Compare Percent, Fraction, and Decimal Claims | Grade 6 Math | LevelKite

Which is greatest: $\frac{3}{5}$ , $0.7$ , or $65\%$ ?

You can't tell from the digits alone. The three numbers use three different forms, and here each one counts parts of a different size.

In this lesson you will learn how to compare percents, fractions, and decimals fairly. Then you will use that skill to solve word problems where the numbers come in mixed forms.

A fraction, a decimal, and a percent can each name a share: a part of some whole. The whole that a share is a part of is called its reference whole. When a number stands alone, with no story around it, its reference whole is the number $1$ .

Each form tells you two things: how big the equal parts are, and how many of those parts you have.

$\frac{3}{5}$ : the whole is cut into 5 equal parts, called fifths. You have 3 of them.
$0.7$ : the whole is cut into 10 equal parts, called tenths. You have 7 of them.
$65\%$ : percent means per hundred. The whole is cut into 100 equal parts, called hundredths. You have 65 of them.

For the same whole, a fifth is bigger than a tenth, and a tenth is bigger than a hundredth. So the counts 3, 7, and 65 do not tell you which number is greatest. Three big pieces can be more or less than 65 tiny ones. You can't know until the pieces are the same size.

Here is the fix: rewrite every number so that it counts parts of the same size. You already know how to convert between percents, decimals, and fractions. The new step is to pick one part size and put every number into it.

Hundredths are often a handy choice, because a percent already counts hundredths. The table below rewrites four numbers in hundredths. Notice the two decimals: $0.4$ counts tenths, but $0.25$ counts hundredths. Even two decimals can count parts of different sizes.

Once every number counts hundredths of the same whole, the counts are fair to compare. $\frac{1}{4}$ and $0.25$ are both 25 hundredths, so they are equal. $0.4$ and $40\%$ are both 40 hundredths, so they are equal too. And 25 hundredths is less than 40 hundredths, so $0.25$ is less than $0.4$ .

Four numbers, each rewritten as a count of hundredths of the same whole, 1

Number	Size of its parts	How many parts	Rewritten in hundredths
$\frac{1}{4}$	fourths	1	$\frac{25}{100}$ , or 25 hundredths
$0.4$	tenths	4	$0.40 = \frac{40}{100}$ , or 40 hundredths
$0.25$	hundredths	25	$\frac{25}{100}$ , or 25 hundredths
$40\%$	hundredths	40	$\frac{40}{100}$ , or 40 hundredths

Key concept

Compare counts of same-sized parts of the same whole

To compare percents, fractions, and decimals, rewrite each number so that it counts parts of the same size, such as hundredths. Then the number with more of those parts is the greater number.

Two conditions make this comparison fair:

Same part size. Putting every number in the same form is not enough by itself, because two decimals can still count parts of different sizes. Line up the place values (tenths with tenths, hundredths with hundredths), as in $0.25$ and $0.40$ , before you compare counts.
Same reference whole. When both shares are parts of the same whole, the greater share is also the greater amount. When the wholes are different, that does not have to be true, so work out each amount before you compare amounts.

Worked example

Ordering three numbers in three forms

Put $\frac{3}{5}$ , $0.7$ , and $65\%$ in order from least to greatest.

Pick one part size for all three numbers. They count fifths, tenths, and hundredths. Fifths and tenths can both be cut into hundredths, and the percent already counts hundredths, so use hundredths.
Rewrite three fifths. Cut each fifth into 20 equal pieces, because 5 times 20 is 100. To do that, multiply the numerator (top number) and the denominator (bottom number) by 20. This is the same as multiplying by 20 over 20, which equals 1, so the value does not change.
$\frac{3}{5} = \frac{3 \times 20}{5 \times 20} = \frac{60}{100}$
Rewrite 0.7. It is 7 tenths, and each tenth is 10 hundredths, so it is 70 hundredths. Writing a zero in the hundredths place shows the same value.
$0.7 = 0.70 = \frac{70}{100}$
Rewrite 65 percent. Percent means per hundred, so it already counts hundredths.
$65\% = \frac{65}{100}$
All three numbers now count hundredths of the same whole, the number 1. That makes the counts fair to compare: 60 is less than 65, and 65 is less than 70.
$\frac{60}{100} < \frac{65}{100} < \frac{70}{100}$

Answer

From least to greatest, the order is $\frac{3}{5}$ , $65\%$ , $0.7$ . So $0.7$ is the greatest of the three.

