % of 55?
e)
30% of what number is 80?
f)
What number is 42% of 50?
ANSWERS
97
100
b)
41
50
c) 1
1
2
1.
a)
2.
a) .42
b) .0937
3.
a) 95%
b) 66. 6 % or 66
g) 86.5%
h) 237%
i) .92%
4.
a) 25%
b) 20%
c) 61. 6 %
5.
a) 7.2
b) 4.983
c) 9.75
6.
a) 30
b) 32.72
c) 80
7.
a) 20%
b) 10.5
c) 425
c) .02
2
%
3
d)
453
1000
e)
37
400
d) 2.439
e) .0095
c) 24%
d) 8. 3 %
f)
2
5
e) 55. 5 %
j) 175%
d) 163. 63 %
e) 266. 6
f) 21
g)
11
200
f) 95%
EXPONENTS AND ROOTS
A. EXPONENTS
In math, many symbols have been developed to simplify certain types of number expressions. One of
these symbols is the “exponent”.
Rule: An exponent indicates how many times a base number is used as a factor.
Example:
104 = 10 x 10 x 10 x 10 or (10) (10) (10) (10)
35 = 3 x 3 x 3 x 3 x 3
x2 = (x) (x)
The exponent is written smaller and is placed above the base number (the number to be multiplied).
The first example can be read “ten exponent four” or “ten to the power of four”.
Second and third power have special names: second power is usually called “squared”, and third power
is usually called “cubed”.
Example:
52 is “five squared”
53 is “five cubed”
A simple way to work out exponents is to write the base digit the same number of times as the value of
the exponent, and put a multiplication sign between each digit.
Example:
45 = 4 x 4 x 4 x 4 x 4 (the exponent is 5, so write 4 five times)
= 1024
PRACTICE A:
Write as an exponent or power (also called “exponential notation”):
1.) 5  5  5  5 =
2.) 2  2  2  2  2 =
3.) 10  10 =
4.) 6  6  6 =
Evaluate the following:
5.) 24 =
6.) 53 =
7.) 34 =
8.) 25 =
9.) 102 =
10.) 17 =
B. SQUARE ROOTS
Rule: Taking the square root of a number is the reverse of squaring the number.
The symbol for square root is
9 (the square root of 9)
Example:
= 3 because (3)2
= 9
144 (the square root of 144) = 12 because (12)2 = 144
PRACTICE B:
Evaluate the following:
1.)
4 =
2.)
1 =
3.)
121 =
4.)
81 =
5.)
100 =
6.)
10000 =
ANSWERS
Practice A (exponents):
1.) 54
2.) 25
3.) 102
4.) 63
5.) 2 x 2 x 2 x 2 = 16
6.) 5 x 5 x 5 = 125
7.) 3 x 3 x 3 x 3 = 81
8.) 2 x 2 x 2 x 2 x 2 = 32
9.) 10 x 10 = 100
10.) 1 x 1 x 1 x 1 x 1 x 1 x 1 = 1
Practice B (square roots)
1.) 2
6.) 100
2.) 1
3.) 11
4.) 9
5.) 10
BASIC ALGEBRA REVIEW
SIGNED NUMBERS
In the REAL NUMBER SYSTEM, numbers can be either positive or negative.
Positive 5 can be written as: +5, or (+5), or just 5.
Negative 5 can be written as: –5, or (–5).
ADDING AND SUBTRACTING SIGNED NUMBERS
When adding numbers of the same sign, simply put the numbers together and carry the sign.
(–4) + (–2) = –6
(–9) – 5 = –14
Note: This reads –9 combined with –5
( + 4) + ( + 2) = +6
– 4 – 2 = –6
When adding numbers of different signs, take the difference between the two numbers; carry the sign of the number
with the largest absolute value.
For example: –9 + 2. The difference is 7. Take the sign of the 9. Answer = (–7)
4–2=2
NOTE: These two examples are the
same, just switched around.
–2 + 4 = 2
–7 + 3 = –4
12 + (–3) = 9
–4 + 4 = 0
These last two examples are important, as we use
this concept to isolate variables on one side of the =
sign when doing algebra.
4 + (–4) = 0
Practice:
1.
2.
3.
4.
5.
