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O Level · International Mathematics 0607

Converting Fractions, Decimals & Percentages

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Summary — What This Chapter Covers

  • Converting percentage → decimal (÷100) and decimal → percentage (×100)
  • Converting a decimal → fraction using place value (tenths, hundredths, thousandths...)
  • Converting a percentage → fraction by writing it over 100
  • Converting a fraction → decimal using known facts, powers of 10, or short division
  • Converting a fraction → percentage (fraction → decimal → ×100)
  • Understanding recurring decimals and writing them with dot notation
  • Turning a recurring decimal into a fraction using the algebra "x = ..." method
  • Ordering a mix of fractions, decimals, and percentages from smallest to largest
  • Using inequality symbols (<, >, ≤, ≥, =, ≠) to compare values

1. The Basic FDP Conversions

Percentage ⇄ Decimal

Think of "per cent" literally — it means "per hundred". So a percentage is already a fraction out of 100 in disguise. That's why moving between percentage and decimal is just about shifting the decimal point two places.

Percentage → Decimal Divide by 100 (move digits two places right)
Worked Examples
6% ÷ 100 = 0.06
40% ÷ 100 = 0.4
350% ÷ 100 = 3.5 (percentages over 100% just mean "more than one whole")
0.2% ÷ 100 = 0.002
Decimal → Percentage Multiply by 100 (move digits two places left, add a % sign)
Worked Examples
0.35 × 100 = 35%
1.32 × 100 = 132%
0.004 × 100 = 0.4%
Why this works
Multiplying or dividing by 100 shifts every digit two "columns" — because 100 has two zeros. It's exactly the same skill as ×10 shifting one column, just doubled up.
Practice Question 1

Convert 0.045 to a percentage, and convert 215% to a decimal.

Decimal → Fraction

Every decimal is secretly a fraction with a denominator that's a power of 10 (10, 100, 1000, ...). The trick is just figuring out which power of 10 to use — and that depends on how many digits come after the decimal point.

Rule If a decimal has n digits after the decimal point, write those digits (as a whole number) over 10n. Then simplify if possible.
DecimalDigits after pointWrite overFraction
0.31103/10
1.111011/10
0.0721007/100
30.0121003001/100
0.51331000513/1000
0.0007410 0007/10 000
Watch out
Count digits after the decimal point only — not the whole number. 30.01 has two digits after the point (0 and 1), so it goes over 100, giving 3001/100 (the whole "3001" is the numerator, including the 30 whole part combined with 01).

Two special recurring decimals are worth memorising outright rather than working out each time:

0.333... (0.3 recurring) = 1/3   |   0.666... (0.6 recurring) = 2/3

And remember — any whole number can be written as a fraction just by putting it over 1. For example, 5 = 5/1.

Practice Question 2

Write 0.048 as a fraction in its simplest form.

Percentage → Fraction

Rule Write the percentage number over 100, then simplify.
Worked Example
37% is already 37/100 — and since 37 and 100 share no common factors, that's the final answer.
Practice Question 3

Write 65% as a fraction in its simplest form.

Fraction → Decimal

There are two routes here, and knowing which to use saves you loads of time in an exam.

Route A — Recognise it (fastest)

Certain fraction-to-decimal facts come up constantly. Learn these so well you can recall them instantly:

FractionDecimal
1/20.5
1/40.25
3/40.75
1/50.2
1/100.1
1/30.333... (0.3 recurring)
2/30.666... (0.6 recurring)

Route B — Rewrite over a power of 10

If the denominator can be scaled up (or down) to 10, 100, or 1000, do that — it's much faster than long division.

Worked Examples
3/5 = 6/10 (multiply top & bottom by 2) = 0.6
7/20 = 35/100 (multiply top & bottom by 5) = 0.35
1/500 = 2/1000 (multiply top & bottom by 2) = 0.002

Route C — Short division (when nothing else works)

Divide the numerator by the denominator directly — adding zeroes after the decimal point as needed.

Worked Example: 5/8
Do 5 ÷ 8 using short division. Add zeroes: 5.000...
8 goes into 50 → 6 remainder 2; into 20 → 2 remainder 4; into 40 → 5 remainder 0.
Result: 5/8 = 0.625
Examiner Tip
A calculator can always be used to check your fraction-decimal conversions — even on questions that say "show your working without a calculator." Just don't skip the written method, since that's where the marks are.
Practice Question 4

Convert 7/8 to a decimal (a) using the "scale to a power of 10" method if possible, and (b) using short division if not.

