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ON THE JOB MATHEMATICS
FOR
CONSTRUCTION TRADES
Reference Guide
By Mary Fudge
What Are Fractions?
A fraction is a part of something. A penny is a fraction of a dollar. It is one of the 100 equal
1
parts of a dollar or
(one hundredth) of a dollar. An inch is a fraction of a foot. It is one of the 12
100
1
equal parts of a foot or
(one twelfth) of a foot.
12
The two numbers in a fraction are called the:
numerator - which tell how many parts you have
denominator - which tells how many parts in the whole
Forms of Fractions
Proper fraction - The top number is less than the bottom number.
EXAMPLES:
1
4
3
8
5
16
A proper fraction is less than all the parts the whole is divided into.
The value of a proper fraction is always less than one.
Improper fraction - The top number is equal to or larger than the bottom number.
EXAMPLES:
5
4
11
8
19
16
8
, or it is more than the
8
total parts in the whole. The value of an improper fraction is either equal to one or more than
one.
An improper fraction is all the parts that a whole is divided into such as
2
Mixed number - A whole number is written next to a proper fraction.
EXAMPLES:
1
1
4
3
5
8
2
7
16
Reducing Fractions
The coin we call a quarter stands for 25 pennies out of the total of 100 pennies in a dollar. You
5
25
could call 25
of a dollar, or thinking of the 5 nickels in 25,
of a dollar. The easiest way is to say
20
100
5
1
1
25
that 25 is
of a dollar.
is the reduced form of
and
. Reducing a fraction means writing it an
20
4
4
100
easier way B with smaller numbers.
Study the following examples to see how fractions are reduced.
EXAMPLE 1.
Reduce
12
16
Step 1.Find a number that goes evenly into the top
12  4 3

16  4 4
and bottom numbers of the fraction. 4 goes
evenly into both 12 and 16.
Step 2.
EXAMPLE 2.
Step 1.
Check to see whether another number goes
evenly into both the top and bottom numbers
of the fraction. Since no other number goes
evenly into both 3 and 4, the fraction is reduced
as far as it will go.
Reduce
24
32
Find a number that goes evenly into the top
and bottom numbers of the fraction. 4 goes
evenly into both 24 and 32.
Step 2.
Check to see whether another number goes
evenly into both the top and bottom numbers
of the fraction. 2 goes evenly into both 6
and 8.
24  4 6

32  4 8
62 3

82 4
3
Step 3.
3
4
Check to see whether another number goes
evenly into both the top and bottom numbers
of the fraction. In this case, the fraction is
reduced as far as it will go.
When you reduce a fraction, the value does not change. A reduced fraction is equal to the original
fraction.
When you have reduced a fraction as much as possible, the fraction is then in lowest terms.
Raising Fractions to Higher Terms
An important skill in addition and subtraction of fractions is raising a fraction to higher terms.
This is the opposite of reducing a fraction to lowest terms.
EXAMPLE:
Raise
3
to 24th.
8
3
8 24
Step 1.
Divide the old bottom number into the new one.
Step 2.
Multiply the answer (3) by the old top number (3).
3 3 9

8  3 24
Check.
Reduce the new fraction to see if you get the
93 3

24  3 8
original fraction.
Changing Improper Fractions to Whole or Mixed Numbers
An improper fraction is a fraction with a top number that is as big or bigger than the bottom
number. An improper fraction is equal to or larger than one whole.
9
into a mixed number by dividing the bottom
8
number into the top number and writing the remainder, if any, over the original bottom number.
You can change any improper fraction, such as
4
EXAMPLE:
Step 1.
Step 2.
Change
24
to a mixed number.
16
Divide the bottom into the top.
Write the remainder as a fraction over
the original bottom number.
Step 3.
Reduce the remaining fraction.
The answer becomes 1
1
16 24
16
8
8
1
16
88 1

16  8 2
1
2
Changing Mixed Numbers to Improper Fractions
Study the following examples to see how mixed numbers are changed to improper fractions.
1
EXAMPLE 1.
Change 3 to an improper fraction.
8
Step 1.
Multiply the bottom number by the whole
number.
8  3  24
Step 2.
Add the result to the top number.
Step 3.
Place the total over the bottom number.
24  1  25
25
8
EXAMPLE 2.
Change 2
3
to an improper fraction.
4
Step 1.
2 4  8
Step 2.
8  3  11
Step 3.
Place 11 over 4.
Answer:
2
11
4
3 11

