Download Math Review

Math Review
This arithmetic review will help you solve many of
the end-of-chapter problems in this text as well as
common arithmetic problems you may encounter
in business.
Examples
• Estimate the answer to 24,432 1 15,000.
24,432
1 15,000
Estimating Sums
and Differences
In some situations, an estimate may be useful.
Sometimes an exact answer is not needed, so you
can estimate. Other times, estimation can be used
to check mathematical calculations, especially
when using a calculator.
Most people estimate by rounding numbers. Rounded numbers are easier to work with.
Rounded numbers usually contain one or two
non-zero digits followed by all zeros.
Option 1:
Round to the
nearest ten
thousand.
20,000
120,000
40,000
Option 2:
Round to
the nearest
thousand.
24,000
115,000
39,000
• Estimate the answer to $32.23 2 $17.54.
$32.23
217.54
Option 1:
Round to the
nearest ten
dollars.
$30.00
220.00
$10.00
Option 2:
Round to
the nearest
dollar.
$32.00
218.00
$14.00
• Estimate how much change you should get
Examples
The U.S. Bureau of the Census estimated the 2009
population of Texas at 24,782,302.
• Round 24,782,302 to the nearest ten million.
24,782,302
20,000,000
Because 4 is less than 5, round down to
20,000,000.
if you give the clerk $20 to pay for a bill of
$6.98.
$20.00
Round to the
26.98
nearest dollar.
$20.00
27.00
$13.00
• Round 24,782,302 to the nearest million.
dollars.
$25,209.50
$25,000
Because 2 is less than 5, round down to
$25,000
• Round $25,209.50 to the nearest dollar.
Examples
7
2
7
1,400
14
{
2 zeros
200
{
{
$25,209.50
$25,210
Because 5 is greater than or equal to 5, round
up to $25,210.
When you multiply numbers that have final zeros,
you can use this shortcut:
Multiply the numbers by using only the digits
that are not zeros. Then write as many final
zeros in the product as there are zeros in the
numbers being multiplied.
{
{
• Round $25,209.50 to the nearest thousand
Multiplying Numbers
Ending in Zeros
{
24,782,302
25,000,000
Because 7 is larger than 5, round up to
25,000,000.
One new car has a list price of $25,209.50.
2 zeros
From DLABAY/BURROW/KLEINDL. Principles of Business, 8E. © 2012 SouthWestern, a part of Cengage Learning, Inc. Reproduced by permission. www.cengage.com/permissions
676
Math Review
{
5
20
{
4
{
{
{
1 zero 3 zeros
200,000
{
5,000
{
40
4 zeros
When multiplying larger numbers, use an
imaginary line to separate zeros from the rest of
the digits.
36
2,500
36
25 00
180
72
900 00
Dividing Whole
Numbers
Division is the opposite of multiplication. Division
is shown in several ways. To show that 18 divided
by 3 is 6, you may use any of these forms:
6
18
18 4 3 5 6 5 6 3q18
3
In each case, 18 is the dividend, 3 is the divisor,
and 6 is the quotient.
2 zeros
dividend 4 divisor 5 quotient
90,000 Answer
quotient
dividend
5 quotient divisorqdividend
divisor
3,600 25,000
36 00
2 zeros
25 000
3 zeros
180
2 3 5 zeros
72
900 00000
5 zeros
90,000,000 Answer
Estimating Products
There are various ways to estimate the answer to a
multiplication problem.
• Option 1: Round both numbers up. Estimate
will be greater than the actual product.
• Option 2: Round both numbers down. Estimate
will be less than the actual product.
• Option 3: Round each number to the nearest
Dividing Numbers
Ending in Zeros
When you divide multiples of 10, there are several
shortcuts you can use. Try either of the shortcuts
discussed below:
• Write the numbers as a fraction. Cross out the
same number of zeros in both the numerator
and denominator of the fraction.
• Move the decimal point in the dividend to the
left the same number of places as there are
zeros in the divisor.
Examples
1,000,000,000
10,000
5 100,000
• 1,000,000,000 4 10,000 5
unit with one nonzero digit. The estimate will
be close to the actual product.
Examples
Estimate 82,543 3 653.
Option 1: 90,000 3 700 5 63,000,000
Option 2: 80,000 3 600 5 48,000,000
Option 3: 80,000 3 700 5 56,000,000
• 1,000,000,000 4 10,000
5 100000.0000. 4 1.0000. 5 100,000
Math Review
677
Estimating Quotients
One way to estimate the answer to a division
problem is to start by rounding the divisor to a
number with one nonzero number followed by all
zeros. Then round the dividend to a multiple of
that rounded divisor.
