Percents as Rates per 100: Convert and Compare | Grade 7 Math | LevelKite

A phone battery shows $40\%$ . That number only helps because you know what it measures against: a full charge.

A percent is a special kind of rate. You already know rates such as $3$ goals per $2$ games, where one quantity goes with a set amount of another. In a percent, that set amount is always $100$ . The word means "per hundred," and its symbol is $\%$ .

So $40\%$ means $40$ for every $100$ equal parts of a whole. The whole is the full amount you measure against, such as a full charge, all the seats in a hall, or the full length of a trail.

A percent always needs a stated whole, and that whole must be an amount greater than $0$ . A whole of $0$ has nothing to cut into parts.

One whole cut into

100

equal parts with

40

shaded:

40

per

100

, or

40\%

of the whole.

In the grid above, the whole is the full grid. It has $10$ rows of $10$ equal parts, which makes $100$ equal parts, and $40$ of them are shaded.

You can name the shaded part of that whole in three ways:

as a percent: $40$ per $100$ is $40\%$
as a fraction: $40$ of $100$ equal parts is $\frac{40}{100}$
as a decimal: $40$ hundredths is $0.40$ , which equals $0.4$

All three names describe the same shaded part of the same whole. They agree because the parts are equal, so each part is exactly one hundredth of the whole.

A whole does not always come in $100$ parts, though. The bar below has only $5$ equal parts.

One bar cut into

5

equal parts with

2

shaded.

Worked example

From 2 out of 5 to a percent

The bar above is one whole. It has $5$ equal parts, and $2$ of them are shaded. What percent of the bar is shaded?

Write the shaded part as a fraction of the bar. The figure shows 2 shaded parts out of 5 equal parts.
$\frac{2}{5}$
A percent is a rate per 100, so you need 100 equal parts. Because 5 times 20 is 100, imagine cutting each of the 5 parts into 20 equal smaller parts. The bar and its shading stay the same.
$5 \times 20 = 100$
Each of the 2 shaded parts becomes 20 smaller shaded parts. Multiply to count them.
$2 \times 20 = 40$
The same bar now has 40 shaded parts out of 100 equal parts. Multiplying both numbers by 20 keeps the rate, because it only renames the same shaded share using smaller parts.
$\frac{2}{5} = \frac{2 \times 20}{5 \times 20} = \frac{40}{100} = 40\%$

Answer

$40\%$ of the bar is shaded, because $2$ of $5$ equal parts is the same rate as $40$ per $100$ .

Key concept

A percent is a rate per 100 of a stated whole

A percent tells how many for every $100$ equal parts of a stated whole. The whole must be an amount greater than $0$ .

Let $p$ be the number in front of the percent sign. Then $p\%$ of a whole is the same amount as $\frac{p}{100}$ of that same whole. The whole does not need to come in $100$ parts: $2$ of $5$ equal parts is the rate $40$ per $100$ , so it is $40\%$ .

A percent, a fraction, and a decimal can name the same part of the same whole. You can change the form without changing the value.

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

To convert a number is to write it in a different form without changing its value. Every conversion in this lesson rests on two facts.

First, percent means per $100$ parts of the whole, and "per $100$ " means divided by $100$ . So you divide by $100$ to leave percent form. To enter percent form, you multiply by $100$ and write the $\%$ sign. The sign divides by $100$ again, so the value stays the same.

Second, a fraction is a division. For a numerator $a$ (the top number) and a denominator $b$ (the bottom number) that is not $0$ , the fraction $\frac{a}{b}$ means $a \div b$ .

Some conversions end with a fraction that you can make simpler. A common factor is a whole number that divides both the numerator and the denominator exactly. To simplify a fraction, divide both by a common factor greater than $1$ . The fraction is in simplest form when the only common factor left is $1$ .

The table below shows all six conversions for one example: a tank that is $60\%$ full. The whole is the full tank, and every form in the table names the same filled part of that tank.

Six conversions for a tank that is 60% full

Conversion	What you do	Why the value stays the same	Example
Percent to fraction	Write the number in front of $\%$ over $100$ , then simplify.	Percent means per $100$ .	$60\% = \frac{60}{100} = \frac{60 \div 20}{100 \div 20} = \frac{3}{5}$
Percent to decimal	Divide the number in front of $\%$ by $100$ .	Per $100$ means divided by $100$ .	$60\% = 60 \div 100 = 0.6$
Decimal to percent	Multiply by $100$ and write the $\%$ sign.	The $\%$ sign divides by $100$ , which undoes the multiplying.	$0.6 = (0.6 \times 100)\% = 60\%$
Decimal to fraction	Read it as a number of tenths, hundredths, or thousandths, then simplify.	The place of the last digit tells the size of the equal parts.	$0.6 = \frac{6}{10} = \frac{6 \div 2}{10 \div 2} = \frac{3}{5}$
Fraction to decimal	Divide the numerator by the denominator.	A fraction means its numerator divided by its denominator, which is not $0$ .	$\frac{3}{5} = 3 \div 5 = 0.6$
Fraction to percent	Rewrite it with a denominator of $100$ , or change it to a decimal first.	Multiplying the numerator and denominator by the same number (not $0$ ) multiplies by $1$ .	$\frac{3}{5} = \frac{3 \times 20}{5 \times 20} = \frac{60}{100} = 60\%$

Worked example

Write a percent as a decimal and a fraction

A download is $45\%$ complete. Write the completed part of the download as a decimal and as a fraction in simplest form.

