Cambridge Lower Secondary CheckpointStage 6

Number (6Nf)

Mathematics Stage 6 Chapter Notes

What this chapter covers

Number — Fractions, decimals, percentages, ratio and proportion
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Number (6Nf) notes

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In this chapter

  1. 1Finding a Fraction of an Amount
  2. 2Simplifying and Comparing Fractions
  3. 3Mixed Numbers and Improper Fractions
  4. 4Fractions, Decimals and Division

1. Finding a Fraction of an Amount

To find a fraction of any number or quantity, you follow a simple two-step process. First, divide the total amount by the denominator (the bottom number of the fraction). This tells you the value of one 'part'. Second, multiply this result by the numerator (the top number of the fraction) to find the value of the number of 'parts' you want. This is often remembered as 'divide by the bottom, times by the top'. For example, to find 3/5 of 100, you would calculate (100 ÷ 5) = 20, and then 20 × 3 = 60.

(a/b) of Quantity = (Quantity ÷ b) × a

Key term

Unit Fraction: A fraction where the numerator is 1 (e.g., 1/4, 1/8), representing one part of a whole that has been divided into equal sections.

Examiner insight

Examiners look for clear working. Show the division step and the multiplication step separately to maximise your chances of earning method marks, even if you make a calculation error.

Common pitfall

A common mistake is to perform the operations in the wrong order, for example, multiplying by the denominator and dividing by the numerator.

Worked example 14 marks

In a school of 840 students, 3/7 are in the lower school. Of the remaining students, 1/4 are in the sixth form. How many students are in the sixth form?

  1. 1

    Step 1: Find the number of students in the lower school. Calculate 3/7 of 840.

  2. 2

    840 ÷ 7 = 120 (This is 1/7 of the students)

  3. 3

    120 × 3 = 360 (This is 3/7 of the students)

  4. 4

    Step 2: Find the number of remaining students.

  5. 5

    840 - 360 = 480 students

  6. 6

    Step 3: Find the number of students in the sixth form. Calculate 1/4 of the remaining students.

  7. 7

    480 ÷ 4 = 120 students

  8. 8

    Answer: There are 120 students in the sixth form.

Worked example 23 marks

A recipe for a cake that serves 6 people requires 270g of flour. How much flour is needed if you want to bake a cake that serves 10 people?

  1. 1

    Step 1: Find the amount of flour needed per person.

  2. 2

    270g ÷ 6 = 45g per person.

  3. 3

    Step 2: Calculate the amount of flour needed for 10 people.

  4. 4

    45g × 10 = 450g.

  5. 5

    Alternatively, think of this as finding 10/6 of 270g.

  6. 6

    10/6 of 270 = (270 ÷ 6) × 10 = 45 × 10 = 450g.

  7. 7

    Answer: 450g of flour is needed.

Recap

  • To find a fraction of an amount, divide the amount by the denominator.
  • Then, multiply the result by the numerator.
  • Finding a unit fraction (e.g., 1/5) of an amount is the same as dividing the amount by the denominator.
  • Always read the question carefully to see if you are finding a fraction of the original amount or a remaining amount.

Quick check

  1. Calculate 5/8 of £120.2 marks
  2. Find 2/3 of 96 kg.2 marks

2. Simplifying and Comparing Fractions

Simplifying a fraction means to write it in its 'lowest terms'. This is done by dividing both the numerator and the denominator by their highest common factor (HCF). The simplified fraction is equivalent to the original but uses the smallest possible whole numbers. To compare two fractions, such as 5/8 and 3/5, you need to rewrite them with a common denominator. The lowest common multiple (LCM) of the denominators is the most efficient choice. For 8 and 5, the LCM is 40. So, 5/8 becomes 25/40 (multiplying top and bottom by 5) and 3/5 becomes 24/40 (multiplying top and bottom by 8). Now it's easy to see that 25/40 is greater than 24/40, so 5/8 > 3/5.

To simplify a/b, find HCF(a, b) and calculate (a ÷ HCF) / (b ÷ HCF).

To compare a/b and c/d, find a common denominator (e.g., bd) and compare the new numerators (ad and cb).

Key term

Equivalent Fractions: Fractions that represent the same value or proportion of the whole, even though they have different numerators and denominators (e.g., 1/2, 2/4, 5/10).

Examiner insight

Examiners award marks for correctly finding a common denominator when comparing or ordering fractions, so always show this step clearly in your working.

Worked example 12 marks

Write the fraction 54/72 in its simplest form.

  1. 1

    Step 1: Find a common factor of the numerator (54) and the denominator (72). Both are even, so we can divide by 2.