4 4
5
Adding Fractions with the Same Bottom Numbers
To add fractions with the same bottom numbers, add the top numbers, and put the total over the
bottom number.
EXAMPLE:
Step 1.
3
8
Add the top numbers. 3  2  5
+
Step 2.Place the total (5) over the bottom number:
2
8
5
8
If the total of an addition problem is an improper fraction, change the total to a mixed number.
Step 1.
Add the top numbers. 5  7  12
Add the whole numbers
Step 2.
5
8
7
1
8
3
Place the total over the bottom
number:
12
8
12
4
4
1
4  4 1  5  5
8
8
8
2
4
Step 3.
Change the improper fraction to a mixed
12
4
1
number:
8
8
and add the whole number to
4
4
the whole number column. 4  1  5
8
8
Step 4.
Reduce the remainder:
5
4
1
5
8
2
6
Finding a Common Denominator
Here are some ways of finding a common denominator when the largest denominator in an
addition problem doesn’t work:
1.
2.
Multiply the denominators together.
Go through the multiplication table of the largest denominator.
EXAMPLE:
Step 1.
Multiply the denominators.
2
3
3
+
8
5
8
3  8  24
24 is the LCD.
Step 2.
2 8 16
 
3 8 24
Raise each fraction to 24ths.
+
Step 3.
Step 4.
3 3 9
 
8 3 24
_________________
25
1
1
Change the answer to a mixed number if needed.
24
24
Add
Adding Fractions with Different Bottom Numbers
If the fractions in an addition problem do not have the same bottom numbers (denominators),
you must rewrite the problem so that all of the fractions have the same bottom number (called a
common denominator). This will mean raising at least one of the fractions to higher terms.
A common denominator is a number that can be divided evenly by all of the denominators in the
problem. The smallest number that can be divided evenly by all of the denominators in the problem is
called the lowest common denominator or LCD. Sometimes, the largest denominator in the problem
will work as the LCD.
7
EXAMPLE:
Step 1.
Since 4 divides evenly into 8, 8 is the LCD.
Step 2.
Raise
Step 3.
Add the new fractions
1 2

4 8
3 3

8 8
5
8
1
to 8ths.
4
Subtracting Fractions with the Same Bottom Numbers
To subtract fractions with the same bottom numbers, subtract the top numbers and put the total
over the bottom number.
EXAMPLE:
Step 1.
Subtract the top numbers. 7  3  4
Step 2.
Place the answer over the bottom number:
Step 3.
Reduce the answer:
44 1

16  4 4
7
16
3

16
4 1

16 4
4
16
Subtracting Fractions with Different Bottom Numbers
If the bottom numbers in a subtraction problem are different, find the LCD and raise the fractions to
higher terms.
EXAMPLE:
Step 1.
The LCD is 8  3  24.
Step 2.
Raise each fraction to 24ths.
Step 3.
Subtract 21  16  5.
7 21

8 24
2 16
 
3 24
??
8
Borrowing and Subtracting Fractions
In order to have a fraction to subtract from, you sometimes have to borrow from a whole number.
EXAMPLE 1
84
3
4
Since there is nothing to subtract the
Step 1.
3
from, you have to borrow.
4
Borrow 1 from the 8 and change the 1 to 4ths because
4 is the LCD. 1 
Step 2.
4
4
4
4
3
3
4  4
4
4
1
3
4
87
Subtract the top numbers and the whole numbers.
EXAMPLE 2.
3
7
8 2
8
8
Since you cannot take
Step 1.
3
7
from you have to borrow.
8
8
Borrow 1 from 8 and change the 1 to 8th because
8
8
8
3 8 3 11
 
Add the to .
8 8 8 8 8
8 is the LCD. 1 
Step 2.
Step 3.
Subtract the top numbers and the whole numbers
and reduce to lowest terms.
3
8 3
8 7 
8
8 8
7
2
8
8 3
11
7  7
8 8
8
7
7
2
2
8
8
5
4
1
5
8
2
9
Multiplying Fractions
To multiply fractions, multiply the top numbers together and the bottom numbers together.
EXAMPLE:
3 1  3
Step 1.
Multiply the top numbers.
Step 2.
Multiply the bottom numbers. 8  4  32
3 1 3
 
8 4 32
With three fractions, multiply the top numbers of the first two fractions together. Then multiply
that answer by the third top number. Do the same for the bottom numbers.
Canceling and Multiplying Fractions
Canceling is a shortcut in multiplication of fractions. It is just like reducing. It means dividing a
top and a bottom number by a number that goes evenly into both before actually multiplying. You don=t
have to cancel to get the right answer, but it makes the multiplication easier.
7 8

