Fractions, Decimals and Percentages

Mathematics (Core) revision notes · Cambridge (CIE) · 0580 · Core

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1. Percentages and decimals

Percentage to decimal

  • A percentage counts hundredths, so dividing by 100 recovers the decimal
  • Dividing by 100 shifts every digit two columns to the right
    • 8% becomes 0.08
    • 65% becomes 0.65
    • 240% becomes 2.4
    • 0.5% becomes 0.005
  • A percentage above 100 always gives a decimal above 1

Decimal to percentage

  • Multiplying by 100 reverses the process and shifts every digit two columns left
    • 0.47 becomes 47%
    • 2.06 becomes 206%
    • 0.009 becomes 0.9%
  • The per cent sign must be written, since 47 and 47% are different quantities

2. Decimals and fractions

Decimal to fraction

  • The number of decimal places fixes the denominator: one place gives tenths, two gives hundredths, and n places gives 10ⁿ
    • 0.9 is 9/10
    • 0.23 is 23/100
    • 0.041 is 41/1000
  • Write the digits after the point as the numerator, then simplify
    • 0.44 is 44/100, which cancels to 11/25
  • A whole number is written as a fraction by placing it over 1

Percentage to fraction

  • Every percentage is already a number of hundredths, so it goes straight over 100 and is then cancelled
    • 44% is 44/100, which cancels to 11/25

Fraction to decimal

  • Where the denominator divides into a power of ten, scale up to that power and read the decimal straight off
    • 25 goes into 100 four times, so 7/25 becomes 28/100, or 0.28
  • Otherwise divide the numerator by the denominator
  • Denominators that are powers of 2 can be reached by repeated halving
    • 3/8: halving 3 gives 1.5, then 0.75, then 0.375
  • Some conversions are worth knowing on sight: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2 and 1/10 = 0.1
  • 1/3 = 0.333… and 2/3 = 0.666…, where the digit repeats for ever
  • Any other fraction is a short division with zeros written after the point: 5/8 is 5.000 ÷ 8 = 0.625

Fraction to percentage

  • Convert to a decimal first, then multiply by 100
    • 9/20 scales to 45/100, which is 0.45, and therefore 45%

Worked example

The four conversions

Write 0.36 as a fraction in its simplest form, and write 3/8 as a decimal and as a percentage.

Solution:

  • 0.36 has two decimal places, so it is 36 hundredths: 0.36 = 36/100
  • Cancel by 4: 36/100 = 9/25
  • For 3/8, divide the numerator by the denominator: 3 ÷ 8 = 0.375
  • To reach a percentage, multiply the decimal by 100: 0.375 × 100 = 37.5%
  • Check the other way round: 37.5 ÷ 100 = 0.375, and 0.375 × 8 = 3

3. Ordering fractions, decimals and percentages

Ordering fractions only

  • Rewrite every fraction over the lowest common denominator, then compare numerators
    • For 2/3, 5/8, 7/12 and 3/4 the lowest common denominator is 24
    • These become 16/24, 15/24, 14/24 and 18/24
    • Ascending order is therefore 7/12, 5/8, 2/3, 3/4
  • Give the answer in the form the values were originally written in

Ordering a mixed list

  • Convert every value to a decimal, since decimals are the one form that can be compared column by column
    • To order 0.31, 2/3, √0.09, 32% and 3⁻¹: 0.31 is already a decimal, 2/3 = 0.666…, √0.09 = 0.3, 32% = 0.32 and 3⁻¹ = 1/3 = 0.333…
  • Pad the decimals to a common number of places before comparing
The same five decimals written in a column with their points lined up, then compared one digit at a time: the tenths digits show that 0.66… is the largest, and the hundredths digits 0, 1, 2 and 3 then order the other four
Source: Ordering FDP by Save My Exams
  • Order the decimals, then rewrite each in its original form

Comparison symbols

  • A chain of values can be joined with the inequality symbols rather than listed
    • 1/5 < 1/4 < 1/3
  • Recall that > is greater than, < is less than, ⩾ is greater than or equal to, and ⩽ is less than or equal to

Exam tip

Write every decimal, to enough places

Writing every value in the list as a decimal, or every one as a percentage, earns one of the two marks even if the final order slips. The decimals must be precise enough to tell apart, and on Paper 3 some of the values usually agree to two places: 6/7 = 0.8571… only beats 0.857 at the fourth place. Write each to three or four places, then order them.

Worked example

Ordering a mixed list without a calculator

Write these values in ascending order: 0.62, 5/8, 63%, 3/5

Solution:

  • Convert each to a decimal
  • 0.62 is already a decimal
  • 5/8 by repeated halving: 5, 2.5, 1.25, 0.625
  • 63% divided by 100 gives 0.63
  • 3/5 is equivalent to 60/100, so 0.6
  • Pad to three places: 0.620, 0.625, 0.630, 0.600
  • Ascending order of the decimals: 0.600, 0.620, 0.625, 0.630
  • Rewritten in their original forms: 3/5, 0.62, 5/8, 63%