Proper fraction
Numerator smaller than denominator.
\(\frac58\)
Chapter 4
Fractions, decimals and percentages
Develop the fluency to move between fraction, decimal and percentage forms, simplify equivalent fractions, find fractions of quantities and handle recurring decimals with clear algebraic working.
By the end of this chapter, you should be able to
- use proper fractions, improper fractions and mixed numbers;
- convert between mixed numbers and improper fractions;
- find a fraction of a quantity;
- recognise equivalent fractions and simplify fully;
- convert terminating decimals to fractions and fractions to decimals;
- distinguish terminating and recurring decimals;
- convert recurring decimals to fractions using algebra;
- convert confidently between fractions, decimals and percentages.
1
Subtopic 1
Fractions
A fraction has the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers and \(b\neq0\). The top number is the numerator; the bottom number is the denominator.
For an improper fraction the numerator is at least as large as the denominator; a mixed number consists of a whole number and a proper fractional part. Convert mixed numbers to improper fractions before longer calculations.
Improper fraction
Numerator greater than or equal to denominator.
\(\frac{11}{4}\)
Mixed number
A whole number plus a proper fraction.
\(2\frac34\)
Convert between forms
Improper → mixed
- Divide numerator by denominator.
- Quotient = whole-number part.
- Remainder goes over the original denominator.
Mixed → improper
- Whole number × denominator.
- Add numerator.
- Keep the same denominator.
Worked example · Easy
Improper fraction to mixed number
Write \(\frac{23}{6}\) as a mixed number.
\(23\div6=3\) remainder 5.
\[\frac{23}{6}=\boxed{3\frac56}\]Worked example · Medium
Mixed number to improper fraction
Write \(4\frac37\) as an improper fraction.
\[4\times7+3=31\]\[4\frac37=\boxed{\frac{31}{7}}\]
Common mistake
The denominator does not change when converting a mixed number to an improper fraction.
Additional worked example 1 · Medium
Improper to mixed fraction
Convert \(\frac{47}{6}\) to a mixed number.
Step 1: Divide 47 by 6: \(47=7\times6+5\).
Step 2: \(\boxed{7\frac56}\).
Additional worked example 2 · Exam-Style
Mixed number from a measurement
A ribbon is \(\frac{53}{8}\text{ m}\) long. Express it as a mixed number and find the number of complete 1 m lengths.
Step 1: \(53\div8=6\) remainder 5, so length \(=6\frac58\text{ m}\).
Step 2: There are \(\boxed6\) complete one-metre lengths, with \(\frac58\text{ m}\) left.
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
1.1
Easy [1 mark]
Convert \(\frac{19}{4}\) to a mixed number.
Show Answer
Step 1: \(19\div4=4\) remainder 3.
Step 2: \(\boxed{4\frac34}\).
1.2
Medium [2 marks]
Convert \(5\frac27\) into an improper fraction.
Show Answer
Step 1: Numerator \(=5\times7+2=37\).
Step 2: \(\boxed{\frac{37}{7}}\).
1.3
Medium [2 marks]
Convert \(\frac{29}{6}\) to a mixed number.
Show Answer
Step 1: \(29=4\times6+5\).
Step 2: \(\boxed{4\frac56}\).
1.4
Exam-Style [3–4 marks]
Write \(3\frac56\) as an improper fraction. How many sixths does it contain?
Show Answer
Step 1: \(3\times6+5=23\).
Step 2: \(\boxed{\frac{23}{6}}\), meaning 23 one-sixth parts.
2
Subtopic 2
Fraction of a quantity
The denominator tells you how many equal parts the whole is divided into; the numerator tells you how many parts are required.
Method: divide by the denominator, then multiply by the numerator. If you need the quantity remaining after taking a fraction, subtract the fraction used from 1 first.
Core method
\[\frac{a}{b}\text{ of a quantity}:\quad \text{divide by }b\text{, then multiply by }a.\]Worked example · Easy
Find a fraction of an amount
Find \(\frac37\) of 84.
\[84\div7=12\]\[12\times3=\boxed{36}\]
Worked example · Medium
Compare two shares
Which is greater: \(\frac58\) of 144 or \(\frac7{10}\) of 120?
\[\frac58\times144=90\]\[\frac7{10}\times120=84\]
So \(\boxed{\frac58\text{ of }144}\) is greater.
Worked example · Exam-style
A fractional discount
A jacket costs $156. Its price is reduced by \(\frac14\). Find the sale price.
\[\frac14\times156=39\]\[156-39=\boxed{\$117}\]
Additional worked example 1 · Medium
Fraction then remainder
\(\frac38\) of 240 books are borrowed. How many remain?