EXAMPLE:
16 28
Step 1.
Cancel 7 and 28 by 7.
71
8


16 28 4
7  7  1and 28  7  4 .
Cross out the 7 and the 28.
Step 2.
Cancel 8 and 16 by 8.
8  8  1 and 16  8  2 .
Cross out the 8 and the 16.
Step 3.
Multiply across by the new numbers.
1  1  1 and 2  4  8 .
71
81
1


16 2 28 4 8
10
Multiplying Fractions by Whole Numbers
Any whole number can be written as a fraction with a bottom number of 1. For example, 6 is the
6
same as .
1
3
6
EXAMPLE:
4
6
6 3 18
2
1
 
4 4
Step 1.
Write 6 as a fraction. 6 
1
1 4 4
4
2
Step 2.
Cancel 6 and 4 by 2.
Step 3.
Multiply across by the new numbers.
Step 4.
Change the improper fraction to a mixed number.
Step 5.
Reduce fraction to lowest terms.
Multiplying with Mixed Numbers
To multiply with mixed numbers, change every mixed number to an improper fraction.
1 3
1 
EXAMPLE:
4 10
Step 2.
1
1 5
to an improper fraction. 1 
4
4 4
1
5
3
3


4 10 2 8
Cancel 5 and 10 by 5.
Step 3.
Multiply across.
Step 1.
Change 1
Dividing Fractions by Fractions
To calculate the answer to any division of fractions problem, there are two rules to remember:
1)
Invert the fraction to the right of the division sign (the divisor). That is, turn the
fraction upside down by writing the top number in the bottom position and the
bottom number at the top.
2)
Change the division sign to a multiplication sign and follow the rules of multiplication.
In other words, the rules for multiplication and division of fractions are exactly the same as
soon as you invert the fraction to the right of the division sign.
11
EXAMPLE:
Step 1.
5 1

8 4
Invert the fraction on the right
1
4
to and
4
1
5 41 5
1
  2
82 1 2
2
change the  sign to  .
Step 2.
Cancel 4 and 8 by 4.
Step 3.
Multiply across.
Step 4.
Change the improper fraction to a mixed number.
Dividing Whole Numbers by Fractions
When dividing whole numbers by fractions, write the whole number as a fraction by putting it
over 1
1
( 3  1 ), invert the fraction to the right of the division sign and multiply:
8
EXAMPLE:
3
6
16
3
1
Step 1.
Write 3 as a fraction:
Step 2.
Invert
Step 3.
Cancel 3 and 6 by 3.
Step 4.
Multiply across.
Step 5.
Change the improper fraction to a mixed number.
6
and change the  sign to  .
16
31 15 16


8
1 62
2
12
Dividing Fractions by Whole Numbers
When dividing a fraction by a whole number, first write the whole number as a fraction over 1.
Then invert that fraction and multiply:
EXAMPLE:
3
4
4
Step 1.
Write 4 as a fraction:
Step 2.
Invert the fraction
Step 3.
Multiply across.
3 4

4 1
4
1
4
1
to and change the  sign to  .
1
4
3 1 3
 
4 4 16
Dividing with Mixed Numbers
To divide with mixed numbers, first change every mixed number to an improper fraction. Also, be sure
to write whole numbers as fractions over 1. Then invert the fraction to the right of the division sign and
finish the problem.
EXAMPLE:
1
5
2 2
4
8
1
to an improper fraction:
4
5
Change 2 to an improper fraction:
8
Step 1.
Change 2
Step 2.
Invert the fraction
9
4
21
8
9 21

4 8
21
8
to
and change the  sign to  .
8
21
Step 3.
Cancel 4 and 8 by 4 and the 9 and 21 by 3.
Step 4.
Multiply across.
93 82
6


