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Number (7Ni)

Mathematics Stage 7 Chapter Notes

What this chapter covers

Number - Integers, powers and roots
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In this chapter

  1. 1Ordering Fractions
  2. 2Adding and Subtracting Mixed Numbers
  3. 3Multiplying Fractions and Mixed Numbers
  4. 4Dividing by Fractions
  5. 5Order of Operations with Fractions

1. Ordering Fractions

To compare or order fractions, you need to make them 'speak the same language'. The easiest way to do this is to give them a common denominator. This means rewriting each fraction so they all have the same number on the bottom. To do this, find the lowest common multiple (LCM) of the original denominators. Once they have a common denominator, the fraction with the largest numerator is the largest fraction. An alternative method, especially if you have a calculator, is to convert each fraction into a decimal by dividing the numerator by the denominator, and then comparing the decimals.

Key term

Common Denominator: A shared multiple of the denominators of two or more fractions, used as a basis for comparing, adding, or subtracting them.

Examiner insight

Examiners look for clear working that shows how you found the common denominator and the equivalent fractions. Simply writing the final ordered list without any steps may not earn full marks, even if correct.

Common pitfall

A very common mistake is to compare fractions by just looking at the numerators or denominators without first finding a common denominator. For example, incorrectly thinking 3/8 is larger than 1/2 because 3 is larger than 1 and 8 is larger than 2.

Worked example 13 marks

Arrange the fractions 2/3, 5/6, and 3/4 in ascending order (smallest to largest).

  1. 1

    Step 1: Find the lowest common multiple (LCM) of the denominators (3, 6, and 4). The multiples of 3 are 3, 6, 9, 12... The multiples of 6 are 6, 12, 18... The multiples of 4 are 4, 8, 12... The LCM is 12.

  2. 2

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

  3. 3

    For 2/3: 12 ÷ 3 = 4. So, multiply the numerator and denominator by 4: (2 × 4) / (3 × 4) = 8/12.

  4. 4

    For 5/6: 12 ÷ 6 = 2. So, multiply the numerator and denominator by 2: (5 × 2) / (6 × 2) = 10/12.

  5. 5

    For 3/4: 12 ÷ 4 = 3. So, multiply the numerator and denominator by 3: (3 × 3) / (4 × 3) = 9/12.

  6. 6

    Step 3: Compare the new fractions: 8/12, 10/12, and 9/12. By comparing the numerators, we can see the order is 8/12, 9/12, 10/12.

  7. 7

    Step 4: Write the final answer using the original fractions. The correct order is 2/3, 3/4, 5/6.

Recap

  • To order fractions, first find a common denominator.
  • The best common denominator is the Lowest Common Multiple (LCM) of the original denominators.
  • Convert each fraction to its equivalent with the common denominator.
  • Compare the numerators to determine the order.
  • Always write the final answer using the original fractions.
  • Alternatively, convert fractions to decimals to compare them.

Quick check

  1. Which fraction is larger: 4/5 or 5/7?2 marks

2. Adding and Subtracting Mixed Numbers

When adding or subtracting mixed numbers, you must first convert them into improper fractions. This creates a standard format that makes calculation straightforward. Once you have improper fractions, the process is the same as for simple fractions: find a common denominator, perform the addition or subtraction on the numerators, and finally, convert the result back into a mixed number if required. Trying to add or subtract the whole numbers and fractions separately can be tricky, especially with subtraction, so converting to improper fractions is the most reliable method.

a b/c = (a × c + b) / c

Key term

Improper Fraction: A fraction where the numerator (top number) is greater than or equal to the denominator (bottom number), such as 7/3.

Examiner insight

Full marks are awarded for a correct final answer, but method marks can be earned for correctly converting to improper fractions and finding a common denominator, even if a calculation error occurs later.

Common pitfall

When subtracting mixed numbers like 3 1/4 - 1 3/4, a common error is to handle the whole numbers and fractions separately and incorrectly. Converting to improper fractions (13/4 - 7/4) avoids this trap completely.

Worked example 13 marks

Calculate 4 1/5 - 2 3/4 and give your answer as a mixed number.

  1. 1

    Step 1: Convert both mixed numbers to improper fractions. For 4 1/5, (4 × 5 + 1) / 5 = 21/5. For 2 3/4, (2 × 4 + 3) / 4 = 11/4.

  2. 2

    Step 2: The calculation is now 21/5 - 11/4.

  3. 3

    Step 3: Find the lowest common multiple of the denominators (5 and 4), which is 20.

  4. 4

    Step 4: Create equivalent fractions with the denominator 20. 21/5 becomes (21 × 4) / (5 × 4) = 84/20. 11/4 becomes (11 × 5) / (4 × 5) = 55/20.

  5. 5

    Step 5: Perform the subtraction on the numerators: 84 - 55 = 29. The result is 29/20.

