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Fractions, Decimals, Percents, & Ratios

Fractions, Decimals, Percents, & Ratios

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In This Chapter…

Four Ways to Express Parts of a Whole

Convert 0.25 to 25%: Move The Decimal Point Two Places Right

Convert 0.25 or 25% to 1/4: Put 25 over 100 and Simplify

Convert 1/4 to 0.25 or 25%: Long-Divide 1 by 4

Multiply a Decimal by a Power of Ten: Shift the Decimal Point

Add or Subtract Decimals: Line Up the Decimal Points

Multiply Two Decimals: Ignore Decimal Points At First

Multiply a Decimal and a Big Number: Trade Decimal Places

Divide Two Decimals: Move Points in the Same Direction To Kill Decimals

“20% Of $55” = 0.2 × $55

Percent Change: Divide Change in Value by Original Value

Percent Of a Percent Of: Multiply Twice

Ratio: One Quantity Divided By Another

Part : Part : Whole Ratios—Write Part + Part: Whole and Use the Unknown Multiplier



Chapter 5:

Fractions, Decimals,

Percents, & Ratios

In This Chapter:

• Relationships among fractions, decimals, percents, & ratios



Four Ways to Express Parts of a Whole

Say you have the shaded part of this orange. You can express how much you have in four ways.



A. You have of the orange.

B. You have 0.25 of the orange.

C. You have 25% of the orange.

D. The ratio of your piece to the whole orange is 1 to 4, or 1:4.



Fraction

Decimal

Percent

Ratio



Any of these four forms can express a part-to-whole relationship. The main difference between the

forms is how you think about the whole.



= 1 out of 4 pieces of the whole.

0.25 = 0.25 of the whole itself.

25% = 25 out of 100 pieces of the whole.

“1 to 4” = 1 out of 4 pieces of the whole.

In other words, what is each form “out of”? What is the whole that you are dividing by?

Fractions are out of the denominator (4 in this case).

Decimals are out of 1 (the whole). You've already done the division.

Percents are out of 100. Percent literally means “per hundred,” or divided by 100.

Ratios are out of the second term in the ratio (4 in this case). So ratios are very similar to

fractions, and you can quickly rewrite any ratio as a fraction. For instance, a ratio of 3 to 7 is

. Another word for ratio is proportion.

Which form is most useful depends on the problem at hand. You might say any of the following:

The container is full.

The container is filled to 0.5 of its capacity.

The container is 50% full.

The ratio of the contents of the container to its capacity is 1 to 2.

By the way, the “part” can be greater than the whole.

I ate of a box of cereal. (I ate more than one box.)

I ate 1.25 boxes of cereal.

I ate 125% of a box of cereal.

The ratio of what I ate to a whole box of cereal was 5 to 4.



Convert 0.25 to 25%: Move The Decimal Point Two Places Right

Decimals are out of 1. Percents are out of 100. So, to convert a decimal to a percent, move the

decimal point two places to the right. Add zeroes if necessary.

0.53 = 53%



0.4 = 40%



0.03 = 3%



1.7 = 170%



A percent might still contain a visible decimal point when you're done.

0.4057 = 40.57%



0.002 = 0.2%



0.0005 = 0.05%



Just keep track of which decimal is part of the percent and which one is the “pure” decimal.

To convert a percent to a decimal, go in reverse. That is, move the decimal point two places to the

left. If the decimal point isn't visible, it's actually just before the % sign. Add zeroes if necessary as

you move left.

39% = 39.% = 0.39

60% = 0.60 = 0.6

8% = 0.08

225% = 2.25

13.4% = 0.134

0.7% = 0.007

0.001% = 0.00001

If you…

Want to convert a decimal to a

percent

Want to convert a percent to a

decimal



Then you…

Move the decimal point two places to the

right

Move the decimal point two places to the left



Like this



0.036 = 3.6



41.2% =

0.412



Check Your Skills

1. Convert 0.035 to a percent.

Answer can be found on page 225.



Convert 0.25 or 25% to 1/4: Put 25 over 100 and Simplify

The decimal 0.25 is twenty-five one-hundredths. So rewrite that as 25 over 100:



Now simplify by cancelling common factors from top and bottom.