Fraction → Percentage

Rule Change the fraction into a decimal first, then multiply by 100.
Worked Example
4/5 = 8/10 = 0.8 as a decimal
0.8 × 100 = 80%
Practice Question 5

Convert 9/20 to a percentage.

2. Recurring Decimals

What Are They?

When you turn a fraction into a decimal, one of two things always happens:

  • The decimal stops — this is called a terminating decimal. Example: 1/4 = 0.25
  • The decimal repeats a pattern forever — this is called a recurring decimal. Example: 32/99 = 0.32323232...

Writing out "0.32323232..." forever is impractical, so mathematicians use a shorthand: a dot above the digit that repeats. If more than one digit repeats, you put a dot above the first and last digit of the repeating block.

Full decimalDot notationWhat repeats
0.3333...0.3 (dot over 3)just the "3"
0.121212...0.12 (dots over 1 and 2)the block "12"
0.325632563256...0.32563 (dots over first "2" and last "3" of the block)the block "2563"
Important fact
Recurring decimals are rational numbers — not irrational! Even though they go on forever, they follow a completely predictable repeating pattern, which is exactly why they can always be converted back into a fraction. (Irrational numbers, like π, never repeat and never terminate — that's the real difference.)

Turning a Recurring Decimal Into a Fraction

This is the classic "algebra trick" question. The idea: if you multiply a recurring decimal by the right power of 10, you can line up two versions of it so that when you subtract one from the other, the infinite recurring tail cancels out completely — leaving you with a normal, solvable equation.

Method — 4 Steps Step 1: Let x = the recurring decimal (write out several repeats to see the pattern clearly)
Step 2: Multiply both sides by 10, 100, 1000... repeatedly until two lines have matching recurring tails
Step 3: Subtract the two matching lines — the recurring part cancels to zero
Step 4: Divide to solve for x, and simplify the fraction if possible
Worked Example: Convert 0.35353535... to a fraction
Step 1: x = 0.35353535...
Step 2: 10x = 3.5353535... and 100x = 35.353535...
Notice: x and 100x both have "35" repeating right after the decimal point — that's our matching pair.
Step 3: 100x − x = 35.353535... − 0.35353535... → 99x = 35
Step 4: x = 35/99 (can't simplify further)
Worked Example: Convert 0.307307307... to a fraction
Step 1: x = 0.307307307307...
Step 2: Try 10x, 100x, 1000x. The repeating block is 3 digits long ("307"), so 1000x lines it up: 1000x = 307.307307307...
Step 3: 1000x − x = 307.307307307... − 0.307307307... → 999x = 307
Step 4: x = 307/999 (this can't be simplified — final answer)
Common mistake
Students often multiply by the wrong power of 10. The rule of thumb: if the repeating block is n digits long, multiply by 10ⁿ to shift the decimal point exactly one full repeating block to the right. A 1-digit repeat → ×10. A 2-digit repeat → ×100. A 3-digit repeat → ×1000.
Practice Question 6

Write 0.1666... (0.16 recurring, where only the 6 repeats) as a fraction in its lowest terms.

3. Ordering Fractions, Decimals & Percentages

Ordering Fractions Only

Rule Rewrite all fractions over their lowest common denominator (LCD), then just compare the numerators.
Worked Example
Order 3/5, 1/2, 13/20, 7/12 — smallest to largest.
LCD of 5, 2, 20, 12 is 60.
Rewrite: 3/5 = 36/60,   1/2 = 30/60,   13/20 = 39/60,   7/12 = 35/60
Order numerators smallest → largest: 30, 35, 36, 39
Answer: 1/2, 7/12, 3/5, 13/20