Name the whole and use the meaning of percent. The whole is the full download, and 45% means 45 for every 100 equal parts of it.
$45\% = \frac{45}{100}$
For the decimal, divide 45 by 100, because per 100 means divided by 100. Dividing by 100 moves each digit two places to the right, so 4 tens become 4 tenths and 5 ones become 5 hundredths.
$45 \div 100 = 0.45$
For the fraction, simplify 45 over 100. Both numbers have 5 as a factor, so divide both by 5. Dividing the numerator and denominator by the same number (not 0) keeps the value, because it is the same as dividing by 1.
$\frac{45}{100} = \frac{45 \div 5}{100 \div 5} = \frac{9}{20}$
Check for simplest form. The factors of 9 are 1, 3 and 9. Neither 3 nor 9 divides 20 exactly, so 1 is the only common factor left.

Answer

As a decimal, $0.45$ of the download is complete. As a fraction, $\frac{9}{20}$ of the download is complete.

Worked example

Write a fraction as a decimal and a percent

You have hiked $\frac{3}{8}$ of a trail. Write the part of the trail you have hiked as a decimal and as a percent.

Name the whole: it is the full trail, and you have hiked 3 of its 8 equal parts. Because 8 is not a factor of 100, change the fraction to a decimal first. A fraction means its numerator divided by its denominator.
$\frac{3}{8} = 3 \div 8$
Divide one place at a time, trading what is left for 10 of the next smaller place. 3 ones are 30 tenths, and 30 tenths divided by 8 is 3 tenths with 6 tenths left. 6 tenths are 60 hundredths, and 60 divided by 8 is 7 hundredths with 4 hundredths left. 4 hundredths are 40 thousandths, and 40 divided by 8 is 5 thousandths with nothing left.
$3 \div 8 = 0.375$
To enter percent form, multiply the decimal by 100 and write the percent sign. The percent sign divides by 100 again, so the value does not change.
$0.375 = (0.375 \times 100)\% = 37.5\%$
Check by converting back. 37.5% means 37.5 divided by 100, and that returns the decimal from step 2.
$37.5 \div 100 = 0.375$

Answer

You have hiked $0.375$ of the trail, which is $37.5\%$ of the trail.

The same whole with

4

of its

100

equal parts shaded:

4\%

, which is

0.04

, not

0.4

.

To compare two numbers is to decide which is greater, or whether they are equal. To order a list is to arrange it from least to greatest or from greatest to least.

Numbers in different forms can be hard to compare, so convert them all to one form first. Any one form works. Percent form is a good choice, because then every number counts the same thing: parts per $100$ .

One condition matters. All the numbers must describe parts of the same stated whole, cut into equal parts. Then a greater percent is a greater amount. If the wholes differ, the percents alone cannot tell you which amount is greater: $50\%$ of a small bottle is less water than $50\%$ of a large bottle.

Here is a quick comparison. One runner has finished $\frac{3}{4}$ of a race, and another has finished $70\%$ of the same race. The whole is the full race for both runners, so you can compare.

Because $4 \times 25 = 100$ , multiply the numerator and denominator by $25$ . That multiplies the fraction by $1$ , so its value stays the same: $\frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100} = 75\%$ .

Since $75\% > 70\%$ , the runner at $\frac{3}{4}$ has finished more of the race.

Worked example

Order numbers written in different forms

Four markers stand along one trail. Measured from the start, they are at $0.4$ , $\frac{3}{8}$ , $36\%$ , and $\frac{7}{20}$ of the trail's full length. Order the four numbers from least to greatest.

Check the whole. All four numbers describe parts of the same whole, the full length of one trail, so you can compare them. Choose percent form, so that every number counts parts per 100 of the trail.
Convert 0.4 by multiplying by 100 and writing the percent sign. The percent sign divides by 100 again, so the value stays the same.
$0.4 = (0.4 \times 100)\% = 40\%$
Convert three eighths. The hiking example already showed that 3 divided by 8 is 0.375, and that 0.375 times 100 is 37.5.
$\frac{3}{8} = 0.375 = 37.5\%$
Convert seven twentieths. Because 20 times 5 is 100, multiply the numerator and denominator by 5. This multiplies the fraction by 1, so its value stays the same.
$\frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100} = 35\%$
36% is already in percent form. Now compare. All four percents count parts per 100 of the same trail, so fewer parts per 100 means a smaller amount.
$35\% < 36\% < 37.5\% < 40\%$
Write each number in its original form. Each percent equals the number it came from, so the order stays the same.
$\frac{7}{20} < 36\% < \frac{3}{8} < 0.4$

Answer

From least to greatest, the numbers are $\frac{7}{20}$ , $36\%$ , $\frac{3}{8}$ , and $0.4$ .

Takeaway: A percent is a rate per $100$ of a stated whole that is greater than $0$ . Because percent means divided by $100$ , a percent, a fraction, and a decimal can name the same part of the same whole. Divide by $100$ to leave percent form, and multiply by $100$ to enter it. To compare or order numbers in different forms, first make sure they describe the same whole, then write them all in one form.

Next, you will model percents with strip diagrams and double number lines.