41 217 7
13
What Are Decimals?
Decimals are a type of fraction that you probably work with every day of your life.
Decimal fractions can go from tenths to millionths when they are used for certain types of exact
measurement. Since decimal fractions don=t have bottom numbers, you read them by noting the number
of places they take up to the right of the decimal point.
Decimals are arranged by 10's only B 10ths, 100ths, 1,000ths, 10,000ths, 100,000ths,
1,000,000ths.
Mixed decimals are written with whole numbers to the left of the decimal point and decimal
fractions to the right.
Mixed Decimals
Mixed Number
3.6
3
14.09
14
7.003
7
252.0024
252
6
10
Read
Three and six tenths
9
100
3
1000
24
10,000
Fourteen and nine hundredths
Seven and 3 thousandths
Two hundred fifty two and
twenty
four ten-thousandths
Reading Decimals
First, find the decimal point. Any number to the left of the decimal point is read as a whole
number. Any number to the right of the decimal point is a decimal fraction. The number of places to
the right of the decimal point will tell you the decimal name of that fraction. Remember to use the word
and to separate the whole number from the decimal fraction.
14
EXAMPLE 1.
Read this decimal: .0036
Step 1.
Read the number: thirty-six.
Step 2.
Count the number of places: .0036 has four places. Four places
means ten-thousandths.
Step 3.
Read .0036 as thirty-six ten-thousandths.
EXAMPLE 2.
Read this mixed decimal: 9.004
Step 1.
Read the whole number: nine
Step 2.
Read the decimal fraction number: four
Step 3. Count the number of places the decimal fraction is to the right of the
decimal point: .004 has three places. Three places means thousandths.
Step 4.
Read the whole number first and then the decimal fraction, inserting the word and
between them. Thus, 9.004 is read: nine and four thousandths.
Writing Decimals
When writing decimals from words, be sure that you have the correct number of places, and
watch for the word and.
EXAMPLE 1.
Write six hundredths.
Step 1.
Write the number 6.
Step 2.
Hundredths means two places. Since 6 uses only one place, we hold the first
place to the right of the decimal point with a zero.
ANSWER:
.06
15
EXAMPLE 2.
Write three hundred nine millionths.
Step 1.
Write the number 309.
Step 2.
Millionths means six places. Since 309 uses only three places, we hold the first
three places to the right of the decimal point with zeros.
ANSWER:
.000309
Changing Decimals to Fractions
To change a decimal to a fraction (or a mixed decimal to a mixed number), write the figures in
the decimal fraction as the top number and write the bottom number according to the number of places
used. If you can, reduce the fraction.
EXAMPLE 1:
Change .36 to a common fraction.
Step 1.
Write 36 as the top number.
Step 2.
Two places means hundredths. Write
36
36
100
100 as the bottom number.
Step 3.
36 4 9
 
