Cambridge Lower Secondary CheckpointStage 7

Number (7Nf)

Mathematics Stage 7 Chapter Notes

What this chapter covers

Number - Fractions, decimals, percentages, ratio and proportion
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Number (7Nf) notes

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In this chapter

  1. 1Comparing and Ordering Fractions
  2. 2Adding and Subtracting Fractions
  3. 3Multiplying Fractions
  4. 4Dividing by a Fraction
  5. 5Order of Operations with Fractions

1. Comparing and Ordering Fractions

To compare fractions, you need to make sure they are 'speaking the same language'. The easiest way to do this is to rewrite them so they have the same denominator. This shared denominator is called a common denominator. A good choice for a common denominator is the Lowest Common Multiple (LCM) of the original denominators. Once they have the same denominator, the fraction with the larger numerator is the larger fraction. For improper fractions (where the numerator is bigger than the denominator) and mixed numbers, it's often helpful to convert them to the same format before comparing.

Key term

Common Denominator: A number that is a common multiple of the denominators of two or more fractions, used to enable comparison, addition, or subtraction.

Examiner insight

Examiners look for clear working that shows how you found the common denominator and the resulting equivalent fractions before you state your final order.

Common pitfall

A very common mistake is to find a common denominator but forget to multiply the numerators by the corresponding factor, leading to an incorrect comparison.

Worked example 13 marks

Which is larger, 5/8 or 7/12?

  1. 1

    Step 1: Find a common denominator for 8 and 12. The multiples of 8 are 8, 16, 24, 32... The multiples of 12 are 12, 24, 36... The Lowest Common Multiple (LCM) is 24.

  2. 2

    Step 2: Convert 5/8 to an equivalent fraction with a denominator of 24. To get from 8 to 24, we multiply by 3. So, we must also multiply the numerator by 3: 5/8 = (5 × 3) / (8 × 3) = 15/24.

  3. 3

    Step 3: Convert 7/12 to an equivalent fraction with a denominator of 24. To get from 12 to 24, we multiply by 2. So, we must also multiply the numerator by 2: 7/12 = (7 × 2) / (12 × 2) = 14/24.

  4. 4

    Step 4: Compare the new fractions. Since 15/24 is greater than 14/24, we can conclude that 5/8 is larger than 7/12.

Worked example 23 marks

Write the following fractions in ascending order (smallest to largest): 3/4, 5/6, 2/3.

  1. 1

    Step 1: Find the LCM of the denominators 4, 6, and 3. Multiples of 4: 4, 8, 12. Multiples of 6: 6, 12. Multiples of 3: 3, 6, 9, 12. The LCM is 12.

  2. 2

    Step 2: Convert each fraction to have a denominator of 12.

  3. 3

    3/4 = (3 × 3) / (4 × 3) = 9/12.

  4. 4

    5/6 = (5 × 2) / (6 × 2) = 10/12.

  5. 5

    2/3 = (2 × 4) / (3 × 4) = 8/12.

  6. 6

    Step 3: Compare the numerators of the equivalent fractions: 8/12 < 9/12 < 10/12.

  7. 7

    Step 4: Write the original fractions in this order: 2/3, 3/4, 5/6.

Recap

  • To compare fractions, find a common denominator.
  • The Lowest Common Multiple (LCM) of the denominators is the most efficient common denominator.
  • When you change the denominator, you must multiply the numerator by the same factor.
  • Once denominators are the same, compare the numerators.
  • For mixed numbers, compare the whole number parts first.

Quick check

  1. Place the correct symbol (<, > or =) between these fractions: 2/5 and 3/7.2 marks

2. Adding and Subtracting Fractions

Just like with comparing, you can only add or subtract fractions when they have a common denominator. Once they do, you simply add or subtract the numerators and keep the denominator the same. For mixed numbers, you have two main methods: 1) Convert them to improper fractions first, then perform the calculation. 2) Deal with the whole numbers and the fraction parts separately, but be careful with subtraction if you need to 'borrow'.

