Compare and Order Fractions, Decimals, and Percents | Grade 8 Math | LevelKite

Which is greatest: $\frac{3}{5}$ , $0.58$ or $62\%$ ? It is hard to tell at a glance, because the three numbers use three different forms. By the end of this lesson, you will be able to answer with confidence.

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

Two things are true of every number in this lesson. Each number is positive. And each fraction has a denominator that is not zero, because division by zero is undefined.

You already know how to convert between fractions, decimals and percents. Converting changes how you write a number. It does not change how big the number is. For example, $\frac{1}{2}$ , $0.5$ and $50\%$ are three ways to write one number, so they sit at the same point on the number line. Forms with the same value are equivalent forms.

This is what makes comparing possible. When you replace a number with an equivalent form, the number does not move on the number line, so the order of the numbers stays the same. That lets you choose one form for all of the numbers, a common form, in which the order is easy to see.

Key concept

Rewrite in one common form, then compare

To compare or order fractions, decimals and percents, first rewrite them in one common form: all decimals, all percents, or all fractions with the same positive denominator. Equivalent forms have the same value, so the order you find is the order of the original numbers.

This works under three conditions:

Every fraction has a nonzero denominator, because division by zero is undefined.
The fractions and percents are parts of the same whole, the full amount that each part is taken from. With no context, the whole is the number $1$ .
You compare exact values. Put a bar over digits that repeat without end, as in $\frac{1}{3} = 0.\overline{3}$ , and write $\approx$ (approximately equal) for a rounded value, as in $\frac{1}{3} \approx 0.33$ .

The three opening numbers, each written in three common forms

Number as given	Fraction out of 100	Decimal	Percent
$\frac{3}{5}$	$\frac{60}{100}$	$0.60$	$60\%$
$0.58$	$\frac{58}{100}$	$0.58$	$58\%$
$62\%$	$\frac{62}{100}$	$0.62$	$62\%$

Read down any of the last three columns of the table. Each of those columns uses one common form, and in each one the three entries count the same unit, hundredths. Reading down, the counts are $60$ , $58$ and $62$ . When numbers count the same unit, the greater count is the greater number. All three columns give the same order, because all three hold the same values.

So $62\%$ is the greatest of the three numbers and $0.58$ is the least. Here is the full order, with each number in its original form:

0.58 < \frac{3}{5} < 62\%

Worked example

Compare a fraction and a percent

Which is greater, $\frac{3}{8}$ or $37\%$ ?

Choose a common form. The denominator $8$ is not zero, and with no context both numbers are parts of the same whole, $1$ . Rewrite both numbers as decimals.
Rewrite $\frac{3}{8}$ . Multiply the numerator and the denominator by $125$ , so that the denominator becomes $1000$ . This is the same as multiplying the fraction by $\frac{125}{125} = 1$ , so the value does not change.
$\frac{3}{8} = \frac{3 \times 125}{8 \times 125} = \frac{375}{1000} = 0.375$
Rewrite $37\%$ . Percent means per hundred, so $37\%$ is $37$ hundredths.
$37\% = \frac{37}{100} = 0.37$
Make the two decimals count the same unit. Write a zero after the last decimal place of $0.37$ . The value stays the same, because $37$ hundredths equals $370$ thousandths.
$0.37 = 0.370$
Compare the counts. Both decimals now count thousandths, and $375$ thousandths is more than $370$ thousandths.
$0.375 > 0.370$

Answer

$\frac{3}{8}$ is greater than $37\%$ , because $0.375$ is greater than $0.37$ .

So far the numbers had no context, so each one was a part of the same whole, the number $1$ . In a real situation, the whole is a real amount, such as a full tank of water. The greater fraction or percent is sure to be the greater amount only when both are parts of the same stated whole.

Here is why. $50\%$ of $40$ liters is $0.5 \times 40 = 20$ liters, but $\frac{1}{4}$ of $200$ liters is $200 \div 4 = 50$ liters. The number $50\%$ is greater than $\frac{1}{4}$ , yet it gives the smaller amount, because the two wholes are different. Before you compare parts, check that the problem states one whole for all of them.

Worked example

Order three parts of the same whole

Three identical tanks each hold $200$ liters of water when full. Tank A is $\frac{13}{20}$ full, Tank B is $0.7$ full, and Tank C is $68\%$ full. Order the tanks from the least water to the most water.