100 4 25
Reduce the fraction. 36 and 100
can be divided evenly by 4.
Changing Fractions to Decimals
To change a fraction to a decimal, divide the bottom number into the top number. To do this,
add a decimal point and zeros to the top number. Usually, two zeros are enough. Bring the point up into
the answer.
1
EXAMPLE 1:
Change to a decimal.
4
.25
4 1.00
Step 1.
Divide the bottom number (4) into the top
number (1).
Step 2.
Add a decimal point and zeros.
Divide and bring the point up.
8
20
20
0
16
Comparing Decimals
When you look at a group of decimal fractions, it is sometimes difficult to tell which one is the
largest. Here is a trick you can use to compare decimals.
To compare decimals, give each decimal you are comparing the same number of places by
adding zeros. This is the same as giving each decimal a common denominator.
EXAMPLE 1.
Which of the following is larger
.03 or .6
Step 1.
Add one zero to .6
By adding the zero, we have .03 and .60
Step 2.
Compare
Since sixty hundredths is larger than three hundredths, the answer is .6.
Adding Decimals
To add decimals, first line them up with point under point. Remember: Any whole number is
understood to have a decimal point at its right.
EXAMPLE: Add 3.27 + .007 + 25
Step 1.
Line up point under point. Notice the decimal
point after the whole number 25.
Step 2.
Add.
3.27
.007
25.
28.277
Subtracting Decimals
To subtract decimals: put the larger number on top; line up the decimal points, add zeros to the
right so that each decimal has the same number of places; and subtract as you would for whole numbers,
bringing down the decimal point.
17
EXAMPLE: 24.3 - .196
Step 1.
Put the larger number on top and line up the
decimal points.
Step 2.
Add zeros to give the top number the same number
of places as the bottom number.
Step 3.
Subtract and bring down the decimal point.
24.300
- .196
24.104
Multiplying Decimals
To multiply decimals, multiply the two numbers the same way you would whole numbers. Then
count the number of decimal places in both numbers you are multiplying. Decimal places are numbers
to the right of the decimal point. Put the total number of places in your answer.
EXAMPLE:
3.76
x 4
14.04
two decimal places
no decimal places
two decimal places
Dividing Decimals by Whole Numbers
To divide a decimal by a whole number, bring the point up in the answer directly above its
position in the problem. Then divide as you would whole numbers.
EXAMPLE:
4.57
6 27.42
24
34
30
42
42
18
Dividing Decimals by Decimals
To divide by a decimal, you must change the problem to a problem in which you are dividing by
a whole number.
EXAMPLE: .05 1.2675
Step 1.
Move the point in the number outside the bracket
.05 1.2675
(the divisor) to the right as far as it will go.
Step 2.
Move the point in the number inside the bracket
.05 1.2675
(the dividend) the same number of places that
you moved the point in the divisor.
Step 3.
Bring the point up in the answer directly above
25.35
.05 1.2675
its new position in the dividend and divide.
Dividing Whole Numbers by Decimals
When dividing a decimal into a whole number, put a point after the whole number and add zeros
in order to move the point enough places. Remember: A whole number is understood to have a decimal
point at its right.
EXAMPLE: .008 48
Step 1.
Move the point in the divisor three places to the right.
Step 2.
Place a point to the right of the whole number and move
.008 48
.008 48.000
it three places to the right, holding each place with a zero.
Step 3.
Bring the point up and divide.
6000
.008 48.000 .
19
What Are Percents?
Percent is a very common term in the everyday world. Commission, interest, mark-up, and tax
rates are all written with percents. Discounts, raises, paycheck deductions, and credit card charges are
all figured with percents.
Percent is another way to describe a part or fraction of something, but it is an even more special
type of fraction. The only denominator (bottom number) it can have is 100. This denominator is not
written; it is shown by a percent sign (%).
Changing Decimals to Percents
To change a decimal to a percent, move the decimal point two place to the right and write the
percent sign (%). If the point moves to the end of the number, it is not necessary to write the point.
Decimal
Move two places to the right
Percent
.25
=
.25
=
25%
.09
=
.09
=
9%
=
.06
=
6
.06
1
2
1
2
1
%
2
Changing Percents to Decimals
To change a percent to a decimal, drop the percent sign and move the point two places to the left.
EXAMPLES:
Percent
Move two places to the left
Decimal
5%
=
05
=
.05
50%
=
50
=
.5
125%
=
125
=
1.25
20
Changing Fractions to Percents
There are two ways to change a fraction to a percent.
1
EXAMPLE: Change to a percent.
4
Method 1.
Method 2.
Multiply the fraction by 100%
Divide the bottom number of the fraction
1 100% 25 25%


 25%
41
1
1
.25
4 1.00  25%
into the top number and move the point
two places to the right.
Changing Percents to Fractions
To change a percent to a fraction, write the percent as a fraction with 100 as the bottom number
and reduce.
EXAMPLE 1.
Step 1.
Change 75% to a fraction.
Write the percent as a fraction with 100 as
75
100
the bottom number.
Step 2.
Reduce.
75  5 15

100  5 20
21
Changing Percents to Fractions Cond’t
EXAMPLE 2.
1
Change 4 % to a fraction.
2
the bottom number.
1
2
100
Step 2.
You can rewrite this fraction as a division problem.
4
Step 3.
Change the mixed number to an improper fraction.
9 100

2
1
Step 4.
Invert the divisor and multiply, canceling where possible.
Step 1.
Write the percent as a fraction with 100 as
4
1
 100
2
9 1
9


2 100 200
Finding a Percent of a Number
To find a percent of a number, change the percent to a decimal or to a fraction and multiply.
EXAMPLE: Find 25% of 40.
Method 1.
Step 1.
Change the percent to a decimal.
Step 2.
Multiply.
25% = .25
40
x .25
200
80
10.00
Method 2.
1
4
Step 1.
Change the percent to a fraction.
25% =
Step 2.
Multiply.
1 4010 10


 10
41
1
1
22
Finding What Percent One Number Is of Another
To find what percent one number is of another, make a fraction by putting the part (usually the
smaller number) over the whole. Reduce the fraction and change it to a percent.
EXAMPLE: 6 is what % of 54?
Step 1.
Put the part over the whole and reduce.
Step 2.
Divide the bottom number into the top
Step 3.
6 1

