Comparing and Ordering Fractions, Decimals and Percentages

1. Comparing common fractions

Comparing fractions is easy in two special cases. If the denominators are the same, the fraction with the bigger numerator is bigger: $\frac{5}{8}$ > $\frac{3}{8}$ because 5 pieces of the same size are more than 3. If the numerators are the same, the fraction with the smaller denominator is bigger: $\frac{2}{3}$ > $\frac{2}{5}$ because thirds are bigger pieces than fifths.

When both numerator and denominator differ, make them the same kind of piece first. Find the LCM of the denominators (the lowest common denominator), change each fraction to an equivalent fraction with that denominator, and then compare the numerators.

1. 1Find the LCM of the denominators. For $\frac{5}{6}$, $\frac{3}{4}$ and $\frac{7}{8}$: multiples of 8 are 8, 16, 24…; 24 is the first one that 6 and 4 also divide. LCM = 24.
2. 2Rewrite each fraction over 24: $\frac{5}{6}$ = $\frac{20}{24}$, $\frac{3}{4}$ = $\frac{18}{24}$, $\frac{7}{8}$ = $\frac{21}{24}$.
3. 3Compare the numerators: 21 > 20 > 18.
4. 4Write the answer with the ORIGINAL fractions: in descending order, $\frac{7}{8}$, $\frac{5}{6}$, $\frac{3}{4}$.

▶ Once every bar is cut into 24 equal pieces, you only need to count the shaded pieces.

2. Comparing decimals up to thousandths

Decimals are compared using place value, just like whole numbers. Write the numbers one under the other with the decimal points in a line. Fill empty places with zeros so every number has the same number of decimal places — adding zeros at the end of a decimal does not change its value (0.99 = 0.990).

Now compare from the left, one place at a time. The first place where the digits differ decides the answer. For 0.990 and 0.977 the ones (0) and tenths (9) are the same, but in the hundredths place 9 > 7. So 0.99 > 0.977, even though 0.977 has more digits.

▶ Numbers further right on the number line are bigger. Notice that 0.99 beats 0.977 even though it has fewer digits.

3. Ordering a mixture of fractions, decimals and percentages

When numbers come in different forms, you cannot compare them directly. Change them all into one form. Decimals or percentages are usually the easiest, because they are already "out of" 10, 100 or 1000. Your benchmark facts ($\frac{1}{4}$ = 25%, $\frac{3}{8}$ = 37.5%, $\frac{1}{3}$ ≈ 33.3%) make this quick.

Look at the percentage column: 37.5% < 38% < 83.2%. So from least to largest the order is $\frac{3}{8}$, 38%, 0.832. Notice that $\frac{3}{8}$ and 38% are very close — only half a percent apart — so a careful conversion matters.

▶ Even when two numbers look alike, putting them in the same form shows which is bigger.

4. Symbols and words for order

We use < for "is less than" and > for "is greater than". The open mouth of the symbol always faces the bigger number: 0.4 < 0.45 and $\frac{7}{8}$ > $\frac{5}{6}$. If two numbers are equal in value, such as $\frac{2}{5}$ and 0.4, we use =.

• Ascending order (least to greatest): 0.098, 0.123, 0.985 — like climbing up stairs.
• Descending order (greatest to least): 0.985, 0.123, 0.098 — like coming down stairs.
• Check your answer: the first number in ascending order must be the smallest, and every number must appear exactly once.

Worked examples

Try each one yourself first, then open the solution and compare every step.