FSCE 11+ Maths: 10 Worked Examples with Step-by-Step Solutions
The FSCE 11+ maths is not about speed-drilling arithmetic. It is about applying KS2 mathematical thinking to unfamiliar, often multi-step, often contextual problems. The questions reward children who can read a worded problem carefully, identify the relevant operation or technique, and work through it step by step — showing logical reasoning along the way.
This post walks through ten worked examples that mirror the style of FSCE-format questions. Each example shows the working in full, names the technique being used, and ends with a brief note on what the question is testing and how to approach similar ones in the exam.
Use these alongside our FSCE 11+ Mathematics course for structured practice.
Example 1 — Multi-step ratio with money
Question. Three friends, Aisha, Ben and Carla, share £84 in the ratio 2 : 3 : 7. Carla then gives a quarter of her share to Ben. How much money does each person now have?
Working.
Step 1: Find the value of one share. Total parts = 2 + 3 + 7 = 12. One part = £84 ÷ 12 = £7.
Step 2: Calculate each person's initial share.
- Aisha: 2 × £7 = £14
- Ben: 3 × £7 = £21
- Carla: 7 × £7 = £49
Step 3: Calculate Carla's transfer. A quarter of Carla's £49 = £49 ÷ 4 = £12.25.
Step 4: Adjust Ben's and Carla's totals.
- Ben: £21 + £12.25 = £33.25
- Carla: £49 − £12.25 = £36.75
- Aisha: unchanged, £14
Quick check: total still = £14 + £33.25 + £36.75 = £84 ✓.
What this tests. Ratio sharing in a non-trivial ratio, then a fraction-of-an-amount transfer. The trap is forgetting to update both Ben's and Carla's totals after the transfer.
Example 2 — Percentage decrease then increase (the classic trick)
Question. A jacket costs £80. In a sale it is reduced by 25%. After the sale, the price is increased by 25% from the sale price. What is the final price? Is it £80?
Working.
Step 1: Sale price = £80 − 25% of £80 = £80 − £20 = £60.
Step 2: Final price = £60 + 25% of £60 = £60 + £15 = £75.
No — the final price is £75, not £80.
What this tests. The crucial insight that a percentage applied to a smaller amount gives a smaller absolute change than the same percentage applied to a larger amount. A 25% increase from £60 (£15) is smaller than the original 25% reduction from £80 (£20).
A common error: assuming "25% off then 25% on" returns you to the start. It doesn't.
Example 3 — Average word problem
Question. Maya scored 18, 22, 15 and 25 in four spelling tests. What does she need to score in her fifth test to bring her average up to exactly 22?
Working.
Step 1: For her average across 5 tests to be 22, her total across 5 tests must be 5 × 22 = 110.
Step 2: Her current total is 18 + 22 + 15 + 25 = 80.
Step 3: She needs 110 − 80 = 30 in her fifth test.
What this tests. Working backwards from a target average. The technique of "total = average × number of items" is more flexible than the standard "average = total ÷ number of items".
Example 4 — Algebra in disguise
Question. I think of a number, multiply it by 3, add 7, then divide the result by 2. The answer is 14. What was my original number?
Working.
Step 1: Work backwards from the answer using inverse operations.
| Forward | Inverse |
|---|---|
| Multiply by 3 | Divide by 3 |
| Add 7 | Subtract 7 |
| Divide by 2 | Multiply by 2 |
Step 2: Start with 14 and reverse:
- 14 × 2 = 28 (undoing the divide by 2)
- 28 − 7 = 21 (undoing the add 7)
- 21 ÷ 3 = 7 (undoing the multiply by 3)
Check: 7 × 3 = 21; 21 + 7 = 28; 28 ÷ 2 = 14 ✓.
What this tests. Inverse operations. This is algebra without the algebraic notation — solving for an unknown by reversing each operation in order. Once a child internalises this, formal algebra (e.g. solving 3x + 7 = 28) becomes intuitive.
Example 5 — Area of a composite shape
Question. A garden is L-shaped. The full footprint is a rectangle 12 m by 9 m, but a smaller rectangle 4 m by 3 m has been cut out of one corner to make space for a shed. What is the area of the garden (excluding the shed)?
Working.
Step 1: Area of full rectangle = 12 × 9 = 108 m².
Step 2: Area of cut-out rectangle = 4 × 3 = 12 m².
Step 3: Garden area = 108 − 12 = 96 m².
What this tests. Composite area by subtraction. The same problem can be solved by splitting the L-shape into two rectangles and adding their areas — both methods should give 96 m². Doing both is a strong way to check the answer.
Example 6 — Time and rate
Question. A bus leaves Cheltenham at 09:35 and arrives in Gloucester at 10:08. Gloucester is 14 km from Cheltenham along the bus route. What is the bus's average speed in km/h?
Working.
