Number · KS3

Fractions, Decimals, Percentages

¾, 0.75 and 75% are the same number wearing three outfits. Once you can change outfits without thinking, half of KS3 Number becomes easy: comparing, ordering, discounts, probabilities and test scores all come down to picking the most convenient form.

National curriculum KS3: interchange terminating decimals and their corresponding fractions; define percentage as "number of parts per hundred"; interpret fractions and percentages as operators; recurring decimals at GCSE

1 · One amount, three names

Here is a square cut into 100 equal pieces. Click or drag across it to shade some. The three boxes underneath describe the same shaded amount, just in three notations. Nothing about the quantity changes; only the way it is written.

Fraction
Decimal
Percentage

Each form says something slightly different about the same amount:

2 · The six conversion routes

Three forms give six directions. Four are one-step; two need a little more. Percentages always go through 100.

F → DDivide top by bottom. ⅜ = 3 ÷ 8 = 0.375. (A fraction is a division.) If the denominator is 10, 100 or 1000, just read the place value: 7/100 = 0.07.
D → FSay it out loud, then simplify. 0.375 is "375 thousandths" → 375/1000 → ÷25 → 15/40 → ÷5 → 3/8. The number of decimal places tells you the denominator: 1 d.p. → tenths, 2 d.p. → hundredths, 3 d.p. → thousandths.
D → PMultiply by 100 (move the point 2 places right). 0.375 → 37.5%. 1.2 → 120%.
P → DDivide by 100 (point 2 places left). 37.5% → 0.375. 4% → 0.04 (not 0.4!).
P → FWrite over 100, simplify. 37.5% → 37.5/100 → ×2 → 75/200 → ÷25 → 3/8. Whole-number percentages are easier: 40% = 40/100 = 2/5.
F → PTwo options. If the denominator goes into 100, scale it up: ⅗ = 60/100 = 60%. Otherwise go via the decimal: ⅜ = 0.375 = 37.5%.

Type anything in any box and the other two update, with the working shown.

Fraction
Decimal
Percentage

Ones to know by heart

These come up so often that converting them should be instant. Notice the families: halves, quarters, eighths double each other; fifths and tenths; thirds.

FractionDecimalPercentageFamily

3 · Simplifying and equivalent fractions

75/100 and 3/4 are the same number because you can divide top and bottom by the same thing (25) without changing the value. Dividing both by their highest common factor gets you to the simplest form in one go (see the prime factorisation page for HCF). The bar below shows why: cutting the same length into fewer, bigger pieces doesn't change the length.

4 · Decimals that never end

Divide 1 by 3 and you get 0.333…, with the 3s going on forever. That's a recurring decimal, written with a dot over the repeating digit: 0.3̇. A fraction gives a terminating decimal only when its simplified denominator has no prime factors other than 2 and 5 (the primes in 10). Anything else recurs.

Decimal
Denominator's primes
Verdict

Why does the remainder pattern repeat? When you divide by 7, the only possible remainders are 0 to 6. After at most 7 steps a remainder must come round again, and from then on the digits repeat. That's why the repeating block for /7 is six digits long (142857) and never longer.

The famous one. 0.9̇ (0.999… forever) is exactly 1, not "just under". If ⅓ = 0.3̇, then multiply both sides by 3: 1 = 0.9̇. There is no gap between them to fit another number in.

5 · Comparing and ordering

Which is bigger, ⅝ or 0.6 or 62%? You can't compare across forms. Convert everything to one form (decimals are usually easiest), then put them on a line.

6 · Where it shows up

Shopping

"20% off" means pay 80%, so multiply by 0.8. "⅓ off" means pay ⅔. Comparing a ⅓-off deal with a 30%-off one needs a conversion: ⅓ = 33.3%, so ⅓ off is the better discount.

Test scores

38 out of 50 is 38/50 = 76/100 = 76%. Percentages exist so that scores out of 50, 60 and 80 can be compared on one scale.

Probability

A probability is a fraction (1 in 4), a decimal (0.25) or a percentage (25%) interchangeably. Weather apps use %, bookmakers use fractions, statisticians use decimals.

Computing

A progress bar at 0.75 is drawn 75% wide. Opacity 0.5 is 50% transparent. CSS accepts both width: 0.75-style numbers and 75%, and converts for you.

Money and interest

4.5% interest is a multiplier of 1.045 per year. Converting the percentage to a decimal multiplier is the first step of every compound-interest question.

Cooking and measuring

¾ of a cup, 0.75 litres, 75 cl: recipes switch form depending on the tool. Half of ¾ is ⅜, which is easier to see as a fraction than as 0.375.

Worked examples

Write 0.45 as a fraction in its simplest form
Say it. 0.45 is "45 hundredths": 45/100.
Simplify. HCF of 45 and 100 is 5. 45 ÷ 5 = 9, 100 ÷ 5 = 20. So 9/20.
Check. 9 ÷ 20 = 0.45 ✓
Write 7/8 as a percentage
8 doesn't go into 100 neatly, so go via the decimal: 7 ÷ 8 = 0.875.
× 100: 87.5%. (Or: ⅛ = 12.5%, so ⅞ = 7 × 12.5% = 87.5%.)
Order from smallest to largest: 0.7, ¾, 72%, ⅔
Convert to decimals: 0.7, 0.75, 0.72, 0.666…
Order the decimals: 0.666… < 0.7 < 0.72 < 0.75.
Write the answer in the original forms: ⅔, 0.7, 72%, ¾. (Examiners want the originals, not your decimals.)
Exam-style (GCSE): write 0.4̇5̇ as a fraction
Let x = 0.454545…
Two digits repeat, so multiply by 100: 100x = 45.454545…
Subtract: 100x − x = 45.4545… − 0.4545… → 99x = 45.
Solve and simplify: x = 45/99 = 5/11. Check with the long-division tool above: 5/7 was a six-digit cycle, try 5/11.

Quick check

What you've learned

A fraction, a decimal and a percentage are three notations for one amount, and the 100-square shows all three at once. Fraction to decimal is a division; decimal to fraction is reading the place value then simplifying by the HCF; percentages are hundredths, so moving between percentage and decimal is a shift of two places. Fractions whose denominators contain primes other than 2 and 5 give recurring decimals, which repeat because the remainders must. To compare mixed forms, convert to one form first, then answer in the originals.