  6. 6

    Step 6: Convert the improper fraction back to a mixed number. 29 ÷ 20 = 1 with a remainder of 9. So, the answer is 1 9/20.

Recap

  • Always convert mixed numbers to improper fractions before adding or subtracting.
  • Find a common denominator for both fractions.
  • Only add or subtract the numerators; the denominator stays the same.
  • If the answer is an improper fraction, convert it back to a mixed number.
  • Show all your working steps clearly.

Quick check

  1. Work out 1 2/3 + 2 1/6. Give your answer as a mixed number.2 marks

3. Multiplying Fractions and Mixed Numbers

Multiplying fractions is more direct than adding them as you do not need a common denominator. The rule is simple: multiply the numerators together and multiply the denominators together. If you are working with mixed numbers, it is essential that you convert them into improper fractions before you multiply. A useful technique is to simplify before multiplying by 'cross-cancelling' any common factors between a numerator and a diagonal denominator. This keeps the numbers smaller and easier to manage.

a/b × c/d = (a × c) / (b × d)

Key term

Cross-cancellation: A shortcut for simplifying fractions before multiplying, by dividing a numerator and a diagonal denominator by a common factor.

Examiner insight

Examiners reward students who simplify before multiplying (cross-cancel). It demonstrates a deeper understanding of number properties and often prevents calculation errors with large numbers.

Common pitfall

The most common error is to multiply the whole number parts and the fraction parts of mixed numbers separately. This gives the wrong answer and must be avoided. Always convert to improper fractions first.

Worked example 13 marks

Calculate 2 1/7 × 1 3/4.

  1. 1

    Step 1: Convert the mixed numbers to improper fractions. 2 1/7 = (2 × 7 + 1) / 7 = 15/7. 1 3/4 = (1 × 4 + 3) / 4 = 7/4.

  2. 2

    Step 2: The calculation is now 15/7 × 7/4.

  3. 3

    Step 3: Look for common factors to simplify before multiplying (cross-cancellation). The 7 in the numerator of the second fraction and the 7 in the denominator of the first fraction can both be divided by 7, becoming 1.

  4. 4

    Step 4: This simplifies the problem to 15/1 × 1/4.

  5. 5

    Step 5: Multiply the numerators (15 × 1 = 15) and the denominators (1 × 4 = 4). The result is 15/4.

  6. 6

    Step 6: Convert back to a mixed number: 15 ÷ 4 = 3 with a remainder of 3. The final answer is 3 3/4.

Recap

  • Convert mixed numbers to improper fractions before multiplying.
  • Multiply the numerators together and the denominators together.
  • Simplify the final answer, or simplify before multiplying (cross-cancel).
  • The word 'of' in maths usually means multiply, e.g., '1/2 of 10' is 1/2 × 10.

Quick check

  1. Work out 5/9 × 3/10 and give your answer in its simplest form.2 marks

4. Dividing by Fractions

To divide by a fraction, you use a simple three-step rule often remembered by the phrase 'Keep, Change, Flip'. First, you Keep the first fraction the same. Second, you Change the division sign to a multiplication sign. Third, you Flip the second fraction upside down to find its reciprocal. After that, you just follow the rules for multiplication. As with multiplication, any mixed numbers must be converted to improper fractions at the very beginning of the process.

a/b ÷ c/d = a/b × d/c

Key term

Reciprocal: The reciprocal of a fraction is found by inverting it; the reciprocal of a/b is b/a.

Common pitfall

A frequent error is flipping the first fraction instead of the second, or forgetting to flip any fraction at all and just changing the division sign to multiplication. Remember to only flip the fraction you are dividing by.

Fun fact

Dividing by a fraction is the same as asking 'how many of this fraction fit into the other number?'. This is why dividing by a fraction smaller than 1, like 1/2, gives a bigger answer (you are multiplying by 2).

Worked example 13 marks

A baker has 7 1/2 kg of flour. A recipe for one loaf of bread requires 3/4 kg of flour. How many full loaves of bread can the baker make?

  1. 1

    Step 1: This is a division problem: How many times does 3/4 fit into 7 1/2? The calculation is 7 1/2 ÷ 3/4.

  2. 2

    Step 2: Convert the mixed number to an improper fraction: 7 1/2 = (7 × 2 + 1)/2 = 15/2.

  3. 3

    Step 3: The calculation is now 15/2 ÷ 3/4.

  4. 4

    Step 4: Apply the 'Keep, Change, Flip' rule. Keep 15/2, change ÷ to ×, and flip 3/4 to 4/3. The new calculation is 15/2 × 4/3.

  5. 5

    Step 5: Simplify before multiplying. 15 and 3 share a common factor of 3. 4 and 2 share a common factor of 2. The calculation becomes (15÷3)/(2÷2) × (4÷2)/(3÷3) = 5/1 × 2/1.