• Find the difference between 952.1 and 34.2517.
952.1
2 34.2517
Add 0s to the right
after the decimal point
952.1000
2 34.2517
917.8483
Examples
• Estimate 609 4 19.
Round 19 to 20 and 609 to 600.
600 4 20 5 30
• Estimate 19,876,548 4 650.
650 rounds up to 700.
Multiples of 7 are 7, 14, 21, 28, 35, and so on.
Use the closest multiple, 21.
21000000
21,000,000 4 700 5
700
210000
5
5 30,000
7
Adding and
Subtracting Decimals
When adding and subtracting decimals, align the
decimal points. Then add or subtract as for whole
numbers. Place the decimal point in the answer
directly below where it is located in the computation. A number like 532 can also be written as
532. or 532.0. When writing decimals less than
one, a zero is placed before the decimal point to
show that there are no ones.
Examples
• Find the sum of 33.67, 72.84, 0.75, and 43.34.
33.67
72.84
0.75
1 43.34
150.60
• Find the sum of 320.5471, 1.4, and 82.352.
320.5471
1.4
1 82.352
404.2991
678
Math Review
Multiplying Decimals
When multiplying decimals, align the numbers
at the right. Multiply as if you are multiplying
whole numbers. To locate the decimal point
in the answer, count all digits to the right of
the decimal point in each number being multiplied and place the decimal point so there
are that many digits after the decimal point in
the answer.
Remember: Estimation can be used to check
that your answer is reasonable and that you
have correctly located the decimal point in the
answer.
Examples
• Multiply 7.46 by 3.2.
7.46
3 3.2
1492
2238
23.872
2 decimal places
1 decimal place
21153
3 decimal places
• Multiply 0.193 by 0.2.
0.193
3 0.2
0.0386
3 decimal places
11 decimal place
4 decimal places
If needed to get enough decimal places,
add a zero before the numeric answer but
after the decimal point.
Remember that the zero before the decimal
point shows that there are no ones in the
product. The answer is less than one.
Estimate to check your answer.
0.2 3 0.2 5 0.04
0.04 is close to 0.0386, so the answer is
reasonable.
Multiplying by
Powers of 10
Numbers like 100,000,000, 10,000, 100, 0.1, 0.01,
and 0.00001 are powers of 10. To multiply by
these, simply move the decimal point in the number being multiplied.
When multiplying by a power of 10 greater
than one, move the decimal point to the right. The
answer is larger than the number you started with.
When multiplying by a power of 10 less than
one, move the decimal point to the left. The answer is smaller than the number you started with.
Examples
• Multiply 25,397 by ten thousand.
25,397 3 10,000 5 253,970,000
Think: 25397.0000.
• Multiply 0.078 by one hundred.
0.078 3 100 5 7.8
Think: 5 0.07.8
• Multiply 192,536 by one ten-thousandth.
192,536 3 0.0001 5 19.2536
Think: 19.2536.
• Multiply 0.293 by one hundredth.
0.293 3 0.01 5 0.00293
Think: 0.00.293
Shortcuts When
Multiplying with
Money
Businesses price items at amounts such as 49¢,
$5.98, or $99.95. A price of $99.95 seems less
to the buyer than an even $100. When finding
the cost of several such items, you can use a
mathematical shortcut.
Examples
• Find the cost of 27 items at 98¢ each.
normal multiplication
$0.98
3 27
686
196
$26.46
Shortcut: Think: 98¢ 5 $122¢
(27 3 $1) 2 (27 3 2¢) 5 $27254¢
$27.00
54¢ 5 $0.54
20.54
$26.46
• Find the cost of 32 items at $6.95 each.
Shortcut: Think $6.95 5 $7 2 $0.05
(32 3 $7) 2 (32 3 $0.05)
5 $224 2 $1.60 5 $222.40
• Find the cost of 101 items at $9.99 each.
Shortcut: Think $9.99 5 $10 2 $0.01
(101 3 $10) 2 (101 3 $0.01)
5 $1,010 2 $1.01 5 $1,008.99
Alternate Shortcut:
Think: 101 5 100 1 1
(100 3 $9.99) 1 (1 3 $9.99) 5
$999 1 $9.99 5 $1,008.99
Multiplying a Whole
Number by a Fraction
To multiply a whole number by a fraction, multiply the whole number by the numerator (top
number) and then divide that answer by the
denominator (bottom number).