Step 1: Borrowed \(=240\div8\times3=90\).
Step 2: Remaining \(=240-90=\boxed{150}\) books.
Additional worked example 2 · Exam-Style
Two successive fractions
A club spends \(\frac25\) of its \$900 funds on equipment, then \(\frac13\) of the remainder on transport. How much remains?
Step 1: Equipment \(=\frac25\times900=\$360\); remaining \(=\$540\).
Step 2: Transport \(=\frac13\times540=\$180\).
Step 3: Remaining \(=540-180=\boxed{\$360}\).
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
2.1
Easy [1 mark]
Find \(\frac34\) of 80.
Show Answer
Step 1: \(80\div4=20\).
Step 2: \(20\times3=\boxed{60}\).
2.2
Medium [2 marks]
\(\frac7{12}\) of 144 students take a bus. How many students is that?
Show Answer
Step 1: \(144\div12=12\).
Step 2: \(12\times7=\boxed{84}\).
2.3
Medium [2 marks]
A bag holds 96 marbles; \(\frac5{8}\) are blue. How many are not blue?
Show Answer
Step 1: Blue \(=96\div8\times5=60\).
Step 2: Not blue \(=96-60=\boxed{36}\).
2.4
Exam-Style [3–4 marks]
A phone priced at \$320 is reduced by \(\frac3{16}\) of its price. Find the sale price.
Show Answer
Step 1: Discount \(=320\div16\times3=\$60\).
Step 2: Sale price \(=320-60=\boxed{\$260}\).
3
Subtopic 3
Equivalent fractions
Equivalent fractions have the same value. Multiplying or dividing numerator and denominator by the same non-zero number does not change the fraction's value.
Equivalent fractions have the same value. Multiply or divide both numerator and denominator by the same non-zero integer. The simplest form uses the highest common factor to cancel completely.
\(\frac12\)
\(\frac24\)
\(\frac48\)
Simplest form
A fraction is in simplest form when numerator and denominator have no common factor greater than 1. Dividing both by their HCF is usually the fastest method.
Worked example · Easy
Simplify fully
Simplify \(\frac{42}{56}\).
HCF = 14.
\[\frac{42}{56}=\frac{42\div14}{56\div14}=\boxed{\frac34}\]Worked example · Exam-style
Build a fraction from data
A club has 48 adults and 32 children. What fraction of all members are children?
\[48+32=80\]\[\frac{32}{80}=\boxed{\frac25}\]
Additional worked example 1 · Medium
Reduce using the HCF
Write \(\frac{84}{126}\) in simplest form.
Step 1: HCF of 84 and 126 is 42.
Step 2: \(\frac{84\div42}{126\div42}=\boxed{\frac23}\).
Additional worked example 2 · Exam-Style
Missing numerator
Find \(x\) if \(\frac{14}{21}=\frac{x}{45}\).
Step 1: \(\frac{14}{21}=\frac23\).
Step 2: \(\frac{x}{45}=\frac23\), so \(x=45\times\frac23=\boxed{30}\).
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
3.1
Easy [1 mark]
Simplify \(\frac{12}{18}\).
Show Answer
Step 1: Divide top and bottom by 6.
Step 2: \(\boxed{\frac23}\).
3.2
Medium [2 marks]
Write \(\frac{42}{56}\) in simplest form.
Show Answer
Step 1: HCF \(=14\).
Step 2: \(\boxed{\frac34}\).
3.3
Medium [2 marks]
Find the missing numerator: \(\frac5{12}=\frac{x}{60}\).
Show Answer
Step 1: 12 is multiplied by 5.
Step 2: 5 multiplied by 5 gives \(\boxed{x=25}\).
3.4
Exam-Style [3–4 marks]
Of 150 learners, 45 chose science. What fraction chose science, in simplest form?
Show Answer
Step 1: Fraction \(=\frac{45}{150}\).
Step 2: Divide by 15: \(\boxed{\frac3{10}}\).
4
Subtopic 4
Fractions and decimals
Use place value to convert a terminating decimal to a fraction. To convert a fraction to a decimal, divide the numerator by the denominator.
To convert a terminating decimal into a fraction, place its digits above 10, 100, 1000, and so on, depending on the last decimal place, then simplify. Conversely, a fraction represents numerator divided by denominator.
Decimal → fraction
Write over \(10\), \(100\), \(1000\), etc. depending on the last decimal place, then simplify.
Fraction → decimal
Calculate numerator \(\div\) denominator.