Key term

Improper Fraction: A fraction in which the numerator (top number) is greater than or equal to the denominator (bottom number).

Common pitfall

When subtracting mixed numbers like 3 1/5 - 1 4/5, students often subtract the whole numbers (3-1=2) and then try to subtract the fractions (1/5 - 4/5), which leads to confusion. Converting to improper fractions avoids this trap.

Worked example 13 marks

Calculate 2 1/4 + 1 2/3.

  1. 1

    Method 1: Converting to improper fractions.

  2. 2

    Step 1: Convert the mixed numbers to improper fractions. 2 1/4 = (2×4+1)/4 = 9/4. 1 2/3 = (1×3+2)/3 = 5/3.

  3. 3

    Step 2: Find a common denominator for 4 and 3, which is 12.

  4. 4

    Step 3: Convert the fractions: 9/4 = 27/12 and 5/3 = 20/12.

  5. 5

    Step 4: Add the improper fractions: 27/12 + 20/12 = 47/12.

  6. 6

    Step 5: Convert the answer back to a mixed number. 47 ÷ 12 = 3 with a remainder of 11. So, the answer is 3 11/12.

Worked example 23 marks

Work out 4 1/5 - 2 3/4. Give your answer as a mixed number.

  1. 1

    Step 1: Convert to improper fractions. 4 1/5 = 21/5. 2 3/4 = 11/4.

  2. 2

    Step 2: Find a common denominator for 5 and 4, which is 20.

  3. 3

    Step 3: Create equivalent fractions: 21/5 = 84/20. 11/4 = 55/20.

  4. 4

    Step 4: Subtract the fractions: 84/20 - 55/20 = 29/20.

  5. 5

    Step 5: Convert the result back to a mixed number: 29/20 = 1 9/20.

Recap

  • You must have a common denominator to add or subtract fractions.
  • Add or subtract the numerators only; the denominator stays the same.
  • Always simplify your final answer if possible.
  • When working with mixed numbers, converting to improper fractions is often the safest method.
  • If subtracting mixed numbers directly, you may need to 'borrow' from the whole number part.

Quick check

  1. Calculate 3/8 + 1/2.2 marks
  2. Calculate 3 - 1 1/4.2 marks

3. Multiplying Fractions

Multiplying fractions is more straightforward than adding or subtracting because you don't need a common denominator. You simply multiply the numerators together and multiply the denominators together. A very useful technique is 'cancelling' or 'simplifying' before you multiply. This involves finding a common factor between any numerator and any denominator and dividing them both by it. This keeps the numbers smaller and makes the calculation easier. Remember that finding a fraction 'of' a number means you need to multiply.

a/b × c/d = (a × c) / (b × d)

Key term

Cancelling: Simplifying fractions in a multiplication problem before calculating by dividing a numerator and a denominator by a common factor.

Examiner insight

Examiners reward students who simplify by cancelling before multiplying, as it demonstrates number sense and leads to simpler calculations.

Common pitfall

Students sometimes try to find a common denominator before multiplying, which is unnecessary and makes the calculation much harder.

Fun fact

The fraction bar (the line between the numerator and denominator) is also called a 'vinculum'.

Worked example 12 marks

Calculate 4/5 × 7/8.

  1. 1

    Step 1: Write out the multiplication: 4/5 × 7/8.

  2. 2

    Step 2: Look for common factors to cancel. The numerator 4 and the denominator 8 share a common factor of 4.

  3. 3

    Step 3: Divide both by 4: 4 ÷ 4 = 1, and 8 ÷ 4 = 2.

  4. 4

    Step 4: The problem becomes 1/5 × 7/2.

  5. 5

    Step 5: Multiply the new numerators (1 × 7 = 7) and the new denominators (5 × 2 = 10).

  6. 6

    Step 6: The answer is 7/10.

Worked example 22 marks

Find 3/5 of 200.

  1. 1

    Step 1: 'Of' means multiply. The calculation is 3/5 × 200. We can write 200 as 200/1.

  2. 2

    Step 2: The calculation is 3/5 × 200/1.

