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Math Review
Knowing basic math concepts and knowing when to apply them are essential skills.
You should know how to add, subtract, multiply, divide, calculate percentages, and
manipulate fractions. This section will help you accurately apply basic math concepts
and skills necessary to your success at school, work, and to manage your personal
finances at home.
Numbers
Numbers are expressed in different forms:
■
■
■
■
Whole numbers, which are the counting numbers and zero. Whole numbers do
not contain decimals or fractions. Examples: 1, 2, 3, 10, 15, 18, 0. Ignore zeros
before whole numbers. For example, 025 is the same as 25.
Nonwhole numbers, which are numbers that have decimals, such as 6.25 or 9.85.
Mixed numbers, which are numbers that combine whole numbers and a fraction,
such as 6 1/4 or 7 2/3.
Percentages, which are portions in relation to a whole, such as 65% or 22%.
Place Value
Numbers that have more than one digit are defined by their place value. Place value
is the value of a digit based on where it is in a number. For example, the number
7777 is given the following values:
Place value
7
7
7
7
▲
▲
▲
▲
tens
ones
thousands
hundreds
This number is described by saying “seven thousand, seven hundred seventy-seven.”
Numbers are written with a comma placed to the left of every third digit. The number 7777 is properly written “7,777.”
whole numbers
(HOHL NUHM bers)
Numbers that do not contain
decimals or fractions
(e.g., 1, 2, 3).
nonwhole numbers
(non HOHL NUHM bers)
Numbers with decimals
(e.g., 6.25, 9.85).
mixed numbers
(mikst NUHM bers)
Numbers with whole
numbers and a fraction
(e.g., 61/4, 72/3).
percentages
(per SEN tij es)
Portions in relation to a whole
(e.g., 65%, 22%).
Rounding Whole Numbers
▲
1. Find the place to be rounded.
2. If the digit to the right is 5 or more, add 1 to the place to be rounded. If the digit
to the right is 4 or less, leave the place to be rounded unchanged.
3. Change all digits to the right of the rounded place to zeroes.
For example, to round 687 to the nearest ten:
1. The place to be rounded is the number in the tens column.
2. The digit to the right of the tens column is greater than 5, so you add 1 to the
digit in the place to be rounded, making it 9.
3. Change the 7 to a zero. The result is 690.
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Math Review
Working with Decimals and Percentages
Four turtles = 100%.
Recall that a percentage is a portion of a whole. 100 percent = the whole or all of
something. The symbol for percent is %. In the figure to the right, four turtles =
100% or all of the turtles. One turtle is 1/4 of the total, or 25%. Two turtles = 1/2 of
the total, or 50%. Three turtles = 3/4 or 75%. Numbers to the left of a decimal are
whole numbers. Numbers to the right of a decimal are less than one.
To divide decimals by whole numbers:
.
24.5 ÷ 4 = 4 24.5
One turtle or 25% of the
turtles are sold.
6.
4 24.5
24
Place the decimal point in the answer directly
above the decimal point in the dividend.
6.125
4 24.500
00
24
5
4
10
8
20
20
6.1
4 24.5
24
5
4
1
Three turtles or 75% of
the turtles are left.
If there were five turtles
in the figure, what percentage would one turtle
represent?
Add 0s to carry answer.
Numbers to the right of a decimal point indicate less than one whole. For example,
1.5 is the same as 11/2.
Examples:
Decimal
0.10
0.25
6.25
=
=
=
Percentage
10%
25%
625%
=
=
=
Fraction(s)
10/100 or 1/10
25/100 or 1/4
6 25/100 or 6 1/4
To change a decimal number into a percentage: Move the decimal two places to the
right.
Examples:
0.5 = 0.5.0. = 50%
30.0 = 30.0.0. = 3000%
0.04 = .0.4. = 4%
To change a percentage into a decimal number: Replace the percent sign with a decimal and move the decimal two spaces to the left.
To find the percentage of a number:
1. Change the percentage to a decimal.
2. Multiply that decimal by the number.
Examples:
15% of 63 change 15% to 0.15 then multiply 0.15 × 63 = 9.45
9.45 is 15% of 63
20% of 100 change 20% to 0.20 then multiply 0.20 × 100 = 20
20 is 20% of 100
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3
Addition
Addition is the totaling of two or more numbers. Each number you are adding is
called an addend. The result is called the sum or total. For example, 2 computers in
the reception area + 3 computers in accounting = a total of 5 computers in the office.
To add:
1. Write the addends in columns, making sure each digit lines up correctly according to its place value. That means putting ones in the ones column, tens in the
tens column, hundreds in the hundreds column, and so on.