When you convert a decimal to a fraction, put a power of 10 (10, 100, 1,000, etc.) in the

denominator of the fraction. Which power of 10? It depends on how far the decimal goes to the



right.

Put as many zeroes in your power of 10 as you have digits to the right of the decimal point.

0.3



=



Zero point

three



is



0.23

three tenths, or three

over ten



=



Zero point two

is

three



0.007

twenty-three onehundredths



=



Zero point zero zero

is

seven



seven onethousandths



Don't forget to cancel.



In the second case, you cancel 125 from top and bottom, leaving 3 and 8.

When you put the digits on top, keep any zeroes in the middle, such as the 0 between the 1 and the 2 in

0.0102. Otherwise, drop any zeroes (such as the 0's to the left of the 1).



To convert a percent to a fraction, write the percent “over one hundred.” Remember that

percent literally means “per hundred.”



Alternatively, you can first convert the percent to a decimal by moving the decimal place. Then

follow the process given earlier.



If you don't convert to a decimal first, be sure to write the fraction over 100.



That fraction ultimately reduces to

further on.

If you…



, but we'll look at the process of dividing decimals a little



Then you…



Like this



Want to convert a

decimal to a fraction



Put the digits to the right of the decimal point over the

appropriate power of 10, then simplify



Want to convert a

percent to a fraction



Write the percent “over 100,” then simplify

OR

Convert first to a decimal, then follow the process for

converting decimals to fractions



3.6% 0.0



Check Your Skills

2. Convert 0.375 to a fraction.

3. Convert 24% to a fraction.

Answers can be found on page 225.



Convert 1/4 to 0.25 or 25%: Long-Divide 1 by 4

A fraction represents division. The decimal equivalent is the result of that division.

To convert a fraction to a decimal, long-divide the numerator by the denominator.

Divide 1 by 4.



Divide 5 by 8.



In some cases, the decimal never ends because the long division never ends. You get a repeating

decimal.



If the denominator contains only 2's and/or 5's as factors, the decimal will end. In that case, you can

take a shortcut to find the decimal equivalent: make the denominator a power of 10 by

multiplication.



Since 4 = 22, you multiply by 25 (= 52) to get 100 (= 102). Likewise, you multiply 8 (= 23) by 125 (=

53) to get 1,000 (= 103).

To convert a fraction to a percent, first convert it to a decimal, then convert the decimal to a

percent.



If you…

Want to convert a

fraction to a decimal



Then you…

Do long division

OR

Convert the denominator to a power of 10, if the denominator

only contains 2's and 5's as factors



Know the following conversions.

Fraction

1/100

1/20

1/10

1/8

1/5

1/4

3/10

1/3

3/8



Decimal

0.01

0.05

0.1

0.125

0.2

0.25

0.3

0.3333…

0.375



Percent

1%

5%

10%

12.5%

20%

25%

30%

33.33…%

37.5%



Like th



2/5

1/2

3/5

5/8

2/3

7/10

3/4

4/5

7/8

9/10

1

6/5

5/4

3/2



0.4

0.5

0.6

0.625

0.6666…

0.7

0.75

0.8

0.875

0.9

1

1.2

1.25

1.5



40%

50%

60%

62.5%

66.66…%

70%

75%

80%

87.5%

90%

100%

120%

125%

150%



Check Your Skills

4. Change 3/5 to a decimal.

5. Convert 3/8 to a percent.

Answers can be found on page 225.



Multiply a Decimal by a Power of Ten: Shift the Decimal Point

Decimals are tenths, hundredths, thousandths, and so on. One tenth is a power of 10—namely, 10 –1.

One hundredth is also a power of 10—namely, 10–2.

You can write any decimal as a fraction with a power of 10 in the denominator, or as a product

involving a power of 10. The power of 10 determines where the decimal point is.

So if you multiply or divide a decimal by a power of 10, you move the decimal point.

If you multiply by 10 itself, you shift the decimal point one place to the right.

0.004 × 10 = 0.04

The 10 cancels with one power of 10 in the denominator.



You can also see it in terms of exponents. The additional 10 increases the overall exponent from –3 to

–2.