–9 + 2 =
–3 + (–5) =
–6 + (–8) =
14 – 14 =
–11 + 11 =
Answers: 1. –72. –8
3. –14
4. 0
5. 0
ALGEBRA
In basic algebra, letters represent numbers. It is important to collect same letters together when possible.
For example:
3x + 2x + 6x
should be written as 11x (there are 11 x’s altogether)
5y – 3y
should be written as 2y
1x
is usually written as x (the 1 is assumed)
If you are given the value (number) for the letter, you can substitute that value for the letters to answer the equation.
For example:
Solve
3x  2
when
x4
Simply substitute 4 for the x and solve.
3x + 2
3 (4) + 2
12 + 2
= 14
An equation is solved when the unknown letter is isolated on one side of the equal sign. When isolating x, the
equation must be kept balanced. To maintain balance, you must always do the same thing to both sides of the
equation.
For example:
x + 3 = 10
3 is being added to x , so do the opposite to both sides and subtract 3 from both sides to
isolate x . On the left side, 3 - 3 is 0, leaving just the x on the left.
x + 3 = 10
–3
–3
x = 7
Practice:
a) Solve
x–6 = 4
6 is being subtracted from x so add 6 to both sides to isolate x . Again,
–6 +6 = 0, leaving just x on the left.
x–6 = 4
x–6+6 = 4+6
x = 10
b) Solve
4x = 20
x is being multiplied by 4 so the opposite of multiply is divide (by 4) on both sides.
4x = 20
4 x = 20
4
4
x = 5
c) Solve
y
= 5
6
y is being divided by 6 so the opposite of divide by 6 is multiply by 6 on both sides.
y
= 5
6
y
(6) = 5(6)
6
y = 30
d) Solve 4x + 3x + 2
= 5+4
Collect like terms first!
7x + 2 = 9
Now isolate the x by subtracting 2 from both sides
7x + 2 = 9
7x + 2 – 2 = 9 – 2
7x = 7
Divide by the number of x ’s to isolate the x on the left
7x
7
=
7
7
x = 1
Algebra Practice
1.
2.
3.
4.
5.
6.
7.
8.
3x + 9x – 8x =
7y – 3y + 2y =
Z – 3 = 25
3x + 4 = 13 (isolate 3x first)
5x + 6 = 31 (isolate 5x first)
2x + 4 , when x = 3
M – 2s = 40, when M = 4s
N  5 = 60
Answers:
1.
2.
3.
4.
5.
6.
7.
8.
4x
6y
Z = 28
X=3
X=5
10
M = 20
N = 300
BASIC GEOMETRY
PERIMETER OF POLYGONS
A polygon is a geometric figure with 3 or more sides. The perimeter of a polygon is the distance around the
outside of the figure, or the sum of the length of each of its sides. Sometimes formulae are used in
calculating the perimeter to make things easier. The most common formulae used are as follows:
Perimeter of a Rectangle or a Parallelogram: P = 2 • (l + w) or P = 2 • l + 2 • w
Perimeter of a Square:
P=4•s
Example 1: Find the perimeter of a rectangle that is 6 mm by 9 mm
P = 6 mm + 6 mm + 9 mm + 9mm
= 30 mm
or P
or P
= 2 • (l + w)
= 2 • (6 mm + 9 mm)
= 2 • (15 mm)
= 30 mm
= 2l + 2w
= 2 (9 mm) + 2 (6 mm)
= 18 mm + 12 mm
= 30 mm
Example 2: Find the perimeter of a square with a side that is 10 cm long
P = 10 cm + 10 cm + 10 cm + 10 cm
= 40 cm
or
P
= 4s
= 4 (10 cm)
= 40 cm
Example 3: Find the perimeter of a parallelogram that has a length of 12m and a width of 5m
P = 2 (l + w)
= 2 (12 m + 5 m)
= 2 (17 m)
= 34 m
or
P
or
P
= 2/ + 2w
= 2 (12 m) + 2 (5 m)
= 24 m + 10 m
= 34 m
= 5 m + 12 m + 5 m + 12 m
= 34 m
Example 4: Find the perimeter of a triangle that has the sides 3 mm, 6.5 mm, and 8.6 mm
P
= 3 mm + 8.6 mm + 6.5 mm
= 18.1 mm
PRACTICE
Find the perimeter of the following shapes:
ANSWERS
a) 30 m
g) 67.2 ft
m) 40.51 cm
b)
h)
n)
17 cm
32.4 m
510 m
c)
i)
20 m
48 ft
d)
j)
28 mm
47 ft
e)
k)
54.8 in
21 km
f)
l)
44 cm
66.62 ft
CIRCLE GEOMETRY – CIRCUMFERENCE
Circumference is the name for the perimeter (or distance around the outside) of a circle.