Worked example · Easy
Decimal to fraction
Write \(0.36\) as a fraction in simplest form.
\[0.36=\frac{36}{100}=\boxed{\frac9{25}}\]
Worked example · Medium
Four decimal places
Write \(0.1375\) as a fraction in simplest form.
\[0.1375=\frac{1375}{10000}=\boxed{\frac{11}{80}}\]
Worked example · Easy
Fraction to decimal
Convert \(\frac7{20}\) to a decimal.
\[7\div20=\boxed{0.35}\]
Additional worked example 1 · Medium
Decimal to a fraction
Write \(0.056\) in simplest fractional form.
Step 1: \(0.056=\frac{56}{1000}\).
Step 2: Divide numerator and denominator by 8: \(\boxed{\frac7{125}}\).
Additional worked example 2 · Exam-Style
Interpret a measured proportion
\(0.375\) of a 64 kg shipment is flour. Find the mass of flour using a fraction.
Step 1: \(0.375=\frac{375}{1000}=\frac38\).
Step 2: Flour mass \(=\frac38\times64=\boxed{24\text{ kg}}\).
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
4.1
Easy [1 mark]
Convert \(0.6\) to a simplest fraction.
Show Answer
Step 1: \(0.6=\frac6{10}\).
Step 2: \(\boxed{\frac35}\).
4.2
Medium [2 marks]
Write \(0.045\) as a simplest fraction.
Show Answer
Step 1: \(0.045=\frac{45}{1000}\).
Step 2: Cancel 5: \(\boxed{\frac9{200}}\).
4.3
Medium [2 marks]
Convert \(\frac7{16}\) into a decimal.
Show Answer
Step 1: Compute \(7\div16\).
Step 2: \(\boxed{0.4375}\).
4.4
Exam-Style [3–4 marks]
A fraction of the total votes is \(0.3125\). Express it as a simplest fraction.
Show Answer
Step 1: \(0.3125=\frac{3125}{10000}\).
Step 2: Divide by 625: \(\boxed{\frac5{16}}\).
5
Subtopic 5
Terminating and recurring decimals
A terminating decimal ends. A recurring decimal contains a digit or block of digits that repeats forever. Both are rational numbers.
A terminating decimal ends; a recurring decimal repeats forever and can be written with dot or bar notation. To turn a recurring block into a fraction, multiply by a suitable power of ten and subtract to cancel the repeating digits.
Terminating
\(\frac38=0.375\)
The decimal stops.
Recurring
\(\frac7{11}=0.636363\ldots\)
The block 63 repeats forever.
Dot notation
A dot marks a recurring digit. For example \(0.\dot3=0.3333\ldots\). If a block repeats, dots can mark the start and end of the recurring block.
Worked example · Medium
Fraction to recurring decimal
Convert \(\frac5{12}\) to a decimal.
\[5\div12=0.416666\ldots\]
So the decimal is \(0.41\dot6\).
Worked example · Exam-style
Recurring decimal to a fraction
Convert \(0.272727\ldots\) to a fraction.
Let \(x=0.272727\ldots\).
\[100x=27.272727\ldots\]\[99x=27\]\[x=\frac{27}{99}=\boxed{\frac3{11}}\]Worked example · Exam-style
Recurring part starts later
Convert \(0.1666\ldots\) to a fraction.
Let \(x=0.1666\ldots\).
\[10x=1.6666\ldots\]\[100x=16.6666\ldots\]\[90x=15\]\[x=\boxed{\frac16}\]High-grade method
Choose multipliers so the recurring tails line up before subtracting. One recurring digit usually needs a factor of 10; two recurring digits usually need 100.
Additional worked example 1 · Medium
Recurring block of two digits
Convert \(0.\overline{27}\) to a fraction.
Step 1: Let \(x=0.272727\ldots\). Then \(100x=27.272727\ldots\).
Step 2: Subtract: \(99x=27\), so \(x=\frac{27}{99}=\boxed{\frac3{11}}\).
Additional worked example 2 · Exam-Style
One non-recurring digit
Convert \(0.1\overline{6}=0.16666\ldots\) to a fraction.
Step 1: Let \(x=0.1666\ldots\), so \(10x=1.6666\ldots\) and \(100x=16.6666\ldots\).
Step 2: Subtract: \(90x=15\).
Step 3: \(x=\frac{15}{90}=\boxed{\frac16}\).
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
5.1
Easy [1 mark]
Is \(\frac38\) terminating or recurring as a decimal?
Show Answer
Step 1: \(3\div8=0.375\).
Step 2: It is \(\boxed{\text{terminating}}\).