Check the whole. Each number is a part of the same stated whole, a full tank of $200$ liters, so the greater part means more water. Choose percents as the common form.
Rewrite $\frac{13}{20}$ . Its denominator, $20$ , is not zero. Multiply the numerator and the denominator by $5$ to make the denominator $100$ . This multiplies the fraction by $\frac{5}{5} = 1$ , so the value stays the same.
$\frac{13}{20} = \frac{13 \times 5}{20 \times 5} = \frac{65}{100} = 65\%$
Rewrite $0.7$ . Seven tenths equals $70$ hundredths, and percent means per hundred.
$0.7 = 0.70 = \frac{70}{100} = 70\%$
Compare. All three percents count hundredths of the same full tank, so the greater count is the greater part.
$65\% < 68\% < 70\%$
Check in liters. Use $68\% = 0.68$ , and multiply each part by the whole, $200$ liters. The whole is the same positive number each time, so the amounts keep the order of the parts: $130$ liters, then $136$ liters, then $140$ liters.
$\frac{13}{20} \times 200 = 130, \quad 0.68 \times 200 = 136, \quad 0.7 \times 200 = 140$

Answer

From the least water to the most, the order is Tank A ( $\frac{13}{20}$ full, $130$ liters), Tank C ( $68\%$ full, $136$ liters), then Tank B ( $0.7$ full, $140$ liters).

Decimals are often the quickest common form, but not every decimal stops. Every number in this lesson is rational. That means you can write it as a fraction $\frac{p}{q}$ , where $p$ and $q$ are integers (whole numbers or their opposites) and $q \neq 0$ . The decimal form of a rational number either terminates, which means it stops, like $0.375$ , or repeats, which means a digit or a block of digits repeats without end, like $\frac{1}{3} = 0.333\ldots = 0.\overline{3}$ . A decimal that neither terminates nor repeats belongs to an irrational number, and this lesson uses none.

You cannot write out every digit of a repeating decimal, so you might round it. If you round $\frac{1}{3}$ to $0.33$ , you get a value that is close to $\frac{1}{3}$ but not equal to it, so you write $\frac{1}{3} \approx 0.33$ . That small gap can hide the difference between two numbers that are very close together, so a close comparison needs exact values. The next example shows how.

Worked example

Order four numbers when one decimal repeats

Order these numbers from least to greatest: $\frac{2}{3}$ , $0.66$ , $67\%$ , $\frac{17}{25}$ .

Choose a common form. The denominators $3$ and $25$ are not zero, and with no context all four numbers are parts of the same whole, $1$ . Start with decimals. Multiplying the numerator and the denominator of $\frac{17}{25}$ by $4$ keeps its value, because $\frac{4}{4} = 1$ , and $67\%$ means $67$ hundredths.
$\frac{17}{25} = \frac{17 \times 4}{25 \times 4} = \frac{68}{100} = 0.68, \quad 67\% = \frac{67}{100} = 0.67$
Rewrite $\frac{2}{3}$ by dividing $2$ by $3$ . Every step of the division leaves the remainder $2$ again, so the digit $6$ repeats without end. Write it exactly with a bar. Rounding gives $\frac{2}{3} \approx 0.67$ , but that value is not exact. It would make $\frac{2}{3}$ look equal to $67\%$ .
$\frac{2}{3} = 2 \div 3 = 0.666\ldots = 0.\overline{6}$
$\frac{2}{3}$ is very close to both $0.66$ and $0.67$ , so compare those three exactly, as fractions with the same denominator, $300$ . Multiply $\frac{2}{3}$ by $\frac{100}{100}$ , and multiply $\frac{66}{100}$ and $\frac{67}{100}$ by $\frac{3}{3}$ . Each multiplier equals $1$ , so no value changes. Here $66 \times 3 = 198$ and $67 \times 3 = 201$ .
$\frac{2}{3} = \frac{200}{300}, \quad 0.66 = \frac{66}{100} = \frac{198}{300}, \quad 0.67 = \frac{67}{100} = \frac{201}{300}$
Compare. The three fractions have the same positive denominator, so their numerators count pieces of the same size, and the greater numerator is the greater fraction.
$\frac{198}{300} < \frac{200}{300} < \frac{201}{300}, \quad \text{so} \quad 0.66 < \frac{2}{3} < 0.67$
Place the last number, and write $\frac{2}{3}$ in its exact decimal form. $0.68$ is $68$ hundredths and $0.67$ is $67$ hundredths, so $0.68$ is greater than $0.67$ . That makes $0.68$ the greatest of the four.
$0.66 < 0.\overline{6} < 0.67 < 0.68$
Write the order with each number in its original form. Here $0.\overline{6}$ is $\frac{2}{3}$ , $0.67$ is $67\%$ and $0.68$ is $\frac{17}{25}$ .
$0.66 < \frac{2}{3} < 67\% < \frac{17}{25}$

Answer

From least to greatest, the numbers are $0.66$ , $\frac{2}{3}$ , $67\%$ and $\frac{17}{25}$ .

Takeaway. A fraction, a decimal and a percent can look very different and still be close in value, or even equal. So compare values, not the digits you see. Rewrite every number in one common form, compare, and then write the order with each number in its original form. Keep the conditions in mind: no denominator is zero, the fractions and percents are parts of the same whole, and a close comparison needs exact values, with a bar on each repeating decimal and $\approx$ on each rounded value.