In this circle, the centre is Z. A, B, and C are points on the circle.
Radius: The distance from the centre of the circle to any point on the circle is
called the radius (r). (ZA is a radius. ZB and ZC are too).
Diameter: The distance from any point on the circle, passing through the
centrepoint and continuing on to the outer edge of the circle (d). (AB is the
diameter of the circle to the right.)
To find the circumference (or perimeter) of the circle, use one of the following
formulae:
(1) C = π d OR
(2) C = 2π r
22
π is called pi and is about 3.14 or
7
Example A: If the circle has a radius of 5 cm, then
(1)
C =π d
(2)
C = 2π r
The diameter would be twice the radius
The radius is 5 cm
(or 5 cm x 2 = 10 cm)
So
C = (3.14) (10 cm)
C = (2) (3.14) (5)
C = 31.4 cm.
C = 31.4 cm
Both formulae work equally well. You may choose either one.
Now let's practice:
1. A circle has a diameter of 20 m. What is the circumference?
2. A circle has a radius of 7 km. What is the circumference?
3. Find the circumference for the following circles:
a)
1)
3)
radius(r) = 14 cm
22
(use π =
)
7
b)
diameter (d) = 60 mm
c)
radius (r) = 15 m
ANSWERS
C = π d ; C = (3.14)(20); C = 62.8 m
2) C = 2π r; C = (2)(3.14)(7); C = 43.96 km
b) C = π d ; C = 3.14(60) = 188.4 mm
 22 
a) C = 2π r; C = (2)   (14) = 88 cm
 7 
c) C = 2 π r ; C = (2)(3.14)(15); C = 94.2 m
AREA OF POLYGONS
The area of a polygon is the number of squares (of a particular unit) that it takes to cover the surface of the
polygon. Formulae are used to calculate the area. The most common formulae are
1
bh
Area of a square = s2
Area of a triangle = b  h or
2
2
Area of a rectangle = l  w
Area of a parallelogram = b  h
1
Area of a trapezoid =
h (a + b)
2
Example 1: Find the area of a triangle which has a base of 10 mm and a height of 9 mm.
1
bxh
2
1
A=
(10 mm x 9 mm)
2
1
A=
(90 mm2)
2
A = 45 mm2
bxh
2
10 mm x 9 mm
A
2
90 mm 2
A
2
A  45 mm 2
A
A=
or
Example 2: Find the area of a square that has a side with a length of 6 cm.
A = s2
A = (6 cm) 2
A = 36 cm2
Example 3: Find the area of a parallelogram that has a base of 21 cm and a height of 13 cm.
A=bxh
A = (21 cm) x (13 cm)
A = 273 cm2
Example 4: Find the area of the trapezoid below
1
h (a + b)
2
1
A=
(29 m) (66 m + 63 m)
2
1
A=
(29 m) (129 m)
2
A = 1870.5 m 2
A=
*Notice that units in the answers are units2 (squared)
PRACTICE
Find the area of the following shapes:
a)
b)
c)
5 cm
12.6 m
10 cm
5 cm
3 cm
d)
e)
f)
14 mm
1 cm
12.2 km
11 cm
24 mm
g)
8.6 km
h)
2.3 ft
i)
12.3 yd
12 ft
10 ft
4 ft
Note: There are no diagrams for j) to m).
j)
A square 35 ft on a side.
k)
A parallelogram with height of 14 in. and base 23 in.
l)
A rectangle with length of 8.8 m and width of 4.2 m.
m)
A triangle with height of 9 km and base of 5.2 km.