5.2
Medium [2 marks]
Write \(0.\overline{4}\) as a fraction.
Show Answer
Step 1: Let \(x=0.444\ldots\). Then \(10x-x=4\).
Step 2: \(\boxed{x=\frac49}\).
5.3
Medium [2 marks]
Convert \(0.\overline{18}\) to a simplest fraction.
Show Answer
Step 1: \(x=0.1818\ldots\), so \(100x-x=18\).
Step 2: \(x=\frac{18}{99}=\boxed{\frac2{11}}\).
5.4
Exam-Style [3–4 marks]
Convert \(0.2\overline{3}=0.23333\ldots\) to a fraction.
Show Answer
Step 1: Let \(x=0.23333\ldots\), so \(10x=2.3333\ldots\) and \(100x=23.3333\ldots\).
Step 2: Subtract: \(90x=21\).
Step 3: \(\boxed{x=\frac7{30}}\).
6
Subtopic 6
Fractions, decimals and percentages
Per cent means “out of 100”. Fractions, decimals and percentages can represent exactly the same value.
Conversions: percent means per hundred. Fraction to percentage: divide, then multiply by 100. Percentage to fraction: put the percentage number over 100 and simplify. Percentages may exceed 100%.
| Fraction | Decimal | Percentage |
|---|---|---|
| \(\frac12\) | 0.5 | 50% |
| \(\frac14\) | 0.25 | 25% |
| \(\frac34\) | 0.75 | 75% |
| \(\frac15\) | 0.2 | 20% |
| \(\frac1{10}\) | 0.1 | 10% |
Conversion routes
Decimal ↔ percentage
Multiply by 100 to get a percentage; divide by 100 to get a decimal.
Fraction ↔ percentage
Fraction to percentage: divide then ×100. Percentage to fraction: write over 100 then simplify.
Worked example · Easy
Fraction to percentage
Convert \(\frac7{20}\) to a percentage.
\[\frac7{20}=0.35\]\[0.35\times100=\boxed{35\%}\]
Worked example · Medium
Percentage above 100%
Convert \(135\%\) to a decimal and mixed number.
\[135\%=1.35\]\[135\%=\frac{135}{100}=\frac{27}{20}=\boxed{1\frac7{20}}\]
Worked example · Exam-style
Compare different forms
Which is greatest: \(38\%\), \(\frac25\), or \(0.37\)?
\[38\%=0.38,\qquad \frac25=0.4,\qquad0.37=0.37\]
Therefore \(\boxed{\frac25}\) is greatest.
Additional worked example 1 · Medium
Fraction to percentage
Convert \(\frac7{20}\) to a percentage.
Step 1: \(\frac7{20}=0.35\).
Step 2: \(0.35\times100=\boxed{35\%}\).
Additional worked example 2 · Exam-Style
Compare three representations
Three survey choices received \(28\%\), \(\frac3{10}\) and \(0.285\) of the votes. Which is largest?
Step 1: Convert to percentages: \(28\%\), \(30\%\) and \(28.5\%\).
Step 2: The largest is \(\boxed{\frac3{10}=30\%}\).
Practice this subtopic
Attempt the early questions first, then the exam-style question. Open Show Answer only after trying your own working.
6.1
Easy [1 mark]
Convert \(0.48\) to a percentage.
Show Answer
Step 1: Multiply by 100.
Step 2: \(\boxed{48\%}\).
6.2
Medium [2 marks]
Convert \(12.5\%\) to a fraction.
Show Answer
Step 1: \(12.5\%=\frac{12.5}{100}=\frac{125}{1000}\).
Step 2: \(\boxed{\frac18}\).
6.3
Medium [2 marks]
Convert \(\frac9{40}\) to a percentage.
Show Answer
Step 1: \(9\div40=0.225\).
Step 2: \(0.225\times100=\boxed{22.5\%}\).
6.4
Exam-Style [3–4 marks]
A class survey records proportions \(\frac7{25}\), 0.31 and 35%. Which is the largest?
Show Answer
Step 1: Convert \(\frac7{25}=28\%\) and \(0.31=31\%\).
Step 2: Compare 28%, 31% and 35%.
Step 3: \(\boxed{35\%}\) is greatest.
Chapter 4 practice
Begin with routine conversions, then move into recurring-decimal algebra and mixed-form exam questions. Keep the answer closed until you have attempted each question.
A. Build confidence
1Fraction types[3 marks]
Classify \(\frac7{12},\frac{19}{6},3\frac25\).
Show Answer
\(\frac7{12}\): proper. \(\frac{19}{6}\): improper. \(3\frac25\): mixed number.