54 9
1
.11
9
. 9 1.00
9
10
9
1
Change your decimal answer to a percent by
moving two places to the right and adding
1
1
.11  11 %
9
9
a percent (%) sign.
Finding a Number When a Percent of it is Given
If a percent of a number is given and you are looking for the whole number, change the percent
into either a fraction or a decimal and divide it into the number you have.
EXAMPLE: 25% of what number is 19?
Method 1.
Step 1.
Change the percent to a fraction.
Step 2.
Divide the fraction into the number you have.
25% 
19 
1
4
1 19 4
   76
4 1 1
Method 2.
Step 1.
Step 2.
Change the percent to a decimal.
25%=.25
Divide the decimal into the number you have.
76.
.25 19.00
175
150
150
0
23
Finding Interest for One Year
Interest is money that money earns. On a loan, interest is the payment you must make for using
the lender=s money. On a savings account, interest is the money the bank pays you for using your
money.
The formula for finding interest is I = PRT:
I is the interest in dollars
P is the principal, the money borrowed or saved
R is the percent rate, which can be written as either a fraction or a decimal
T is the time in years
The formula is read as: Interest is equal to the principal times the rate times the time.
EXAMPLE 1.
Find the interest on $750 at 6% annual interest for one year.
750 6

 1  $45
1 100
EXAMPLE 2.
Find the interest on $800 at 5.5% for one year.
800
.055
4000
4000
$44.000
x
Finding Interest for More than One Year
If you want to find interest for more than one year, write the total number of months over 12 (the
number of months in one year). Use this improper fraction for T in the formula I = PRT.
EXAMPLE: Find the interest on $800 at 6% annual interest for two years and 6 months.
Step 1.
2 years and 6 months =
30 months
12 months years
=
5
years
2
24
Step 2.
800 6 5

  $120
1 100 2
25
Finding Interest for Less than One Year
Interest rates are usually given for one year. If you want to find interest for less than one year,
write the number of months over 12 (the number of months in one year). Use this fraction for T in the
formula I = PRT.
EXAMPLE: Find the interest on $700 at 6% annual interest for six months.
Step 1.
6 months =
6 months
=
1
year
2
six months =
6 1

