Original exam question

Find the number that is exactly halfway between ¹⁄₁₀ and ³⁄₅.

A common type of question in GCSE maths exams asks you to find the number exactly halfway between two given values. When those values are fractions, some students feel unsure about the method. This post will walk through solving exactly this type of problem, using the specific question: Find the number that is exactly halfway between 110 and 35.

Understanding What's Being Asked

First, let's be clear about the meaning. If you were asked for the number halfway between 2 and 8, you would calculate the average: (2 + 8) ÷ 2 = 5. The same principle applies here, just with fractions. The halfway point is technically the arithmetic mean of the two numbers.

We can visualise this on a number line:

flowchart LR
    A["0"] --- B["1⁄10"] --- C["Halfway Point"] --- D["3⁄5"] --- E["1"]
    style B fill:#f9f,stroke:#333
    style D fill:#f9f,stroke:#333
    style C fill:#ccf,stroke:#333

Our job is to find the value at point C.

Step 1: Write the two fractions clearly

We have:

First number: 110

Second number: 35

Step 2: Apply the 'halfway' formula

The number exactly halfway between any two numbers a and b is:
a + b2

So for our question:
Halfway = 110 + 352

Step 3: Add the two fractions in the numerator

To add 110 and 35, we need a common denominator. The lowest common denominator of 10 and 5 is 10.

110 stays as 110

35 = 3 × 25 × 2 = 610 (multiplying numerator and denominator by 2)

Now we can add:
110 + 610 = 710

So our expression becomes:
Halfway = 7102

Step 4: Divide the fraction by 2

Dividing by 2 is the same as multiplying by 12:
710 ÷ 2 = 710 × 12 = 7 × 110 × 2 = 720

Step 5: State the final answer

The number exactly halfway between 110 and 35 is 720.

Verification: Does this make sense?

Let's check our answer is reasonable:

110 = 0.1

35 = 0.6

Halfway between 0.1 and 0.6 is 0.35

720 = 0.35 ✓

We can also show this on a more detailed number line:

flowchart LR
    A["0"] --- B["1⁄10 = 0.1"] --- C["7⁄20 = 0.35"] --- D["3⁄5 = 0.6"] --- E["1"]
    style B fill:#f9f,stroke:#333
    style D fill:#f9f,stroke:#333
    style C fill:#ccf,stroke:#333

Alternative Method: Convert to decimals first

Some students prefer working with decimals:

Convert: 110 = 0.1 and 35 = 0.6

Find halfway: (0.1 + 0.6) ÷ 2 = 0.7 ÷ 2 = 0.35

Convert back: 0.35 = 35100 = 720 (when simplified)

Both methods are perfectly valid for GCSE exams. The fraction method is often quicker and avoids potential rounding errors.

Common Mistakes to Avoid

Adding denominators: Remember you only add numerators when denominators are the same. 110 + 35 ≠ 415
Forgetting to ÷ by 2: The halfway point is the average, not the sum. Don't stop at 710.
Not simplifying fractions: While 720 is already in its simplest form, always check if your final answer can be simplified (÷ numerator and denominator by their highest common factor).

Why This Question Matters

This type of question tests several key GCSE skills:

Fraction arithmetic (addition, division)

Equivalent fractions

Understanding of averages/midpoints

Number sense (estimating and verifying)

It appears across exam boards (AQA, Edexcel, OCR) in both calculator and non-calculator papers, often as a 2-3 mark question.

Practice Question

Try this yourself: Find the number exactly halfway between 23 and 34.

Solution (try it first before looking!):
23 + 342 = 812 + 9122 = 17122 = 1712 × 12 = 1724

Summary

To find the halfway point between two fractions:

Add the two fractions (using a common denominator)

Divide the result by 2 (or multiply by 12)

Simplify your answer if possible

Verify it makes sense (check decimals or position on number line)

Remember, mathematics is logical and consistent. If you understand why the method works (finding the average), you can apply it to any numbers—fractions, decimals, or even algebraic expressions.

Final answer to the original question: 720

Edexcel November 2019 2F Question 12