  3. 3

    Step 3: Method A: Multiply first. (3 × 200) / (5 × 1) = 600/5. Then divide 600 by 5, which is 120.

  4. 4

    Step 4: Method B: Divide first. Find 1/5 of 200 by doing 200 ÷ 5 = 40. Then multiply by the numerator: 3 × 40 = 120.

  5. 5

    Step 5: Both methods give the answer 120.

Recap

  • To multiply fractions, multiply the numerators and multiply the denominators.
  • Always look for opportunities to simplify by cancelling before you multiply.
  • The word 'of' in mathematics usually means multiply.
  • To multiply a fraction by a whole number, write the whole number as a fraction over 1.
  • If multiplying mixed numbers, convert them to improper fractions first.

Quick check

  1. Calculate 2/3 × 9/10.2 marks

4. Dividing by a Fraction

Dividing by a fraction might seem tricky, but there's a simple three-step rule to follow, often remembered by the phrase 'Keep, Change, Flip'. This means you: 1. Keep the first fraction the same. 2. Change the division sign to a multiplication sign. 3. Flip the second fraction upside down to find its reciprocal. After that, you just follow the rules for multiplying fractions!

(a/b) ÷ (c/d) = a/b × d/c

Key term

Reciprocal: The reciprocal of a number is 1 divided by that number; for a fraction a/b, its reciprocal is b/a.

Examiner insight

Clear working that shows the conversion from a division to a multiplication by the reciprocal is essential for full marks, even if the final answer is correct.

Common pitfall

The most frequent error is flipping the wrong fraction. Always flip the second fraction (the one you are dividing by), never the first.

Worked example 13 marks

Calculate 3/4 ÷ 5/6.

  1. 1

    Step 1: Keep the first fraction: 3/4.

  2. 2

    Step 2: Change the division sign to multiplication: ×.

  3. 3

    Step 3: Flip the second fraction: 5/6 becomes 6/5.

  4. 4

    Step 4: The new calculation is 3/4 × 6/5.

  5. 5

    Step 5: Look for common factors to cancel. The 4 and 6 share a common factor of 2. 4÷2=2 and 6÷2=3.

  6. 6

    Step 6: The calculation is now 3/2 × 3/5.

  7. 7

    Step 7: Multiply the numerators (3 × 3 = 9) and denominators (2 × 5 = 10). The answer is 9/10.

Worked example 23 marks

Calculate 2 1/2 ÷ 3/8.

  1. 1

    Step 1: First, convert the mixed number to an improper fraction. 2 1/2 = (2×2+1)/2 = 5/2.

  2. 2

    Step 2: The calculation is now 5/2 ÷ 3/8.

  3. 3

    Step 3: Apply Keep, Change, Flip. 5/2 × 8/3.

  4. 4

    Step 4: Cancel common factors. The 2 and 8 share a factor of 2. 2÷2=1 and 8÷2=4.

  5. 5

    Step 5: The calculation is now 5/1 × 4/3.

  6. 6

    Step 6: Multiply across: (5 × 4) / (1 × 3) = 20/3.

  7. 7

    Step 7: Convert back to a mixed number. 20 ÷ 3 = 6 with a remainder of 2. The answer is 6 2/3.

Recap

  • To divide by a fraction, multiply by its reciprocal.
  • Remember the rule: Keep, Change, Flip.
  • Always convert mixed numbers to improper fractions before dividing.
  • After changing the calculation to a multiplication, look for opportunities to cancel.

Quick check

  1. What is the reciprocal of 7/11?1 mark
  2. Calculate 2/5 ÷ 3/10.2 marks

5. Order of Operations with Fractions

When a calculation involves multiple operations (like addition, subtraction, multiplication, and division) with fractions, you must follow the standard order of operations, known as BIDMAS or PEMDAS. The rules don't change just because you're using fractions.

  • Brackets
  • Indices (or Orders)
  • Division and Multiplication (from left to right)
  • Addition and Subtraction (from left to right)

Break the problem down into smaller parts, calculating the bit with the highest priority first.

