\(\dfrac{3}{8}\), 0.375 and 37.5% are the same number written three ways. Learn to switch between them, write fractions in their simplest form and (Extended) handle recurring decimals.
Percent means 'out of 100'. To convert, go through the decimal: fraction to decimal by dividing top by bottom, decimal to percentage by multiplying by 100. Always cancel a fraction to its simplest form by dividing top and bottom by their HCF.
Fig. 1: three of eight equal parts is the fraction 3/8; on the number line it sits at 0.375, which is 37.5 per cent of the whole
Section 01
Three ways to write one number.
In the lab, set the numerator and the denominator and read the same value as a fraction in its simplest form, a decimal and a percentage. Try denominators 3, 7 and 11 and watch the decimal start to repeat.
Proper fraction: top smaller than bottom, such as \(\dfrac{3}{8}\). Improper fraction: top at least as big as bottom, such as \(\dfrac{11}{4}\). Mixed number: a whole number and a proper fraction, such as \(2\dfrac{3}{4}\).
Fraction to decimal: divide the top by the bottom: \(\dfrac{3}{8} = 3 \div 8 = 0.375\).
Decimal to percentage: multiply by 100: \(0.375 = 37.5\%\). Percentage to decimal: divide by 100: \(4\% = 0.04\).
Simplest form: divide top and bottom by their HCF: \(\dfrac{18}{24} = \dfrac{3}{4}\) (HCF 6).
Worked example Fraction to decimal and percentage
Write \(\dfrac{3}{8}\) as (a) a decimal (b) a percentage.
(a) \(3 \div 8\): \(8\) into \(3.000\) gives \(0.375\).
(b) \(0.375 \times 100 = 37.5\).
Answer: (a) 0.375 (b) 37.5%
Worked example Percentage and decimal to fraction
Write (a) 45% (b) 0.35 as fractions in their simplest form.
(a) \(45\% = \dfrac{45}{100}\); HCF of 45 and 100 is 5.
Write 165% as a mixed number in its simplest form.
\(165\% = \dfrac{165}{100} = 1\dfrac{65}{100} = 1\dfrac{13}{20}\) (dividing 65 and 100 by 5).
Examiner trap
A percentage over 100 is fine: 165% is more than one whole. Do not 'correct' it to 16.5% or 1.65%; as a decimal it is 1.65.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Write 0.6 as a fraction in its simplest form.
Quick check+10 XP
Write \(\dfrac{7}{20}\) as a percentage.
Quick check+10 XP
Write \(\dfrac{17}{5}\) as a mixed number.
Quick check+10 XP
(Extended) Write \(0.\dot{3}\) as a fraction.
Section 03
Recurring decimals.
Extended. Some fractions give decimals that repeat for ever. A dot over a digit shows it repeats; dots over the first and last digits show a repeating block.
Written
Means
Fraction
\(0.\dot{4}\)
0.444 444...
\(\dfrac{4}{9}\)
\(0.1\dot{7}\)
0.177 777...
\(\dfrac{8}{45}\)
\(0.\dot{4}\dot{5}\)
0.454 545...
\(\dfrac{5}{11}\)
\(0.\dot{1}2\dot{3}\)
0.123 123...
\(\dfrac{123}{999} = \dfrac{41}{333}\)
Worked example One repeating digit (Extended)
Write \(0.1\dot{7}\) as a fraction in its simplest form.
Let \(x = 0.1777\dots\)
Multiply so the repeating parts line up: \(10x = 1.777\dots\) and \(100x = 17.777\dots\)
Subtract to remove the repeating tail: \(100x - 10x = 17.777\dots - 1.777\dots\), so \(90x = 16\).
\(x = \dfrac{16}{90} = \dfrac{8}{45}\).
Answer: \(\dfrac{8}{45}\)
Worked example A repeating pair (Extended)
Write \(0.\dot{2}\dot{7}\) as a fraction in its simplest form.
Let \(x = 0.272727\dots\); the block has 2 digits, so multiply by 100: \(100x = 27.2727\dots\)
Subtract: \(100x - x = 27\), so \(99x = 27\).
\(x = \dfrac{27}{99} = \dfrac{3}{11}\) (dividing by 9).
Answer: \(\dfrac{3}{11}\)
Your turn
(Extended) Write \(0.\dot{4}\) as a fraction.
\(x = 0.444\dots\), \(10x = 4.444\dots\), so \(9x = 4\) and \(x = \dfrac{4}{9}\).
Examiner trap
\(0.1\dot{7}\) (0.1777...) and \(0.\dot{1}\dot{7}\) (0.171 717...) are different numbers. Read the dots carefully before choosing what to multiply by, and show the subtraction: 'show that' questions give no marks for a calculator answer.
Stage 2 · Exam
Exam-style questions
Unlocks once the four checks above are done. Worth more XP, written to IGCSE Paper 2 and Paper 4 standard.
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Exam style+20 XP
Write 0.0875 as a fraction in its simplest form.
Exam style+20 XP
At a school, \(\dfrac{3}{8}\) of the students walk to school, 0.45 of them come by bus and the rest come by car. What fraction come by car? Give it in its simplest form.
Exam style+20 XP
(Extended) Write \(0.4\dot{5}\) as a fraction.
Exam style+20 XP
(Extended) Which shows \(\dfrac{5}{11}\) in recurring decimal notation?
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