2. Draw a line under the last addend in the column.
3. Add all the addends above the line and write the total or sum below the line.
Examples:
2
+3
5
34
+4
38
<---- addends
<---- addends
<---- totals
When adding more than one column of numbers, always start adding numbers in the
right column—place value ones—first.
Adding by Carrying Numbers
When the numbers in a column add up to more than 10, you must carry all digits to
the left of the place value ones column. To carry means to move the digits to the top
of the next place value column. You then add the carried digits with all the numbers
in that column.
For example, to add 81 + 384 + 10 + 9 (refer to the following example):
1. Write the numbers in columns, lining them up according to place value.
2. Add the numbers in the ones column. The total is 14.
3. Write the 4 in the ones column below the total line, and carry the 1 to the top of
the tens column.
4. Add the numbers in the tens column, including the 1 you carried. The total is 18.
5. Write the 8 in the tens column below the total line, and carry the 1 to the top of
the hundreds column.
6. Add the numbers in the hundreds column, including the 1 you carried. The total
is 4.
7. Write the 4 in the hundreds column below the total line. The sum of the numbers
is 484.
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Math Review
Example 1:
1
1
3
8
8
1
1
4
Total
1
4
0
9
8 4
(carry 1)
81
384
10
1 9
484
(carry 1)
= 14; leave the 4 in the total row and carry the 1 to the
top of the tens column.
= 18; leave the 8 in the total row and carry the 1 to top of the
hundreds column.
= 4; place the 4 in the total row.
Example 2:
1
3
9
1
Total
1
2
8
8
9
9
5
1
4
0
9
4
(carry 1)
81
984
90
1 99
1,254
(carry 3)
= 14; leave the 4 in the total row and carry the 1 to the
top of the tens column.
= 35; leave the 5 in the total row, carry the 3 to top of the
hundreds column.
= 12; leave the 2 in the total row, and carry to the 1 to the top
of the thousands column. 1 is the only number in the thousands
columns, so you write the 1 in the total column.
Subtraction
subtraction
(suhb TRAK shuhn)
Taking a number away from
another number; the operation opposite of addition.
Subtraction is the opposite of addition. Subtracting numbers means taking one
number—called the subtrahend—away from another number—called the minuend.
The result is called the difference. Write a subtraction problem in columns, just like
you write an addition problem. Put the minuend on top, and the subtrahend below
it. Make sure you line up the digits in the proper place value column.
Subtract the number in the ones column first, and write the difference below the line.
Then, subtract the number in the tens column, then the hundreds column, and so
on.
84
– 23
61
<---- minuend ---->
<---- subtrahend ---->
136
– 12
<---- difference ---->
124
Check your answer by adding the difference to the subtrahend. If your answer is correct, your total will equal the minuend
For example, you can check your subtraction for the first equation above by adding
61 (the answer) to 23 (the number you subtracted). 61 + 23 = 84, which is the number you started with at the top of the equation. The answer is correct.
Math Review
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Subtracting by Borrowing Numbers
When the number you are subtracting is greater than the number above it in the column, you can borrow a one from the column to the left. Borrowing makes the number at the top of the equation greater, so you can complete the subtraction.
When you borrow the one, you must do two things:
1. Cross out the number in the column to the left, and replace it with a number that
is one less than the number that was there. For example, if there is a 3 in the column to the left, you cross out the 3 and replace it with a 2. If there is a 7, you
cross out the 7 and replace it with a 6.
2. Write the borrowed 1 to the left of the number that is currently in the column
you are subtracting. So, if there is a 5 in the column you are subtracting, you
write the borrowed 1 to the left of the 5; the 5 becomes 15. (Refer to the following
example.)
Examples:
3 15
– 1 8
1 7
2
8 is greater than 5, so you can borrow a one from the tens column.
Cross out the 3 in the tens column and replace it with a 2.
Write the borrowed 1 to the left of the 5, making it 15.
Subtract 8 from 15 and write the result(7) below the line in the ones column.
Subtract 1 from 2 and write the result (1) below the line in the tens column.