Step 1: Find the journey time in minutes. From 09:35 to 10:08 = 25 minutes (to 10:00) + 8 minutes = 33 minutes.
Step 2: Convert minutes to hours. 33 minutes = 33 ÷ 60 = 0.55 hours.
Step 3: Apply speed = distance ÷ time. Speed = 14 ÷ 0.55 ≈ 25.5 km/h (to 1 decimal place).
What this tests. Working with mixed units (minutes and hours), converting cleanly, and applying the speed formula. Many students lose marks by leaving time in minutes and getting a wildly wrong answer.
Example 7 — Fractions in context
Question. A recipe makes 12 muffins and uses 3/4 of a cup of flour. How many cups of flour are needed to make 30 muffins?
Working.
Step 1: Find the flour per muffin. 3/4 cup ÷ 12 muffins = 3/48 = 1/16 cup per muffin.
Step 2: Multiply by 30 muffins. 30 × 1/16 = 30/16 = 15/8 cups = 1 ⅞ cups = 1.875 cups.
What this tests. Scaling fractions in context. A child who can confidently move between fractions and decimals (15/8 = 1.875) has more flexibility in how they express their answer.
Example 8 — Probability with replacement vs without
Question. A bag contains 4 red marbles and 6 blue marbles. Two marbles are drawn out at random, without replacing the first. What is the probability that both are red?
Working.
Step 1: Probability the first marble is red. 4 red out of 10 total = 4/10 = 2/5.
Step 2: Probability the second marble is red, given the first was red. Now 3 red out of 9 total = 3/9 = 1/3.
Step 3: Multiply (because both events must happen). P(both red) = 2/5 × 1/3 = 2/15.
What this tests. Conditional probability without saying "conditional probability". The key insight is that after drawing one red, the bag composition changes — there is one fewer red and one fewer marble overall. Children who don't notice this often calculate (4/10) × (4/10) = 16/100 = 4/25, which is wrong.
Example 9 — Reading a worded problem carefully
Question. Sam is twice as old as his sister Mia. In four years, Sam will be 18. How old is Mia now?
Working.
Step 1: If Sam will be 18 in four years, Sam is 18 − 4 = 14 now.
Step 2: Sam is twice as old as Mia, so Mia is 14 ÷ 2 = 7 now.
Check: Sam is 14, twice Mia's 7 ✓. In four years, Sam will be 18 ✓.
What this tests. Resisting the urge to manipulate the future condition before working out the present. Many students try to write "Sam = 2 × Mia in 4 years" — but the relationship "twice as old" is given for now, not for the future. Read the question twice before starting the maths.
Example 10 — Reasoning about remainders
Question. A box of pens is to be divided equally between a class. If there are 28 children and 4 pens left over, the box contains how many pens (given the box contains between 100 and 130 pens)?
Working.
Step 1: We need a number between 100 and 130 that gives a remainder of 4 when divided by 28.
Step 2: List multiples of 28 in the range:
- 28 × 4 = 112
- 28 × 5 = 140 (too big)
Step 3: The multiple of 28 in range is 112. Add the remainder of 4. 112 + 4 = 116 pens.
Check: 116 ÷ 28 = 4 remainder 4 ✓. And 100 < 116 < 130 ✓.
What this tests. Reasoning with remainders, and the relationship between a quotient, a remainder and the original number (i.e. number = divisor × quotient + remainder). This kind of reasoning often appears in FSCE word problems and is rarely taught explicitly in primary school.
How to use these examples
Working through these examples once is useful. Reworking them a week later, from scratch, is more useful. Reworking them a month later and getting them all right is the goal.
For each example:
- Read the question carefully — twice if needed. Identify what is being asked.
- Identify the technique — ratio, percentage, average, area, etc. Write it down before you start.
- Show working — every step, in order. Examiners reward visible reasoning even when the final answer is wrong.
- Check your answer — does it make sense? Substitute back into the original question if you can.
- Note any errors — keep a log of the kinds of mistakes you make. Patterns emerge quickly: forgetting to update a transfer, misreading a percentage, mixing up time units.
What the FSCE rewards
The FSCE maths paper rewards mathematical thinking, not arithmetic speed. A child who understands what they are doing, shows working clearly, and checks their answers will outperform a child who rushes through and produces unjustified numbers — even if the second child is faster on simple calculations.
Practise this style of reasoning regularly, and the underlying habits will transfer to whatever specific question types appear in the exam.
Related preparation resources
- FSCE 11+ Mathematics course — full structured course covering all the techniques used in these examples
- FSCE 11+ Critical Thinking — for the reasoning skills underpinning multi-step questions
- FSCE 11+ Exam Strategy — for time management and question approach
- FSCE 11+ Complete Guide — for the broader picture of the FSCE format
- FSCE 11+ Creative Writing Model Answers — same approach for the writing component
Practise regularly. Show working. Check answers. The maths takes care of itself.