Fractions, Decimals and Percentages: Convert and Compare

By the end of this lesson you will be able to move between fractions, decimals and percentages, compare fractions without long division, and work with mixed numbers in one-line sums.

Already confident? Skip to practice: Fractions and Decimals quiz

Before this: you should know factors, multiples and prime numbers. If they feel shaky, try the Number System quiz first.

Three ways to write the same amount

A fraction, a decimal and a percentage are three ways of writing the same amount.

The three rules that connect them:

• Fraction to decimal: divide the top number by the bottom number.
• Decimal to percentage: multiply by 100 (move the decimal point two places to the right).
• Percentage to fraction: write the number over 100, then simplify.

Some words used below:

• In 3/8, the top number 3 is the numerator and the bottom number 8 is the denominator.
• A mixed number has a whole part and a fraction part. 2 1/3 means "two and one-third".
• A fraction is in its lowest terms when the top and bottom have no common factor except 1. For example, 6/8 in lowest terms is 3/4.
• A recurring decimal has a digit, or a group of digits, that repeats for ever. In this lesson, "…" after a decimal means the last digit keeps repeating: 0.333… is 0.3333333 and so on. When a group of digits repeats, we say which, as in 0.4545… (45 repeats).

Worked examples

Read each example slowly. The first two show every step. In the later ones, some steps are left for you, with the answer just below.

Example 1: Fraction to decimal, and back

Question: Write 7/40 as a decimal. Then write 0.36 as a fraction in its lowest terms.

1. Look at the denominator: 40 = 2 × 2 × 2 × 5. It has only 2s and 5s, so the decimal will stop (terminate).
2. Make the denominator a power of 10. Multiply 40 by 25 to get 1000, and multiply the top by 25 too: 7/40 = 175/1000.
3. 175/1000 means 175 thousandths, so 7/40 = 0.175.
4. Now the other way. 0.36 has two decimal places, so write it over 100: 0.36 = 36/100.
5. Divide the top and the bottom by their common factor 4: 36/100 = 9/25.

The principle: a decimal is a fraction whose denominator is 10, 100, 1000…; a fraction in lowest terms stops as a decimal only when its denominator has no prime factors other than 2 and 5.

That is why 7/12 never stops: 12 = 2 × 2 × 3, and the 3 makes it recur (7/12 = 0.58333…).

Example 2: Recurring decimal to fraction

Question: Write 0.333… and 0.1666… as fractions.

A bar over a digit is another way to show that it repeats: $0.\overline{3}$ means 0.333… and $0.1\overline{6}$ means 0.1666…

Part (a): 0.333…

1. Let x = 0.333…
2. Multiply by 10, because one digit repeats: 10x = 3.333…
3. Subtract the first line from the second. The repeating parts cancel: 10x − x = 3.333… − 0.333…, so 9x = 3.
4. Divide by 9: x = 3/9 = 1/3.

Part (b): 0.1666… (the 1 does not repeat; only the 6 does)

1. Let x = 0.1666…
2. Move the non-repeating digit to the left of the point: 10x = 1.666…
3. Move one full repeat further: 100x = 16.666…
4. Subtract: 100x − 10x = 16.666… − 1.666…, so 90x = 15.
5. Divide by 90: x = 15/90 = 1/6.

The principle: multiply so that two copies of the number have the same repeating tail, then subtract to make the tail disappear.

Shortcut that comes from this: on top, write (the digits after the decimal point, with the repeating group written once) minus (the digits that do not repeat). On the bottom, write one 9 for each repeating digit, then one 0 for each non-repeating digit after the point. For 0.1666…: (16 − 1)/90 = 15/90 = 1/6. For 0.4545… (45 repeats): 45/99 = 5/11.

Example 3: Which fraction is bigger?

Question: Arrange 4/7, 5/9 and 7/12 from smallest to largest.

These are close, so guessing is risky. Compare two at a time by cross-multiplying: multiply each numerator by the other fraction's denominator.

1. Compare 5/9 and 4/7: 5 × 7 = 35 and 4 × 9 = 36. Since 35 is less than 36, 5/9 is smaller than 4/7.
2. Compare 4/7 and 7/12: 4 × 12 = 48 and 7 × 7 = 49. Since 48 is less than 49, 4/7 is smaller than 7/12.
3. Your turn: put the three in order from smallest to largest.

Check: 5/9, 4/7, 7/12. In decimals they are 0.555…, about 0.571 and 0.583…, which agrees.

The principle: to compare a/b with c/d, compare a × d with c × b. If a × d is bigger, then a/b is the bigger fraction. Turning each fraction into a decimal also works, but it is slower.

Example 4: Adding and subtracting mixed numbers

Question: Find the value of 2 1/3 + 1 3/4 − 1 5/6.