  2. 2

    54 ÷ 2 = 27; 72 ÷ 2 = 36. The fraction is 27/36.

  3. 3

    Step 2: Find a common factor of 27 and 36. Both are in the 9 times table.

  4. 4

    27 ÷ 9 = 3; 36 ÷ 9 = 4. The fraction is 3/4.

  5. 5

    Step 3: Check if 3 and 4 have any common factors other than 1. They do not.

  6. 6

    Answer: The simplest form is 3/4. (Alternatively, the HCF of 54 and 72 is 18. 54÷18=3, 72÷18=4).

Worked example 23 marks

Arrange the following fractions in ascending order: 2/3, 7/12, 3/4.

  1. 1

    Step 1: Find a common denominator for all three fractions. The denominators are 3, 12, and 4.

  2. 2

    The lowest common multiple (LCM) of 3, 4, and 12 is 12.

  3. 3

    Step 2: Convert each fraction to an equivalent fraction with a denominator of 12.

  4. 4

    2/3 = (2×4)/(3×4) = 8/12

  5. 5

    7/12 remains 7/12

  6. 6

    3/4 = (3×3)/(4×3) = 9/12

  7. 7

    Step 3: Compare the numerators of the equivalent fractions: 7, 8, and 9.

  8. 8

    In ascending order, the fractions are 7/12, 8/12, 9/12.

  9. 9

    Step 4: Write the final answer using the original fractions.

  10. 10

    Answer: 7/12, 2/3, 3/4

Recap

  • To simplify a fraction, divide the numerator and denominator by their highest common factor.
  • A fraction is in its simplest form when the numerator and denominator have no common factors other than 1.
  • To compare fractions, find a common denominator.
  • The lowest common multiple of the denominators is the best choice for a common denominator.
  • Once fractions have a common denominator, the one with the larger numerator is the larger fraction.

Quick check

  1. Simplify the fraction 45/60.1 mark
  2. Which is larger: 4/5 or 5/6?2 marks

3. Mixed Numbers and Improper Fractions

Fractions can be written in two main forms. An improper fraction (or 'top-heavy' fraction) is one where the numerator is larger than the denominator, such as 11/4. It represents a quantity greater than one whole. A mixed number combines a whole number and a proper fraction, like 2 3/4. To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and keep the same denominator. For 2 3/4, this is (2 × 4 + 3) / 4 = 11/4. To convert an improper fraction to a mixed number, divide the numerator by the denominator. The result of the division is the whole number, and the remainder becomes the new numerator over the original denominator. For 11/4, 11 ÷ 4 is 2 with a remainder of 3, so it becomes 2 3/4.

Mixed to Improper: a b/c = (a × c + b) / c

Improper to Mixed: p/q = (p ÷ q) whole number, with remainder r, gives (p ÷ q) r/q

Key term

Improper Fraction: A fraction where the numerator is greater than or equal to the denominator, representing a value of one or more.

Common pitfall

When converting from a mixed number to an improper fraction, students sometimes forget to add the original numerator after multiplying the whole number and denominator.

Worked example 12 marks

Convert the mixed number 5 2/7 into an improper fraction.

  1. 1

    Step 1: Multiply the whole number (5) by the denominator (7).

  2. 2

    5 × 7 = 35

  3. 3

    Step 2: Add the numerator (2) to this result.

  4. 4

    35 + 2 = 37

  5. 5

    Step 3: Write this new number over the original denominator (7).

  6. 6

    Answer: 37/7

Worked example 22 marks

Convert the improper fraction 43/6 into a mixed number.

  1. 1

    Step 1: Divide the numerator (43) by the denominator (6).

  2. 2

    43 ÷ 6 = 7 with a remainder.

  3. 3

    Step 2: Determine the remainder. 7 × 6 = 42.

  4. 4

    The remainder is 43 - 42 = 1.

  5. 5

    Step 3: Write the result as a mixed number. The whole number is 7, the remainder (1) is the new numerator, and the denominator (6) stays the same.

  6. 6

    Answer: 7 1/6

Recap

  • An improper fraction has a numerator larger than its denominator.
  • A mixed number has a whole number part and a fraction part.
  • To convert mixed to improper: (Whole × Bottom + Top) / Bottom.
  • To convert improper to mixed: Divide the top by the bottom. The result is the whole number, and the remainder is the new top.