4 × 10–3 × 10 = 4 × 10–2

If you multiply by 100, or 102, you shift the decimal point two places to the right.

0.004 × 100 = 0.4



4 × 10–3 × 102 = 4 × 10–1



That is,



When you multiply by a power of 10, the exponent of that power is the number of places you move the

decimal.

43.8723 × 103 = 43,872.3



Move the decimal 3 places to the right.



If you divide by a power of 10, you just move to the left instead.

782.95 ÷ 10 = 78.295

57,234 ÷ 104 = 5.7234



Move the decimal 1 place to the left.

Move the decimal 4 places to the left.



If you encounter negative powers of 10, flip them to positive powers of 10 and change from

multiplication to division or vice versa.

Multiplying by a negative power of 10 is the same as dividing by a positive power.

0.004 ì 103 = 0.004 ữ 103 = 0.000004



Move 3 places to the left.



Likewise, dividing by a negative power of 10 is the same as multiplying by a positive power.

62 ữ 102 = 62 ì 102 = 6,200



Move 2 places to the right.



All of these procedures work the same for repeating decimals.

× 10 = 0.333…× 10 = 3.33…



Move 1 place to the right.



If you…

Multiply a decimal by a

power of 10



Then you…

Move the decimal point right a number of places,

corresponding to the exponent of 10



0.007 × 1

= 0.7



Divide a decimal by a

power of 10



Move the decimal point left a number of places,

corresponding to the exponent of 10



0.6 ÷ 10

0.0006



Check Your Skills

6. 32.753 × 102 =

7. 43,681 × 10–4 =



Like thi



Answers can be found on page 225.



Add or Subtract Decimals: Line Up the Decimal Points

When you add or subtract decimals, write the decimals vertically, with the decimal points lined up.

0.3 + 0.65 =



0.65 – 0.5 =



You can add zeroes on the right to help you line up. For instance, turn 0.5 into 0.50 before you

subtract it from 0.65.

If you…



Then you…



Add or subtract decimals



Line up the decimal

points vertically



Like this:



Check Your Skills

8. 3.128 + 0.045 =

9. 1.8746 – 0.313 =

Answers can be found on page 225.



Multiply Two Decimals: Ignore Decimal Points At First

Consider this example:

0.25 × 0.5 =

First, multiply the numbers together as if they were integers. In other words, ignore the decimal

points.

25 × 5 = 125

Now count all the digits to the right of the original decimal points.

0.25 has 2 digits to the right.



0.5 has 1 digit to the right.



There were a total of 3 digits originally to the right. So we move the decimal point of our answer 3

places to the left, in order to compensate.

125 becomes 0.125



Therefore, 0.25 × 0.5 = 0.125



You can see why this process works using powers of 10.

0.25 = 25 × 10–2



0.5 = 5 × 10–1



0.25 × 0.5 = (25 × 10–2) × (5 × 10–1) = 125 × 10–3 = 0.125

The powers of 10 just tell you where to put the decimal point. Here is another example:

3.5 × 20 =

We originally had one digit to the right of a decimal point.

Move the final decimal point one place to the left.



35 × 20 = 700

3.5 × 20 = 70.0 = 70



Count the zeroes to the right of the decimal point as well.

0.001 × 0.005 =

We originally had six digits to the right, including zeroes. Move the final

decimal point six places to the left.



1×5=5

0.001 × 0.005 =

0.000005



If you…

Multiply two

decimals



Then you…



Like this:

0.2 × 0.5 = ?

Ignore the decimal points, multiply integers, then place the decimal

2 × 5 = 10

point by counting the original digits on the right

10 → 0.10

0.2 × 0.5 = 0.1



Check Your Skills

10. 0.6 × 1.4 =

11. 0.0004 × 0.032 =

Answers can be found on pages 225–226.



Multiply a Decimal and a Big Number: Trade Decimal Places

Now consider this example:

4,000,000 × 0.0003 =

When one number is very big and the other one is very small, you can trade powers of 10 from the big

one (4,000,000) to the small one (0.0003). In other words, move one decimal point left and the other

one right. Just make sure you move the same number of places.



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