  6. 6

    Step 6: Multiply the simplified fractions: 5/1 × 2/1 = 10/1 = 10.

  7. 7

    Step 7: Answer the question in context. The baker can make 10 full loaves of bread.

Recap

  • Convert all mixed numbers to improper fractions first.
  • To divide, multiply the first fraction by the reciprocal of the second.
  • 'Keep, Change, Flip' is a good way to remember the division rule.
  • Simplify by cross-cancelling where possible before multiplying.
  • For word problems, make sure your final answer makes sense in the context.

Quick check

  1. Calculate 4/5 ÷ 8/15. Give your answer in its simplest form.2 marks

5. Order of Operations with Fractions

When a calculation involves multiple operations (addition, subtraction, multiplication, division) and brackets, you must follow a specific order to get the correct answer. This is known as the Order of Operations, commonly remembered by the acronym BIDMAS or BODMAS. B - Brackets I - Indices (or O - Orders) D - Division and M - Multiplication (equal priority, work from left to right) A - Addition and S - Subtraction (equal priority, work from left to right) You must apply this order rigorously, performing one step at a time and showing your working clearly on a new line for each step.

Key term

Order of Operations: The universally agreed-upon sequence to perform operations in a mathematical expression to ensure a single, correct result.

Examiner insight

In multi-step problems, examiners award method marks for correct procedures even if a final answer is wrong due to an earlier arithmetic slip. That's why showing clear, logical steps that follow BIDMAS is crucial for maximising your score.

Common pitfall

The most common mistake is to simply work from left to right, ignoring the order of operations. For example, in '1/2 + 1/3 × 3/4', a student might incorrectly add 1/2 + 1/3 first, instead of correctly multiplying 1/3 × 3/4 first.

Worked example 14 marks

Calculate (2 1/2 - 1/4) ÷ 7/8.

  1. 1

    Step 1: Follow BIDMAS. We must calculate the expression inside the Brackets first: (2 1/2 - 1/4).

  2. 2

    Step 2: Inside the bracket, convert the mixed number to an improper fraction: 2 1/2 = 5/2. The bracket becomes (5/2 - 1/4).

  3. 3

    Step 3: Find a common denominator for the subtraction, which is 4. 5/2 becomes 10/4. The bracket is now (10/4 - 1/4) = 9/4.

  4. 4

    Step 4: The original problem has now been simplified to 9/4 ÷ 7/8.

  5. 5

    Step 5: Now perform the Division using 'Keep, Change, Flip'. The calculation becomes 9/4 × 8/7.

  6. 6

    Step 6: Simplify before multiplying. The 4 and 8 share a common factor of 4. The calculation becomes 9/1 × 2/7.

  7. 7

    Step 7: Multiply the numerators (9 × 2 = 18) and denominators (1 × 7 = 7). The result is 18/7.

  8. 8

    Step 8: Convert to a mixed number: 18 ÷ 7 = 2 with a remainder of 4. The final answer is 2 4/7.

Recap

  • Always follow the BIDMAS/BODMAS order.
  • Deal with expressions in Brackets first.
  • Then do Division and Multiplication (working from left to right).
  • Finally, do Addition and Subtraction (working from left to right).
  • Show each step of your calculation clearly on a new line.
  • Convert mixed numbers to improper fractions as your first step within any calculation part.

Quick check

  1. What is the value of 1/5 + 4/5 × 1/2?2 marks

End-of-chapter exercise

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

  1. Place these fractions in order of size, starting with the smallest: 3/4, 2/3, 7/12, 5/6.2 marks
  2. Calculate 5/8 × 4/15. Give your answer in its simplest form.2 marks
  3. Work out 5 1/3 - 2 3/5. Give your answer as a mixed number.3 marks
  4. A plank of wood is 3 3/4 metres long. It is cut into smaller pieces, each of length 1/2 metre. How many full pieces can be cut, and what fraction of a metre is left over?4 marks
  5. In a school, 3/5 of the students are girls. Of the girls, 1/4 have school lunch. What fraction of all students are girls who have school lunch?3 marks
  6. Calculate 3/4 + 1/2 ÷ 2/5. Give your answer as a mixed number.3 marks
  7. Amir spends 1/4 of his pocket money on sweets and 1/3 on a magazine. He saves the rest. If he saves £5, how much pocket money did he receive?4 marks
  8. A number is added to 2/3. The result is then multiplied by 1/2. The final answer is 5/6. What was the original number?4 marks
  9. Find the value of 2 1/2 - (1/3 + 5/6).3 marks
  10. Calculate (4/5 - 1/3) × 1 1/2 + 1/4. Give your answer as a single fraction in its simplest form.5 marks

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