Key term

Order of Operations: The specific sequence (BIDMAS/PEMDAS) in which calculations must be performed to ensure a consistent and correct result.

Examiner insight

In multi-step calculations, examiners award marks for each correct stage. Showing that you have identified the correct order of operations is crucial for gaining method marks, even if you make a later arithmetic slip.

Common pitfall

Ignoring BIDMAS and simply working from left to right. In '2/3 + 3/4 × 8/9', many students would incorrectly add 2/3 and 3/4 first.

Worked example 13 marks

Calculate 2/3 + 3/4 × 8/9.

  1. 1

    Step 1: Identify the operations. We have addition (+) and multiplication (×).

  2. 2

    Step 2: According to BIDMAS, multiplication comes before addition. So we must calculate 3/4 × 8/9 first.

  3. 3

    Step 3: Work out 3/4 × 8/9. We can cancel. 3 and 9 have a common factor of 3. 4 and 8 have a common factor of 4. This simplifies to 1/1 × 2/3 = 2/3.

  4. 4

    Step 4: Now substitute this result back into the original problem. The calculation becomes 2/3 + 2/3.

  5. 5

    Step 5: Add the fractions. Since the denominators are already the same, we add the numerators: 2+2=4. The answer is 4/3.

  6. 6

    Step 6: Convert to a mixed number: 1 1/3.

Worked example 24 marks

Calculate (4/5 - 1/2) ÷ 3/5.

  1. 1

    Step 1: According to BIDMAS, we must do the calculation inside the Brackets first: 4/5 - 1/2.

  2. 2

    Step 2: To subtract, find a common denominator, which is 10. 4/5 = 8/10 and 1/2 = 5/10.

  3. 3

    Step 3: Perform the subtraction: 8/10 - 5/10 = 3/10.

  4. 4

    Step 4: Substitute this back into the problem. The calculation is now 3/10 ÷ 3/5.

  5. 5

    Step 5: Apply Keep, Change, Flip: 3/10 × 5/3.

  6. 6

    Step 6: Cancel the 3s, and cancel the 5 and 10 (to 1 and 2). The calculation becomes 1/2 × 1/1.

  7. 7

    Step 7: The final answer is 1/2.

Recap

  • Always follow BIDMAS/PEMDAS for calculations with multiple operations.
  • Do Brackets first, then Indices, then Division/Multiplication, then Addition/Subtraction.
  • Show your working step-by-step to avoid errors.
  • Calculate the highest priority part of the sum separately, then substitute the result back in.

Quick check

  1. Which operation would you perform first in 1/2 + 1/3 ÷ 1/4?1 mark

End-of-chapter exercise

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

  1. Arrange the following fractions in order of size, starting with the smallest: 7/5, 1 1/3, 13/10.3 marks
  2. A baker has a 5 kg bag of flour. He uses 1 3/4 kg to make bread and 2 1/8 kg to make cakes. How much flour is left in the bag?4 marks
  3. Calculate 3 1/5 × 2 1/4. Give your answer as a mixed number in its simplest form.3 marks
  4. A rectangle has an area of 5/6 m². If its width is 2/3 m, what is its length?3 marks
  5. In a school, 3/5 of the students are girls. There are 750 students in total. 1/3 of the girls play a musical instrument. How many girls play a musical instrument?3 marks
  6. Work out 5/6 + 2/3 ÷ 7/6. Give your answer as a fraction in its simplest form.4 marks
  7. Jamie spends 1/3 of his pocket money on snacks and 1/4 on a magazine. He saves the rest. What fraction of his pocket money does he save?3 marks
  8. A bottle is 7/10 full of water. 1/4 of the water is poured out. What fraction of the full bottle is the remaining water?4 marks
  9. Find a fraction that is exactly halfway between 1/3 and 3/5.3 marks
  10. Three friends share a pizza. Alice eats 1/4, Ben eats 2/5 of the remainder, and Chloe eats the rest. What fraction of the whole pizza does Chloe eat?4 marks

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