Examples
2
180 3 2
360
5
5
5 120
3
3
3
3
500 3 3
1500
• 500 3 5
5
5 375
4
4
4
1
768 3 1
768
• 768 3 5
5
5 96
8
8
8
• 180 3
Math Review
679
Simplifying Fractions
When working with fractions, you can simplify
the fractions by dividing the numerator and the
denominator by a common factor (a number that
will divide into both numbers evenly). Division by
such common factors is called canceling or
cancellation.
Examples
• Simplify
10 #
12
5
Think: 2 is a factor of
both 10 and 12.
10 4 2 5 5
10
5
5
12
6
6
You can also use fractional parts of $1.00 to
find the cost of multiple items mentally.
24 items selling for $1.00 would cost $24.00
1
24 items at 50¢ each 5 of $24 or $12.00
2
1
24 items at 25¢ each 5 of $24 or $6.00
4
1
1
24 items at 33 ¢ each 5 of $24 or $8
3
3
Many similar calculations can be made mentally. While there are many fractional parts of $1.00,
a chart of those most commonly used follows.
12 4 2 5 6
75 #
• Simplify
100
3
Fraction
Think: Use 25 as a
common factor.
75 4 25 5 3
75
3
5
100
4
4
100 4 25 5 4
280 #
• Simplify
60
14
Think: Use 20 as a
common factor.
280 4 20 5 14
280
14
5
60
3
3
60 4 20 5 3
1
8
1
6
1
5
1
4
1
3
3
8
2
5
1
2
Examples
Part of
$1.00
1
$0.12
2
2
$0.16
3
$0.20
$0.25
1
3
1
$0.37
2
$0.33
$0.40
Fraction
Part of
$1.00
3
5
5
8
2
3
3
4
4
5
5
6
7
8
$0.60
1
2
2
$0.66
3
$0.62
$0.75
$0.80
1
3
1
$0.87
2
$0.83
$0.50
1
2
• Find the cost of 16 items at 12 ¢ each.
Using Fractional Parts of $1.00 in
Multiplying Mentally
While goods and services may be priced at any
amount, prices are frequently expressed in fractional parts of $1, $10, or $100. For instance,
1
2 items for $1 is the same as of 100¢, or 50¢ per
2
1
item, and 3 items for $10 is the same as of $10,
3
1
or $3.33 per item.
3
680
Math Review
1
1
Think: 12 ¢ 5 of $1
2
8
1
16 3 5 2
8
1
16 items at 12 ¢ each will cost $2.
2
• Find the cost of 33 items at 25¢ each.
1
Think: 25¢ 5 of $1
4
1
33
1
33 3 5 , or 8
4
4
4
1
33 items at 25¢ each will cost $8 , or $8.25.
4
• Find the cost of 48 items at 75¢ each.
Think: 75¢ 5
• Divide 12.96 by 8.
3
of $1
4
12
48 3
3
48 3 3
36
12 3 3
5
5
5 36
5
4
4
1
1
1
48 items at 75¢ each is $36.
Fractional Parts of Other Amounts
You can use the table on the previous page to find
fractional parts of multiples of 10.
Examples
5
• Find of $1,000.
6
5
5
of $1,000 5 1000 3 of $1
6
6
1
5 1000 3 $0.83
3
1
5 $833.33
3
5
So, of $1,000 is $833.33.
6
1
1
dollar 5 $0.33 , which is $0.33 when
3
3
rounded to the nearest cent.
(
7
• Find of $10.
8
7
7
of $10 5 10 3 of $1
8
8
5 10 3 $0.875
5 $8.75
7
So, of $10 is $8.75.
8
Examples
)
Dividing Decimals
Division involving decimals is completed like
division of whole numbers, except for dealing
with the decimal point. When you divide a
decimal by a whole number, you divide as for
whole numbers and place the decimal point
directly above the location of the decimal point
in the dividend.
1.62
8q12.96
8
49
48
16
16
0
To check division,
multiply your quotient
by the divisor. The
result should be the
dividend.
1.62
3 8
12.96 ✓
• Divide 38 by 4.
9.5
4q38.0
36
20
20
0
Add zeros as needed.
Check:.
9.5
3 4
38.0 ✓
To divide by a decimal, move the decimal
point to the right the same number of places
in both the divisor and the dividend so you are
dividing by a whole number.
Examples
• Divide 12.96 by 0.8.
Think: 0.8 has one decimal place, move the
decimal points in the divisor and the dividend
right one place.
1 6.2
0.8.q12.9.6
8
49
48
16
16
0
• Divide 38 by 0.04.
9 50.
Add zeros as needed.
0.04.q38.00.
36
20
20
0
Remember: Estimation can be used to check
that your answer is reasonable and that you have
correctly located the decimal point in the answer.
Math Review
681