ANSWERS
2
b)
25 cm
2
c)
158.76 m2
d)
336 mm2
e)
11 cm2
44 ft2
i)
151.29 yd2
j)
1225 ft2
a)
15 cm
f)
52.46 km2
g)
5.29 ft2
h)
k)
322 in2
l)
36.96 m2
m) 23.4 km2
CIRCLE GEOMETRY – AREA
To find the area of a circle, use the formula
A =π r 2
Where
A = area of the circle: π = pi  3.14 or
r
22
7
= the radius of the circle
so
A=πr2
A = 3.14 x (10)2
A = 3.14 x 100
A = 314 cm2
*notice: answers are units2
r = the radius of the circle = 10 cm
Now let’s practice:
A:
Find the circumference and the area of the following circles:
(1)
the radius = 4 km
(2)
the diameter = 10 m
B:
Find the circumference and the area of the following circles:
(1)
the radius = 14 in.
(2)
the diameter = 20 mm
ANSWERS
A: 1)
2)
B: 3)
4)
C = 2π r; C = 2 (3.14)(4): C = 25.12 km
C= π d ; C = (3.14)(10); C = 31.4m
C = 2 π r; C = 2( 3.14)( 14) = 87.92 in
C= π d ; C = (3.14)(20) = 62.8 mm
A=πr2
A=πr2
A = π r2
A=πr2
A = (3.14)( 4)2 = 50.24 km2
A = (3.14)(5)2 = 78.5 m2
A = (3.14)(14)2 = 615.44 in 2
A = (3.14)(10)2 = 314 mm 2
LINEAR MEASURES : THE METRIC SYSTEM
The metric system is used in most countries of the world, and the United States is now making
greater use of it as well. The metric system does not use inches, feet, pounds, and so on, although
units for time and electricity are the same as those you use now.
An advantage of the metric system is that it is easier to convert from one unit to another. That is
because the metric system is based on the number 10.
The basic unit of length is the metre. It is just over a yard. In fact,
1 metre ≈ 1.1yd.
(comparative sizes are shown)
1 Metre
1 Yard
The other units of length are multiples of the length of a metre:
10 times a metre, 100 times a metre, 1000 times a metre, and so on, or fractions of a metre:
1
1
1
of a metre,
of a metre,
of a metre, and so on.
10
100
1000
Metric Units of Length
1 kilometre (km) = 1000 metres (m)
1 hectometre (hm) = 100 metres (m)
1 dekametre
= 10 metres (m)
(dam)
dam and dm are not used
1 metre (m)
much.
1 decimetre (dm) =
1 centimetre (cm) =
1 millimetre (mm) =
1
metre (m)
10
1
metre (m)
100
1
metre (m)
10
You should memorize these names and abbreviations. Think of kilo- for 1000, hecto- for 100, and so
on. We will use these prefixes when considering units of area, capacity, and mass (weight).
THINKING METRIC
To familiarize yourself with metric units, consider the following.
1 kilometre (1000 metres)
is slightly more than 12 mile (0.6 mi).
1 metre
is just over a yard (1.1 yd).
1 centimetre (0.01 metre)
is a little more than the width of a paper-clip
(about 0.4 inch).
1 cm
1 inch is about 2.54 centimetres
1 millimetre is about the
width of a dime
The millimetre (mm) is used to measure small distances,
especially in industry.
2 mm
3 mm
The centimetre (cm) is used for body dimensions and
clothing sizes, mostly in places where inches were
previously used.
120 cm
(47.2 in.)
3 ft 11in.
53 cm
39 cm
(20.9 in.)
(15.3 in.)
The metre (m) is used to measure larger objects (for
example, the height of a building) and for shorter distances
(for example, the length of a rug)
25
m
(82 ft)
3.7 m
(12 ft)
2.8 m
(9 ft)
The kilometre (km) is used to measure longer distances, mostly in situations in which miles were
previously used.
MENTAL CONVERSION AMONG METRIC UNITS
When you change from one unit to another you can move only the decimal point, because the metric
system is based on 10. Look at the table below:
Units
1000
km
100
hm
10
dam
1
m
0.1
dm
0.01
cm
0.001
mm
Example:
Complete: 8.42 mm = _______ cm
Think:
To go from mm to cm will mean I will have fewer cm than mm because cm are larger than
mm. So I move the decimal point one place to the left.