12 2
12 months per year
Step 2.
700 6 1

  $21.00
1 100 2
Perimeter And Circumference
The Perimeter of a figure is the distance around it
for a rectangle with width W and length L.
l
The formula for Perimeter is P = 2l + 2w
EXAMPLE:
Given a rectangle with a length of 12 and a width of 4, find the Perimeter.
Step 1.
Write down the dimensions
l = 12
w=4
Step 2.
To find the Perimeter
substitute the values
of l and w into the formula.
P = 2l + 2 w
P = 2(12) + 2(4)
P = 24+8
Step 3.
Solve for P
P = 32
w
26
The formula for Perimeter of a triangle is P = a + b + c or sum of the sides
a
c
EXAMPLE:
b
Step 1.
Write down the dimensions
a = 6, b = 8, c = 11
Step 2.
To find the Perimeter
substitute the values
into the formula
P=a+b+c
P = 6 + 8 + 11
P = 25
Step 3.
Solve for P
Circumference of a Circle
The circumference of a circle is the distance around the outside of the circle. The formula for
circumference of a circle is: C =  d. Where d is the diameter of the circle and  = 3.1416.
d
EXAMPLE 1:
Given a circle with a diameter of 6, find the circumference.
 = 3.1416
Step 1.
Write down the dimensions
d=6
Step 2.
Substitute values into the formula
C=  d
C = 3.1416 x 6
C = 18,8496
Step 3.
Solve for C
EXAMPLE 2:
The diameter of a circle is equal to 2 times the radius. d = 2 x r. Given the radius the formula
for circumference would be c = 2  r. Find the circumference of a circle with a radius of 4”.
 = 3.1416
Step 1.
Write down the dimensions
r=4
Step 2.
Substitute valves into the formula
Step 3.
Solve for C
C=2  r
C = 2 x 3.1416 x 4
C = 25.1328
27
Area
The Area of a figure is the size of the region it covers. Area is measured in square units.
Area of a Square
The Area of a square with side S is A = S 2
s
s
EXAMPLE: Find the Area of a square with a side of 5.
Step 1.
Write down the dimensions
S=5
Step 2.
Substitute the value of S
into the formula
A = 52
Step 3.
Solve the equation for A
A = 25
Area of a Rectangle
For a rectangle with length l and width w the formula for Area is A = l  w .
l
w
EXAMPLE:
Find the Area of a rectangle with length 12” and width 8”.
Step 1.
Write down the dimensions
Step 2.
Substitute the dimensions
into the formula
l = 12
w=8
Alw
A  12  8
Step 3.
Solve for A
A = 96 sq. in.
Be sure to put on the correct unit of measure.
.
28
Area of a Triangle
The Area of a triangle is found by multiplying the base (b) times the height (h) and taking ½ of
the product.
1
A = bh
2
h
h
b
b
EXAMPLE: Find the Area of a triangle that has a height of 16” and a base of 10”.
b = 10”, h = 16”
Step 1.
Write down the dimensions
Step 2.
Substitute dimensions into formula A =
1
 10"16"
2
Step 3.
Solve for A
1
 10 516
21
A=
Be sure to label answer correctly
A = 90 sq. in.
Area of a Circle
The radius of a circle is the distance from the center to the outside edge.
The diameter of a circle is the distance from one side of the circle to the other,
passing through the center of the circle.
The formula for the Area of a circle is A =  r
EXAMPLE 1:
2
Find the Area of a circle with a radius of 6 in. Use 3.14 for  .
 = 3.14
Step 1.
Write down the dimensions
r 6
Step 2.
Substitute the dimensions into
the formula
A = 3.14  6 2
Step 3.
Solve for A
A = 113.04 sq. in.
Be sure to put on the correct unit of measure.
r
d
29
EXAMPLE 2:
Find the Area of a circle with a diameter of 12”.
 = 3.14
Step 1.
Write down the dimensions
d  12
Step 2.
Divide the diameter by 2 to find
the radius
d  2  6"
r 6
Step 3.
Substitute the dimensions into
the formula
A  3.14  6 2
Step 4.
Solve for A
A  113.04 sq. in.
Be sure to put on the correct unit of measure.
Pythagorean Theorem
In a right triangle (a triangle that has one side form a right angle with the base), the side opposite
the right angle (90) is called the hypotenuse. The hypotenuse is the longest side. The other two sides are
called the legs.
In words: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of
the other two sides. In symbols c 2  a 2  b 2
If you know the lengths of two sides of a right triangle, you can use the Pythagorean theorem to find the
length of the third side.
hypotenuse
leg
c
b
right angle
a
leg
30
EXAMPLE: Given the triangle at the right with sides 3 and 4 for sides a and b, find side c or the
hypotenuse.
A
a3
Step 1.
Write down the dimensions
Step 2.
Substitute values into the formula
a2  b2  c2
Add the squares
3 4 c
Step 3.
b4
2
2
9  16  c 2
Step 4.
Solve for c
25  c
2
4=b
c
2
B
25  c 2
3=a
C
5c
To find c, take the square root of 25
Volume
Volume (V) is the amount of space inside a three-dimensional figure. Volume is measured in
cubic units. For a three-dimensional figure shaped like a box, the formula is:
In words: The volume is equal to the length times the width times the height.
