Percents as Fractions and Benchmark Strips | Grade 6 Math | LevelKite

You already know that percent means per hundred. On a hundred grid, $25\%$ covers $25$ of the $100$ equal squares.

That picture also shows a fraction. Call the full grid the whole: the full amount that $100\%$ stands for. Shading $25$ of its $100$ equal parts shades $\frac{25}{100}$ of that whole. So a percent and a fraction can name the same part of the same whole.

In this lesson you will change percents into fractions and fractions into percents. Then you will use a strip model, a bar cut into equal parts, to work with easy percents such as $10\%$ and $25\%$ .

Key concept

A percent is a fraction of the same whole

Cut a whole into $100$ equal parts, called hundredths. Let $p$ be the number in a percent, such as the $40$ in $40\%$ . Then $p\%$ of the whole means $p$ hundredths of it, so $p\%$ equals the fraction $\frac{p}{100}$ of the same whole.

Percent to fraction: write the number over $100$ . Then rename it with fewer, larger equal parts when you can.
Fraction to percent: rename the fraction as hundredths. The number of hundredths is the percent.

Add or compare amounts only when they refer to the same whole. Count parts together only when the parts are equal in size. If $p$ is more than $100$ , the amount is more than one whole, and every whole you draw must be the same size.

p\% = \frac{p}{100}

Worked example

Percent to fraction

Write $40\%$ as a fraction in simplest form. A fraction is in simplest form when no whole number greater than $1$ divides both its numerator (top number) and its denominator (bottom number) exactly.

Percent means per hundred. So 40% of a whole is 40 of the 100 equal parts of that whole. Write it as a fraction of the same whole with denominator 100.
$40\% = \frac{40}{100}$
Look for a common factor of 40 and 100. That is a whole number that divides both of them exactly. 20 is one, because 40 is 20 times 2 and 100 is 20 times 5.
$40 = 20 \times 2 \quad \text{and} \quad 100 = 20 \times 5$
Divide the numerator and the denominator by 20. This puts the hundredths into equal groups of 20. The same whole now has 5 larger equal parts, and the 40 hundredths fill 2 of them. Only the size of the parts changes, so the amount stays the same.
$\frac{40 \div 20}{100 \div 20} = \frac{2}{5}$
Check the form. No whole number greater than 1 divides both 2 and 5 exactly, so 2 fifths is in simplest form. Then check the amount by going back. Each fifth holds 20 hundredths, so 2 fifths hold 40 hundredths, which is 40%.
$2 \times 20 = 40$

Answer

$40\%$ of a whole is $\frac{2}{5}$ of that same whole.

Going from a fraction to a percent uses the same idea in reverse. A percent counts hundredths, so you rename the fraction as an equivalent fraction with denominator $100$ . Equivalent fractions name the same amount of the same whole, using equal parts of a different size.

To rename, multiply the numerator and the denominator by the same whole number greater than $1$ . This cuts every part into that many smaller equal pieces. There are more pieces, but the amount does not change. For example, multiplying both by $5$ is the same as multiplying the fraction by $\frac{5}{5}$ , which equals $1$ .

A whole-number multiplier that reaches exactly $100$ exists only when the denominator is a factor of $100$ . A factor of $100$ is a whole number that divides $100$ exactly, such as $2$ , $4$ , $5$ , $10$ , $20$ , $25$ or $50$ . Every fraction you convert in this lesson has a denominator like that.

Multiplying and dividing by the same number, other than zero, are inverse operations: each one undoes the other. So you can check a conversion by changing your answer back.

Worked example

Fraction to percent

Write $\frac{7}{20}$ as a percent.

A percent counts hundredths, so you need an equivalent fraction with denominator 100. The denominator 20 is a factor of 100, so a whole-number multiplier exists. Find it by dividing, because division undoes multiplication.
$100 \div 20 = 5$
Multiply the numerator and the denominator by 5. Each twentieth is cut into 5 equal pieces, so the same whole now has 100 equal pieces and the 7 twentieths become 35 of them. The amount does not change.
$\frac{7 \times 5}{20 \times 5} = \frac{35}{100}$
Read 35 hundredths as a percent, because percent means per hundred.
$\frac{35}{100} = 35\%$
Check by changing back. Dividing by 5 undoes multiplying by 5, so you should return to the fraction you started with.
$\frac{35 \div 5}{100 \div 5} = \frac{7}{20}$

Answer

$\frac{7}{20}$ of a whole is $35\%$ of that same whole.

A strip model shows a percent and its fraction in one picture. The full strip stands for one whole, so it is $100\%$ . When you cut the strip into equal parts, each part is the same fraction and the same percent of that whole.