Quick check

  1. Write 23/5 as a mixed number.1 mark
  2. Write 4 3/8 as an improper fraction.1 mark

4. Fractions, Decimals and Division

The fraction bar is simply another way of writing the division symbol. The fraction a/b means exactly the same as a ÷ b. This is the key to converting any fraction into a decimal. For example, to convert 3/8 to a decimal, you calculate 3 ÷ 8. Some fractions can be converted more easily by finding an equivalent fraction with a denominator of 10, 100, 1000, etc. For example, 3/4 can be written as 75/100, which is 0.75. To convert a terminating decimal to a fraction, use the place value of the last digit as the denominator. For example, 0.65 is 65 hundredths, or 65/100, which simplifies to 13/20.

a/b = a ÷ b

0.abc = abc/1000

Key term

Recurring Decimal: A decimal number that has a digit or group of digits that repeat forever, such as 1/3 = 0.333... (written as 0.3̇) or 1/7 = 0.142857...

Examiner insight

For 'show that' questions involving fraction-to-decimal conversion, you must show either the division calculation or the steps to create an equivalent fraction over 10, 100, or 1000.

Fun fact

A fraction will only produce a terminating (non-recurring) decimal if the prime factors of its denominator (in simplest form) are only 2s and/or 5s. This is because our number system is base-10 (2 x 5).

Worked example 13 marks

Without using a calculator, convert the fraction 5/8 to a decimal.

  1. 1

    Method 1: Bus Stop Division

  2. 2

    Set up the division for 5 ÷ 8. Since 8 is larger than 5, we add a decimal point and zeros: 5.000.

  3. 3

    8 doesn't go into 5, so we write 0 and carry the 5. The calculation becomes 50 ÷ 8.

  4. 4

    50 ÷ 8 = 6 with a remainder of 2.

  5. 5

    The calculation becomes 20 ÷ 8.

  6. 6

    20 ÷ 8 = 2 with a remainder of 4.

  7. 7

    The calculation becomes 40 ÷ 8.

  8. 8

    40 ÷ 8 = 5 with no remainder.

  9. 9

    Answer: 0.625

  10. 10

    Method 2: Equivalent Fraction

  11. 11

    We want to make the denominator a power of 10. We know 8 × 125 = 1000.

  12. 12

    Multiply the numerator and denominator by 125: (5 × 125) / (8 × 125) = 625/1000.

  13. 13

    625/1000 is 0.625.

Worked example 22 marks

Write 0.28 as a fraction in its simplest form.

  1. 1

    Step 1: Write the decimal as a fraction over a power of 10. The last digit (8) is in the hundredths place.

  2. 2

    0.28 = 28/100

  3. 3

    Step 2: Simplify the fraction by finding the highest common factor of 28 and 100.

  4. 4

    The HCF of 28 and 100 is 4.

  5. 5

    28 ÷ 4 = 7

  6. 6

    100 ÷ 4 = 25

  7. 7

    Answer: 7/25

Recap

  • The fraction bar simply means 'divide'.
  • To convert a fraction to a decimal, divide the numerator by the denominator.
  • To convert a decimal to a fraction, use the place value of the final digit as the denominator and simplify.
  • Memorising common conversions (e.g., 1/8 = 0.125, 1/3 = 0.3̇) can save time in exams.

Quick check

  1. What is 3/5 as a decimal?1 mark
  2. Write 0.45 as a fraction in its simplest form.2 marks

End-of-chapter exercise

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

  1. Calculate 3/8 of £480.2 marks
  2. Convert the mixed number 7 4/5 into an improper fraction.2 marks
  3. Write the fraction 9/12 in its simplest form and then convert your answer to a decimal.3 marks
  4. A library has 1,500 books. 2/5 of the books are fiction. 1/3 of the remaining books are non-fiction and the rest are reference books. How many reference books are there?4 marks
  5. Arrange the following values in ascending order, showing your working: 3/5, 0.65, 13/20, 0.605.3 marks
  6. Calculate 5 1/4 - 2 2/3. Give your answer as a mixed number in its simplest form.4 marks
  7. After a 2/7 reduction in price, a coat costs £150. What was the original price of the coat?3 marks
  8. Amir spends 1/4 of his pocket money on a magazine. He then spends 2/5 of the remaining money on sweets. What fraction of his original pocket money does he have left?4 marks
  9. A 10-metre-long ribbon is cut into 8 equal pieces. What is the length of each piece? Give your answer as a mixed number in its simplest form.3 marks
  10. A farmer's will states his 17 cows should be divided between his three children: the eldest gets 1/2, the middle gets 1/3, and the youngest gets 1/9. A neighbour lends them a cow, making the total 18. Show that with 18 cows, the shares can be given out, and explain what is unusual about the result.5 marks

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