8.42 mm
0.842
so, 8.42 mm = 0.842 cm
Example:
Complete: 1.886 km = _______ cm
Think:
To go from km to cm means that there will be many more cm than there were km because cm
are smaller than km. So I move the decimal place to the right 5 places.
1.886 km
1.88600
so, 1.886 km = 188 600.0 cm
Example:
Complete: 1 m = _______ cm
Think:
To go from m to cm … m are bigger, and cm smaller… so, there will be more cm than I
started with. I can move the decimal place to the right 2 places.
1 m = 1.00 m 1.00 cm
so, 1m = 100 cm
Make metric conversions mentally as much as possible.
The most commonly used units of metric measurement are:
km
m
cm
mm
PRACTICE : CONVERTING BETWEEN METRIC UNITS
Complete. Do as much as possible mentally. Avoid using a calculator!
1. a) 1 km = _____ m
4. a) 1 dm = _____ m
5. a) 1 cm = _____ m
b) 1 m = _____ km
b) 1 m = _____ dm
b) 1 m = _____ cm
6. a) 1 mm = _____ m
b) 1 m = _____ mm
7. 6.7 km = _____ m
8. 9 km = _____ m
9. 98 cm = _____ m
10. 0.233 cm = _____ m
11. 8921 m = _____ km
12. 6770 m = _____ km
13. 56.66 m = _____ km
14. 5.666 m = _____ km
15. 5666 m = _____ cm
ANSWERS
1. a)
1. b)
6. a)
6. b)
7.
10.
13.
1000
0.001
0.001
1000
6700
0.00233
0.05666
4. a)
4. b)
0.1
10
5. a)
5. b)
0.01
100
8.
11.
14.
9 000
8.921
0.005666
9.
12.
15.
0.98
6.77
566 600
USING IMPERIAL AND METRIC RULERS
We have probably all heard the old saying: “Measure twice; cut once.” Using a ruler efficiently to measure
materials and construction is an essential aspect of carpentry.
Many trades use both imperial and metric systems of measurement, so you need to know how to read and
use both types of rulers and tapes. Often both systems are on the same ruler / tape. This is convenient, but
beware of reading the numbers for one system and using the units of measurement for the other.
An Imperial ruler, usually 1 foot (ft or ′) long, is divided into inches (″) and parts of inches. An Imperial
tape is similarly divided, but is much longer. Many Imperial measures of length divide inches into halves,
quarters, eighths, sixteenths - and even thirty-secondths.
Metric measures are divided into multiples of 10, starting with millimetres (mms) and centimetres (cms). A
1 metre (m) rule is divided into 100 centimetres or 1 000 millimetres.
Here is a quick review of past learning, plus an exercise, using that knowledge, with a ruler and a tape.
Practice Exercise
1
2
3
How many of these fractions of an inch are there in one inch?
a) quarters
b) sixteenths
c) halves
d) eighths
Find the answers to how many
a) cms in 3 m b) mms in 55 cms
d) cms in 350 mms
c) mms in 2 m 5 cms
In the boxes below, label the measurements shown on the ruler. Write both numbers and units of
either inches (ins or ″) or millimetres / centimetres (mms / cms), whichever is appropriate.
You will notice that sometimes the arrow is not exactly on the line of measurement. It is as close
as possible.
a)
e)
b)
f)
c)
d)
g)
h)
Practice Exercise Answers
1 a) 4
b) 16
c) 2
d) 8
2 a) 300 cms
b) 550 mms
c) 2050 mms
d) 35 cms
3 a) 1
b) 2
c)
d)
e) = 2.6 cms
f)
=2
7 cms
or 26 mms
4
6
g) = 13.2 cms
h) = 18.9 cms
or 132 mms
or 189 mms
Exercise
1 Label this tape with the measurement points below. Use an arrow and the letter of the measurement
point to show your accurate reading.
inches
mms and cms
a)
3
f) 16.6 cms
b) 11 cms
c)
d)
e) 4
g) 21.5 cms
h) 7
i) 8
j) 18 cm 5 mm
Exercise Answers:
1)
Arrows are as close as possible.
a)
e)
d)
c)
b)
i)
h)
f)
j)
g)