Cut the strip into $4$ equal parts. Each part is $\frac{1}{4}$ of the whole, and $100\% \div 4 = 25\%$ . In the strip below, $3$ of the $4$ parts are shaded. You can count them together because they are equal-sized parts of one whole: $25\% + 25\% + 25\% = 75\%$ . That matches the fraction $\frac{3}{4}$ , because $\frac{3 \times 25}{4 \times 25} = \frac{75}{100}$ .

Percents such as $25\%$ , $50\%$ and $75\%$ are benchmark percents. A benchmark percent matches a simple fraction of the whole, so it is easy to picture and to compute with. The table after the strip lists more of them.

One whole strip (

100\%

) has

4

equal parts of

25\%

each, and

3

are shaded:

\frac{3}{4} = 75\%

.

Cut one whole strip (100%) into equal parts to find benchmark percents.

Equal parts in the strip	Percent of the whole in each part	Fraction of the whole in each part	Benchmarks the strip shows
$2$	$100\% \div 2 = 50\%$	$\frac{1}{2}$	$50\%$ , $100\%$
$4$	$100\% \div 4 = 25\%$	$\frac{1}{4}$	$25\%$ , $50\%$ , $75\%$ , $100\%$
$5$	$100\% \div 5 = 20\%$	$\frac{1}{5}$	$20\%$ , $40\%$ , $60\%$ , $80\%$ , $100\%$
$10$	$100\% \div 10 = 10\%$	$\frac{1}{10}$	$10\%$ , $20\%$ , $30\%$ and so on, up to $100\%$

The full tank of

60

liters is one whole strip (

100\%

) with

10

equal parts of

10\%

each, and

7

are shaded for

70\%

.

Worked example

A benchmark strip with an amount

A full tank holds $60$ liters of water. Right now the tank is $70\%$ full. Use the strip above to find how many liters are in the tank.

Name the whole first, because 70% of a different tank would be a different amount. Here the percent is a percent of the full tank. So the full strip stands for 60 liters, which is 100% of the tank.
$100\% \text{ of the tank} = 60 \text{ liters}$
Use the 10% benchmark. The strip has 10 equal parts, so each part is 100% divided by 10, which is 10% of the tank. So 70% is 7 of the 10 parts. The fraction agrees: dividing the numerator and the denominator of 70 hundredths by 10 gives 7 tenths.
$70\% = \frac{70}{100} = \frac{70 \div 10}{100 \div 10} = \frac{7}{10}$
Find the liters in one part. The 10 parts are equal and together make 60 liters, so divide 60 liters by 10.
$60 \text{ liters} \div 10 = 6 \text{ liters}$
Count the 7 shaded parts. They are equal-sized parts of the same tank, so multiply the liters in one part by 7.
$7 \times 6 \text{ liters} = 42 \text{ liters}$
Check with the 50% benchmark. Half of the same tank is 30 liters. 70% is more than 50% of that tank, and 42 liters is more than 30 liters, so the answer is reasonable.
$60 \text{ liters} \div 2 = 30 \text{ liters}$

Answer

The tank holds $42$ liters of water, which is $70\%$ , or $\frac{7}{10}$ , of the full $60$ liters.

A percent can be greater than $100\%$ . Then the amount is more than one whole, so one strip cannot show it.

Take $125\%$ . It means $125$ hundredths, so $125\% = \frac{125}{100}$ . Divide the numerator and the denominator by their common factor $25$ : $\frac{125 \div 25}{100 \div 25} = \frac{5}{4}$ . That is $5$ fourths: one whole, which is $4$ fourths, and $1$ more fourth.

The model below uses two strips. The two strips are the same size. Each strip is one whole, so each strip is $100\%$ . Each strip is cut into $4$ equal parts, so every part is $25\%$ of one whole. In all, $5$ parts are shaded: $5 \times 25\% = 125\%$ . You may count parts from both strips together only because the two wholes are equal in size and are cut into equal parts.

Two strips of the same size, each one whole (

100\%

) with

4

equal parts of

25\%

, have

5

parts shaded in all:

\frac{5}{4} = 125\%

.

Takeaway: a percent and a fraction are two names for the same part of the same whole. To change a percent to a fraction, write it over $100$ and put it in simplest form. To change a fraction to a percent, rename it as hundredths. On a strip model, the full strip is $100\%$ . Cutting it into equal parts gives benchmark percents such as $10\%$ , $25\%$ and $50\%$ . You can count parts together only when they are equal in size and belong to the same whole, or to wholes of the same size.

Next, you will compare percents, fractions and decimals and use them to solve word problems.