Adding and Subtracting Decimals
For decimals, the process of adding and subtracting numbers is similar to the process
for adding and subtracting whole numbers. The only difference is how the numbers
are aligned in the column. Numbers should be aligned by the decimal point as in the
following examples:
12.136
+ 10.246
22.382
23.453
– 10.37
13.083
Adding and Subtracting Fractions
A fraction is made up of two parts: a numerator, which is the number on top, and a
denominator, or the number on the bottom. To add or subtract a fraction, the
denominator must be the same. Consider the following examples:
Example 1:
1+3=1+3=4=1=1
4 4
4
4 1
Example 2:
1 + 2 + 12 4 = 1 + 2 + 64 = 1 + 2 + 64 = 67 = 13 3
5 5
5 5 5 5
5
5
5
numerators
(NOO muh rey ters)
The top numbers of
fractions.
denominators
(di NOM uh ney ters)
The bottom numbers of
fractions.
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Math Review
Example 3:
2–1=2–1=1
3 3
3
3
Note that Example 1 shows a simplified fraction, or a fraction that does not have any
common factors (other than 1) for the numerator and denominator. Example 2 shows
a mixed number, which is converted to a fraction to be added or subtracted.
In some cases, fractions will need to be converted so each fraction being added has a
common denominator. Consider the following example:
1+1=?
3 2
Step 1: Convert each fraction so they have a common denominator.
The denominators, 2 and 3, are factors of 6, so multiply both the nominator and the
denominator of each fraction by the number that makes the denominator equal to 6.
1×2=2
3 2 6
1×3=3
2 3 6
Step 2: Add fractions.
2+3=5
6 6 6
For subtraction, follow the same process to make fractions that have the same denominator. Simplify fractions after adding and subtracting as needed.
3–1=?
4 4
Step 1: Convert each fraction so they have a common denominator.
3×1= 3
4 3 12
1×1= 1
4 3 12
Step 2: Subtract fractions.
3– 1 = 2
12 12 12
Add or subtract the following:
3.45 + 2.34 + 5.6
33 + 2 + 4
4 3 5
3.4 – 1.28
2 3 – 11
4 3
What will happen if you don’t convert fractions to a common denominator before adding
or subtracting them?
Math Review
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Multiplication
Multiplication, the process of finding the product of two factors, is a quick, easy way
to add. For example, 9 + 9 + 9 = 27, but an easier process is 3 × 9 = 27.
To multiply numbers easily, memorize the multiplication table.
Multiplication Table
1
2
3
4
5
6
7
8
9
10
1
1
2
3
4
5
6
7
8
9
10
2
2
4
6
8
10
12
14
16
18
20
3
3
6
9
12
15
18
21
24
27
30
4
4
8
12
16
20
24
28
32
36
40
5
5
10
15
20
25
30
35
40
45
50
6
6
12
18
24
30
36
42
48
54
60
7
7
14
21
28
35
42
49
56
63
70
8
8
16
24
32
40
48
56
64
72
80
9
9
18
27
36
45
54
63
72
81
90
10
10
20
30
40
50
60
70
80
90
100
Multiplying Fractions
To multiply fractions:
1. Multiply the first numerator—the number above the line in the fraction—by the
second numerator. Write the answer as the numerator in the result.
2 Multiply the first denominator—the number below the line in the fraction—by
the second denominator.
Write the answer as the denominator in the result.
The numerators, or top numbers of the fractions, are multiplied; and the denominators, or bottom numbers of the fractions, are multiplied. For example:
3 × 12 = 36
4 15 60
If possible, reduce the fraction, which means dividing both the numerator and the
denominator by the largest number than goes into both. In the following example, 12
is the largest number that goes into both. 36 divided by 12 equals 3. 60 divided by 12
equals 5. You can reduce the fraction to 3/5.
36 to 3
60
5
multiplication
(muhl tuh pli KEY shuhn)
Finding the product of two
numbers.
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Math Review
Division
division
(di VIZH uhn)
The process of separating into
parts; the operation opposite
of multiplication.
remainder
(ri MEYN der)
The amount left over after
division that is less than a
whole number.
Division is the opposite of multiplication. It is the process of separating a whole into
parts. Knowing the multiplication table will help you when dividing. Consider this
example: A medical office budget allows $600.00 a year for magazines. Each magazine subscription costs $35.00. How many magazines can you buy? Divide the total
budgeted amount—$600.00—by $35.00. The result is 17. You can buy 17 magazine
subscriptions. This problem is written as follows:
600 ÷ 35, or 35 600
17 magazine subscriptions
35 600
To understand how 17 magazine subscriptions can be purchased for $600, review the
following division skills:
Dividing three-digit numbers by one-digit numbers:
812 ÷ 4
Divide hundreds
2
4 812
8
0
Divide tens
20
4 812
8
01} 4 does not divide
1. Place a 0
00 into
next to the 2 in
1 the answer.
Divide ones
203
4 812
8
01
00
12 } Divide 4 into 12.
12