Worked example

Three readers, one book

Ana, Ben, and Cal are reading the same book for class. So far, Ana has read $\frac{3}{8}$ of it, Ben has read $0.4$ of it, and Cal has read $38\%$ of it. Who has read the most pages? Who has read the fewest?

Check the reference whole. All three shares are parts of the same book, with the same number of pages. So here the greatest share is also the most pages.
Pick a part size. One eighth is 12.5 hundredths, which is not a whole number of parts. Thousandths, where the whole is cut into 1000 equal parts, do fit: one eighth is exactly 125 thousandths, because 8 times 125 is 1000. Multiply the numerator and denominator of three eighths by 125. That is multiplying by 1, so the value does not change.
$\frac{3}{8} = \frac{3 \times 125}{8 \times 125} = \frac{375}{1000} = 0.375$
Rewrite Ben's 0.4. It is 4 tenths, and each tenth is 100 thousandths. Zeros in the hundredths and thousandths places show the same value.
$0.4 = 0.400 = \frac{400}{1000}$
Rewrite Cal's 38 percent. It is 38 hundredths, and each hundredth is 10 thousandths.
$38\% = 0.38 = 0.380 = \frac{380}{1000}$
Every number now counts thousandths of the same book, so the counts are fair to compare: 375 is less than 380, and 380 is less than 400.
$\frac{375}{1000} < \frac{380}{1000} < \frac{400}{1000}$

Answer

Ben has read the most pages, with $0.4$ of the book. Ana has read the fewest, with $\frac{3}{8}$ of the book. Cal's $38\%$ falls between them.

In both examples so far, every share had the same reference whole. Word problems do not always work that way.

A share tells you how much of its own whole you have. Half of a short book and half of a long book are the same share, $50\%$ , but they are not the same number of pages. So when two shares come from different wholes, you can compare them as numbers to see which share is greater. That does not tell you which amount is greater. To compare amounts, the wholes must be the same, or you must work out each amount first.

Worked example

Greater share or greater amount?

Ridge School has 100 sixth graders, and $45\%$ of them are in band. Lake School has 200 sixth graders, and $\frac{3}{10}$ of them are in band. Which school has the greater share of its sixth graders in band? Which school has more sixth graders in band?

First compare the two shares as plain numbers. 45 percent is 45 hundredths. Rewrite three tenths in hundredths by multiplying the numerator and denominator by 10, which is multiplying by 1. Both numbers now count hundredths of 1, and 30 is less than 45. So Ridge has the greater share.
$\frac{3}{10} = \frac{3 \times 10}{10 \times 10} = \frac{30}{100}, \qquad \frac{30}{100} < \frac{45}{100}$
Check the reference wholes before you compare amounts. Ridge's share is a part of 100 students. Lake's share is a part of 200 students. The wholes are different, so the greater share does not have to be the greater number of students. Work out each amount.
Find Ridge's amount. 45 percent means 45 for every 100, and Ridge has exactly 100 sixth graders.
$\frac{45}{100} \times 100 = 45$
Find Lake's amount. 30 hundredths means 30 for every 100. Lake's 200 sixth graders make 2 groups of 100, so count 30 for each group.
$\frac{30}{100} \times 200 = 30 \times 2 = 60$
Both amounts now count students, so you can compare them directly: 45 is less than 60.
$45 < 60$

Answer

Ridge has the greater share of its sixth graders in band: $45\%$ against $\frac{3}{10}$ , which is $30\%$ . Lake has more sixth graders in band: $60$ students against $45$ .

Takeaway. Percents, fractions, and decimals are fair to compare by their counts only when the counts are of same-sized parts of the same whole. So rewrite each number in one part size, such as hundredths or thousandths, and line up the place values. Then the greater count is the greater number. In a word problem, check the reference whole too. When the shares are parts of the same whole, the greater share is the greater amount. When the wholes are different, work out the amounts before you compare them.