1. Change each mixed number into an improper fraction (top bigger than bottom): 2 1/3 = 7/3, 1 3/4 = 7/4, 1 5/6 = 11/6.
2. The LCM of 3, 4 and 6 is 12. Rewrite each fraction with 12 at the bottom: 7/3 = 28/12, 7/4 = 21/12, 11/6 = 22/12.
3. Your turn: work out 28/12 + 21/12 − 22/12, and write the answer as a mixed number in lowest terms.

Check: 28 + 21 − 22 = 27, so the answer is 27/12 = 9/4 = 2 1/4.

Multiplying and dividing do not need a common denominator:

• To multiply, multiply the tops and multiply the bottoms: 2/3 × 9/10 = 18/30 = 3/5.
• To divide, turn the second fraction upside down and multiply: 1 1/2 ÷ 3/4 = 3/2 × 4/3 = 2.

The principle: you can only add or subtract fractions that have the same denominator, so change them to a common one first; change mixed numbers to improper fractions before you multiply or divide.

Example 5: Fractions and percentages

Question: Write 7/8 and 2/3 as percentages. Then write 45% and 83⅓% as fractions.

1. 7/8 is seven lots of 1/8, and 1/8 = 12.5%. So 7/8 = 7 × 12.5% = 87.5%.
2. 2/3 × 100 = 200/3 = 66 2/3, so 2/3 = 66⅔%.
3. 45% = 45/100. Divide the top and the bottom by 5: 9/20.
4. Your turn: 83⅓% is five times 16⅔%. Which fraction is 16⅔%? Use it to write 83⅓% as a fraction.

Check: 16⅔% = 1/6, so 83⅓% = 5/6.

The principle: to turn a fraction into a percentage, multiply by 100; to go back, divide by 100. If you know the table below by heart, you can often skip the calculation.

Example 6: A household budget

Question: Sunita earns ₹24,000 a month. She spends 3/8 of it on rent and 1/4 of it on food. What fraction of her salary is left, and how many rupees is that? She puts 2/3 of what is left into a post office savings account. How much does she save?

1. Rent and food together: 3/8 + 1/4 = 3/8 + 2/8 = 5/8 of her salary.
2. Left: 1 − 5/8 = 3/8 of her salary, which is 37.5%.
3. In rupees: 3/8 × 24,000 = 3 × 3,000 = ₹9,000.
4. Your turn: find 2/3 of ₹9,000.

Check: 2/3 × 9,000 = 2 × 3,000 = ₹6,000.

The principle: "of" means multiply: 2/3 of ₹9,000 is 2/3 × 9,000. Find the fraction first and the rupees last; one multiplication at the end is less work than several in the middle. Watch whose fraction it is: here the 2/3 is of what was left, not of her whole salary.

Revision: conversions worth knowing by heart

Learn these and many percentage, ratio and simplification questions become one-step questions.

Two patterns save memory:

• Multiply a row to get its neighbours. 1/8 = 12.5%, so 3/8 = 3 × 12.5% = 37.5%.
• Ninths and elevenths repeat in a pattern. 2/9 = 0.222…, 5/9 = 0.555…; 2/11 = 0.1818… (18 repeats), 3/11 = 0.2727… (27 repeats). For elevenths, the repeating pair is 9 times the numerator, written as two digits: 1 × 9 = 09, so 1/11 = 0.0909… (09 repeats); 2 × 9 = 18; 3 × 9 = 27.

Common mistakes

• Turning a fraction upside down. 5/8 is 0.625, not 1.6 (that is 8/5). The top number is divided by the bottom number.
• Putting the decimal point in the wrong place. 0.85 = 85/100 = 17/20, not 17/2 or 17/200. Count the decimal places: two places means hundredths.
• Using 99 when only part of the decimal repeats. 0.1666… is 15/90 = 1/6, not 16/99. The 9s count only the repeating digits; the 0s count the digits that do not repeat.
• Adding tops and bottoms. 1/2 + 1/3 is not 2/5. Use a common denominator: 3/6 + 2/6 = 5/6.
• Forgetting that "of" comes before ÷ and ×. "Of" is a multiplication, but it is the O in BODMAS (Brackets, Of, Division, Multiplication, Addition, Subtraction): it is done before any ÷ or × written as a sign. In 3/4 ÷ 1/2 of 1/3, work out "1/2 of 1/3" = 1/6 first. Then 3/4 ÷ 1/6 = 3/4 × 6 = 4 1/2. Dividing first gives 1/2, which is wrong.

Next

When these conversions feel easy, move on to percentages. Read the Percentage formula cheat sheet, then try the Percentage quiz.

Ready to test this lesson? Take the Fractions and Decimals quiz

How this was made: Drafted with AI assistance; every formula and worked figure recomputed independently; spot-checked by the editor on 23 October 2026.