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Cambridge IGCSE0580

Fractions, decimals and percentages

Mathematics 0580 Chapter Notes

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Fractions, decimals and percentages
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In this chapter

  1. 1Fractions, mixed numbers and simplest form
  2. 2Fractions, decimals and percentages
  3. 3Fractions of a whole in context
  4. 4Recurring decimal notation
  5. 5Recurring decimals as fractions

1. Fractions, mixed numbers and simplest form

A proper fraction has a numerator smaller than its denominator. An improper fraction has a numerator larger than its denominator. A mixed number is a whole number and a proper fraction together.

Always give a fraction in its simplest form: divide the numerator and the denominator by their highest common factor. To multiply mixed numbers, change them to improper fractions first. If the answer is a whole number, write the whole number, not a fraction over 1.

33/5 = (3 × 5 + 3)/5 = 18/5

Examiner insight

“Changing mixed numbers to improper fractions was well done although many candidates used careless cancelling or multiplication.” “While a correct improper fraction was often seen, the final step of cancelling to the whole number was incorrect or left the whole number answer as a fraction with denominator of 1.”

Worked example 13 marks

Work out 33/5 × 22/9.

Give your answer in its simplest form.

  1. 1

    Mixed numbers to improper fractions: 18/5 × 20/9 [1]

  2. 2

    Cancel: 18 ÷ 9 = 2 and 20 ÷ 5 = 4, so 2/1 × 4/1 [1]

  3. 3

    = 8 [1]

2. Fractions, decimals and percentages

Percent means ‘out of 100’. To change a decimal to a percentage, multiply by 100. To change a percentage to a decimal, divide by 100.

To change a fraction to a decimal, divide the numerator by the denominator. To change a percentage to a fraction, write it over 100 and give it in simplest form.

Examiner insight

“Some thought they just needed to add a % sign so had 0.47%, while 4.7, 0.047, 470 and 4700 were seen occasionally.” “A wide range of incorrect responses were seen, for example 0.9, 9.00, 0.009 and 100.9.”

Worked example 12 marks

(a) Write 0.47 as a percentage.

(b) Write 9/100 as a decimal.

  1. 1

    (a) 0.47 × 100 = 47% [1]

  2. 2

    (b) 9 ÷ 100 = 0.09 [1]

3. Fractions of a whole in context

The whole is 1. Fractions of the same whole add up to 1, so the rest is 1 minus the fractions already used.

To add or subtract fractions, write them over a common denominator. When a part is known, divide to find one part, then multiply to find the whole.

1 - 2/5 - 3/7 = (35 - 14 - 15)/35

Worked example 15 marks

2/5 of the area of a shape is blue, 3/7 is green and the rest is red.

(a) What fraction of the shape is red?

(b) 12 cm² of the shape is green. Find the total area of the shape.

  1. 1

    (a) Red = 1 - 2/5 - 3/7 [1]

  2. 2

    = 35/35 - 14/35 - 15/35 [1]

  3. 3

    = 6/35 [1]

  4. 4

    (b) 3/7 of the area is 12, so 1/7 is 12 ÷ 3 = 4 cm² [1]

  5. 5

    Total area = 4 × 7 = 28 cm² [1]

4. Recurring decimal notation

A recurring decimal has digits that repeat for ever. A dot goes over the digit that repeats.

If a block of digits repeats, a dot goes over the first and last digit of the block.

To write a fraction as a recurring decimal, divide the numerator by the denominator and look for the repeating digits.

0.1dot7 = 0.1777…

0.dot12dot3 = 0.123123…

Worked example 12 marks

Write 5/6 as a decimal, using recurring decimal notation.

  1. 1

    5 ÷ 6 = 0.8333… [1]

  2. 2

    Only the 3 repeats, so the dot goes over the 3

  3. 3

    = 0.8dot3 [1]

5. Recurring decimals as fractions

Let x equal the recurring decimal. Multiply by 10, 100 and so on until the recurring digits line up.

Subtract so that the recurring part cancels. Solve for x and give the fraction in simplest form.

x = 0.1dot7 = 0.1777…

100x - 10x = 17.777… - 1.777… = 16

x = 16/90 = 8/45

Examiner insight

“The most common misconception was to treat the decimal as terminating, using either 0.4, 0.44 or more commonly 0.444.”

Worked example 13 marks

Write 0.7dot3 as a fraction, giving your answer in its simplest form.

  1. 1

    Let x = 0.7333…, so 10x = 7.333… and 100x = 73.333… [1]

  2. 2

    Subtract: 90x = 66 [1]

  3. 3

    x = 66/90 = 11/15 [1]

Recap

  • Proper fraction: numerator smaller. Improper fraction: numerator larger. Mixed number: whole number plus fraction.
  • Give fractions in simplest form. Cancel a whole-number answer down to the whole number.
  • Decimal × 100 = percentage. Fraction: divide the numerator by the denominator.
  • Fractions of the same whole add up to 1.
  • Extended: a dot over a digit marks a recurring digit. 0.1dot7 = 0.1777…
  • Extended: for a recurring decimal, multiply to line the digits up, subtract, solve for x, then simplify.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Do not use a calculator. Write 18/24 in its simplest form.2 marks
  2. Write 0.35 as a percentage, and as a fraction in its simplest form.3 marks
  3. Do not use a calculator. Write 29/8 as a mixed number.1 mark
  4. Write 7/16 as a decimal and as a percentage.2 marks
  5. Do not use a calculator. Work out 31/2 × 13/7. Give your answer in its simplest form.3 marks
  6. Do not use a calculator. In a school, 3/8 of the students walk, 1/4 cycle and the rest take the bus. What fraction of the students take the bus?3 marks
  7. Amir scores 34 out of 40 in a quiz. Bea scores 82%. Who scores higher? Show your working.3 marks
  8. Write 5/12 as a decimal, using recurring decimal notation.2 marks
  9. Do not use a calculator. A bottle holds 0.8dot3 litres of juice. Mia pours 3/8 of a litre out of the bottle. Work out, as a fraction of a litre in its simplest form, how much juice is left in the bottle.4 marks
  10. Do not use a calculator. In a survey, 0.dot3dot6 of the people choose tea, 35% choose coffee and the rest choose water. 63 people choose water. Find the number of people in the survey.5 marks

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