In symbols: V = l X w X h
EXAMPLE: Given a rectangular box with length of 8 ft, a width of 4 ft and a height of 6’, find the
volume of the box.
Step 1.
Write down the dimensions
l  8' , w  4' , h  6'
Step 2.
Substitute the values into the
formula
V  l  w h
V  8 4 6
Step 3.
Multiply the three numbers together V  192
Step 4.
Label the answer as cubic units
V  192 cu. ft.
31
Volume of a Cylinder
A cylinder is a can-shaped container, like a paint can. The top and bottom, which are circles, are
called the bases of the cylinder. To calculate the volume of a cylinder, you can multiply the height times
the area of the base.
In words: Volume of a cylinder is equal to the area of the base times the height.
In symbols:
V  r 2 h where   3.14 and r is the radius of the base.
EXAMPLE 1:
Find the volume of a container that is 12” tall and has a
radius of 4”. Use 3.14 for 
Step 1.
Write down the dimensions
h  12, r  4,   3.14
Step 2.
Write the formula
V  r 2 h
Step 3.
Substitute the values for
, r  h into the formula
V  3.14(4) 2 12
Step 4.
Evaluate 4 2
V  (3.14)(16)(12)
Step 5.
Multiply
V  602.88
Step 6.
Add the correct unit of measure
V  602.88 cu. in.
h
r
A
15”
EXAMPLE 2:
Given triangle ABC above with side
b-12 in. and the hypotenuse or side c equal to 15 in. find side a.
Step 1.
Write down the dimensions
b  12" , c  15"
Step 2.
Substitute values into the formula
a2  b2  c2
Step 3.
Find the square of b and c
a 2  144  225
Step 4.
Subtract b 2 from c 2
a 2  81
Step 5.
Take the square root of both sides
a  9 in.
B
12”
C
32
Measurement
Linear Measure
12 inches
=
1 foot
3 feet
=
1 yard
1
feet
2
=
1 rod
5,280 feet
=
1 mile
1760 yard
=
1 mile
16
To change between Inches, Feet, Yards and Miles
In general, when you change from a smaller unit to a larger one, you will divide. When you change from
a larger unit to a smaller one, you will multiply.
EXAMPLE: Convert 18 yards to feet
Step 1.
Write the problem
18 yd. = ________ ft.
Step 2.
Write the relationship – from the table above
1 yd = 3 ft
Step 3.
Multiply the number of yards by 3
3 x 18 = 54
Step 4.
Write the answer with the correct label
5 ft.
33
Adding and Subtracting Lengths
When you have to add or subtract lengths that contain different units, you may need to regroup
the measurements.
EXAMPLE 1:
Step 1.
Add 2 feet 9 inches
to
3 feet
11 inches.
Line up the measurements putting like units
under like units
Step 2.
Add the inches and add the feet
Step 3.
If the total inches is more than 12, change the
inches to feet and inches
Step 4.
Step 1.
Subtracting Measurements
Subtract 2 ft. 9 in. from
9 inches
11 inches
2 feet
3 feet
5 feet
9 inches
11 inches
20 inches
20 in. = 1 ft. 8 in.
Add the result to the total feet
EXAMPLE 2:
2 feet
3 feet
5 ft.
+ 1 ft. 8 in.
6 ft. 8 in.
8 ft.
4 in.
Line up the measurements putting like units
under like units. Make sure you put the
larger unit on top.
-
Step 2.
Step 3.
As you can’t subtract 9 from 4, you will need
to borrow. Convert 1 ft. to 12 in., add to
original 4 in.
8 ft. 4 in.
2 ft. 9 in.
8 ft. 4 in. =
7 ft. + 1 ft. + 4 in. =
7 ft. + 12 in. + 4 in.
=
7 ft. 16 in.
Rewrite the problem and subtract.
-
7 ft. 16 in.
2 ft. 9 in.
5 ft. 7 in.
34
Multiplying and Dividing Length Measurement
To multiply and divide lengths, you use the same two steps, but in the opposite order. To
multiply a length times a number first perform the operation and then change the units. To divide a
length by a number, first change the units and then perform the operation.
EXAMPLE 1:
Multiply 4 ft.
5 in.
by
6.
Step 1.
Write the problem
4 ft. 5 in.
x 6
Step 2.
Multiply the inches and the feet by 6
4 ft. 5 in.
x 6
24 ft. 30 in.
Step 3.
Change any inch total that is more
than 12 to feet and inches
Step 4.
EXAMPLE 2:
Add the feet and rewrite the total
Divide 5 yr.
1 ft.
30 in. = 24 in. + 6 in.
30 in. = 2 ft. 6 in.
24 ft. + 2 ft. 6 in. =
26 ft. 6 in.
by 4
Step 1.
Write the problem
Step 2.
Change the numerator to feet
5 yd .1 ft.
4
5 yd. = 5 x 3 = 15 ft.
15 ft. + 1 ft. = 16 ft.
16 ft.
4 ft.
Step 3.
Divide
4 ft.
Step 4.
Change back to yards and feet
4
= 1 yd. 1 ft.
3
Step 5.
The answer is 1 yd. 1 ft.
35
STANDARD TABLES OF CUSTOMARY UNITS OF MEASURE
Linear Measure
12 inches (in.)
3 ft.
16 ½ ft.
5 ½ yd.
320 rd.
1760 yd.
5280 ft.
=
=
=
=
=
=
=
1 foot (ft.)
1 yard (yd.)
1 rod (rd.)
1 rd.
1 mile
1 mile
1 mile
Surface Measure
144 sq. in.
9 sq. ft.
30 ¼ sq. yd.
160 sq. rd.
640 acres
43,560 sq. ft.
=
=
=
=
=
=
1 sq. ft.
1 sq. yd.
1 sq. rd.
1 acre
1 sq. mile
1 acre
Cubic Measure
1728 cu. in.
27 cu. ft.
=
=
1 cu. ft.
1 cu. yd.
Time Measure
60 seconds (sec.)
60 min.
24 hr.
7 days
52 weeks
365 days
10 years
=
=
=
=
=
=
=
1 minute (min.)
1 hour (hr.)
1 day
1 week
1 year
1 year
1 decade
36