Ordering a Mixture of Fractions, Decimals & Percentages

Rule Convert everything into decimals — decimals are the easiest format to compare digit-by-digit.
Worked Example — Order 0.31, 2/3, √0.09, 32%, 3⁻¹ (smallest first, no calculator)
Step 1 — Convert everything to decimals:
• 0.31 is already a decimal
• 2/3 = 0.666... = 0.6
• √0.09 = 0.3 (recognise from √9 = 3)
• 32% = 32 ÷ 100 = 0.32
• 3⁻¹ = 1/3 = 0.333... = 0.3 (a negative power means "1 over")
Step 2 — Line the decimals up in a column so the place values match:
0.31, 0.66..., 0.3, 0.32, 0.33...
Step 3 — Compare digit by digit (fill blank places with zeroes if it helps): 0.30, 0.31, 0.32, 0.33, 0.66
Step 4 — Order: 0.3, 0.31, 0.32, 0.33..., 0.66...
Step 5 — Convert back to original form:
Answer: √0.09, 0.31, 32%, 3⁻¹, 2/3
Handy technique
When comparing several decimals of different lengths, write them all to the same number of decimal places by padding with zeroes (e.g. 0.8 becomes 0.800). This makes side-by-side comparison foolproof — just compare like you would with whole numbers.
Worked Example — Order 7/8, 5/6, 0.8, 78% (smallest first)
7/8 = 0.875 (found by halving 7 three times: 7÷2=3.5, ÷2=1.75, ÷2=0.875)
5/6 = 0.8333... = 0.83 (by short division)
0.8 is already a decimal
78% = 0.78
Write to 3 d.p.: 0.875, 0.833, 0.800, 0.780
Answer: 78% < 0.8 < 5/6 < 7/8
Practice Question 7

Without a calculator, write these in order starting with the smallest: 1/3, 2/5, 9/25, 4/15

Practice Question 8

Put these in order, smallest first, and connect them with < symbols: 0.6, 3/5, 58%, 0.6 recurring

Which Symbols Can I Use?

Instead of just listing values in order, exam questions often want you to link them using comparison symbols:

SymbolMeaning
>greater than
≥greater than or equal to
<less than
≤less than or equal to
=equal to
≠not equal to
Example 1/4 < 1/3 < 1/2   (reads: "one quarter is less than one third, which is less than one half")

What to Memorise

% → decimal
Divide by 100 (shift point 2 left)
Decimal → %
Multiply by 100 (shift point 2 right)
Decimal → fraction
n digits after the point → write over 10ⁿ, then simplify
% → fraction
Write the number over 100, then simplify
Fraction → decimal
Recognise it, scale to a power of 10, or use short division
Fraction → %
Fraction → decimal first, then ×100
1/3 and 2/3
1/3 = 0.3 recurring  |  2/3 = 0.6 recurring — learn by heart
Terminating decimal
A decimal that stops (e.g. 0.25)
Recurring decimal
A decimal that repeats forever in a pattern — shown with dot notation
Recurring → fraction
Let x = decimal → multiply by 10ⁿ (n = length of repeat) → subtract → solve
Rational vs Irrational
Recurring decimals ARE rational (predictable pattern); irrational numbers never repeat or terminate
Ordering mixed FDP
Convert everything to decimals first, pad with zeroes, then compare column by column

Concepts Checklist

Exam Tips & Common Mistakes

Mistake: Miscounting decimal places
When converting a decimal to a fraction, students often miscount the digits after the decimal point, especially with numbers like 30.01 (2 digits after the point → over 100) versus 0.513 (3 digits → over 1000). Always count only what's after the decimal point.
Mistake: Wrong multiplier for recurring decimals
The power of 10 you multiply by must match the length of the repeating block, not just be "10 for simple, 100 for everything else." A 1-digit repeat needs ×10; a 3-digit repeat (like 0.307) needs ×1000. Getting this wrong means the recurring parts won't cancel when you subtract.
Mistake: Forgetting to simplify
A fraction answer like 15/90 is correct but incomplete — always check for a common factor and simplify to lowest terms (15/90 → 1/6). Examiners often award the final mark only for the simplified form.
Mistake: Comparing decimals of different lengths incorrectly
Don't assume 0.8 > 0.75 just by "eyeballing" — always pad to equal decimal places first (0.80 vs 0.75) so you're comparing like-for-like columns.
What examiners look for
In "show your working" questions, examiners want to see the conversion process explicitly — not just a final answer. For ordering questions, show each value converted to a decimal before writing the final order. For recurring decimal-to-fraction questions, show all four algebra steps (x = ..., multiply, subtract, divide) even if you can do it in your head — the method marks matter as much as the answer.
Calculator reminder
Even on "non-calculator" papers, you're usually allowed to use a calculator to check your final answer once you've shown the written method — this is a great way to catch careless slips before submitting.
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  • 3. Ordering Fractions, Decimals & Percentages
  • Exam Tips & Common Mistakes
  • Ordering a Mixture of Fractions, Decimals & Percentages
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