2Convert fraction forms[4 marks]
a) Write \(\frac{29}{8}\) as a mixed number. b) Write \(5\frac34\) as an improper fraction.
Show Answer
a) \(29\div8=3\) remainder 5, so \(\boxed{3\frac58}\).
b) \(5\times4+3=23\), so \(\boxed{\frac{23}{4}}\).
3Fraction of a quantity[3 marks]
Find \(\frac7{12}\) of 180.
Show Answer
\[180\div12=15\]\[15\times7=\boxed{105}\]
4Simplify[3 marks]
Simplify \(\frac{54}{72}\).
Show Answer
HCF = 18.
\[\frac{54}{72}=\boxed{\frac34}\]B. Convert fluently
5Decimal to fraction[3 marks]
Write \(0.425\) as a fraction in simplest form.
Show Answer
\[0.425=\frac{425}{1000}=\boxed{\frac{17}{40}}\]
6Fraction to decimal[2 marks]
Convert \(\frac{13}{40}\) to a decimal.
Show Answer
\[13\div40=\boxed{0.325}\]
7Fraction to percentage[3 marks]
Convert \(\frac5{16}\) to a percentage.
Show Answer
\[\frac5{16}=0.3125=\boxed{31.25\%}\]
8Percentage to fraction[3 marks]
Write \(62.5\%\) as a fraction in simplest form.
Show Answer
\[62.5\%=\frac{625}{1000}=\boxed{\frac58}\]
C. Recurring decimals
9Terminating or recurring?[3 marks]
State whether each terminates or recurs: \(\frac7{25},\frac29,\frac{11}{40}\).
Show Answer
\(\frac7{25}=0.28\): terminating.
\(\frac29=0.2222\ldots\): recurring.
\(\frac{11}{40}=0.275\): terminating.
10Recurring decimal to fraction[5 marks]
Convert \(0.454545\ldots\) to a fraction.
Show Answer
Let \(x=0.454545\ldots\).
\[100x=45.454545\ldots\]\[99x=45\]\[x=\boxed{\frac5{11}}\]11Delayed recurring part[6 marks]
Convert \(0.23333\ldots\) to a fraction.
Show Answer
Let \(x=0.23333\ldots\).
\[10x=2.3333\ldots\]\[100x=23.3333\ldots\]\[90x=21\]\[x=\boxed{\frac7{30}}\]D. Exam-style challenge
12Sale price[4 marks]
A tablet costs $288 and is reduced by \(\frac{3}{16}\) of its original price. Find the sale price.
Show Answer
\[\frac{3}{16}\times288=54\]\[288-54=\boxed{\$234}\]
13Compare proportions[5 marks]
Class A has \(\frac7{15}\) of students travelling by bus. Class B has 46%. Which class has the greater proportion?
Show Answer
\[\frac7{15}\times100=46.666\ldots\%\]
So \(\boxed{\text{Class A}}\) has the greater proportion.
14Order three forms[5 marks]
Arrange \(0.58,\frac7{12},57\%\) from smallest to largest.
Show Answer
\[0.58,\quad\frac7{12}=0.58333\ldots,\quad57\%=0.57\]\[\boxed{57\%<0.58<\frac7{12}}\]
15Recurring-decimal reasoning[7 marks]
Let \(x=0.181818\ldots\). a) Express \(x\) as a fraction. b) Hence find \(5x\).
Show Answer
\[100x=18.181818\ldots\]\[99x=18\]\[x=\frac2{11}\]\[5x=\boxed{\frac{10}{11}}\]
16Survey comparison[6 marks]
In a survey, \(\frac38\) prefer A, 36% prefer B and 0.245 prefer C. Which is most popular and by how many percentage points does it exceed the least popular?
Show Answer
\[\frac38=37.5\%,\quad36\%=36\%,\quad0.245=24.5\%\]
A is most popular and C least popular.
\[37.5-24.5=\boxed{13\text{ percentage points}}\]Chapter 4 high-grade checklist
Fraction forms
I can move between proper, improper and mixed forms.
Fractions of quantities
I divide by the denominator and multiply by the numerator.
Equivalent fractions
I simplify fully and recognise equivalent forms.
Decimal conversions
I use place value and division confidently.
Recurring decimals
I can convert recurring decimals to fractions using algebra.
Percentage fluency
I can compare fractions, decimals and percentages by converting to a common form.
Chapter 4 revision: This chapter follows the subtopic headings and core concepts in the supplied Cambridge O Level Mathematics, Second Edition. Figure crops come from the provided textbook. Additional questions are original exam-style practice, not reproduced Cambridge examination questions.