GCSE Maths Practice: place-value-and-rounding

Concept Overview

This higher-tier GCSE Maths question moves beyond simple rounding and asks you to explain why two close numbers can round to different tens. Understanding the logic behind rounding demonstrates mastery of place value reasoning, not just memorising the 5-and-above rule. When two numbers lie on opposite sides of a midpoint (in this case, 255), they round to different multiples of ten because each is closer to a different end of the interval.

When rounding to the nearest ten, every block of ten has a midpoint that determines where the cut-off occurs. For the interval from 250 to 260, the midpoint is 255. Any number below 255 rounds down to 250; any number 255 or above rounds up to 260. Recognising that rounding is about distance from this midpoint is crucial for success in higher-tier reasoning questions.

Step-by-Step Method

1. Identify the rounding interval. When rounding to the nearest ten, intervals are 0–9, 10–19, 20–29, etc. For 253.79 and 257.11, the relevant interval is between 250 and 260.
2. Find the midpoint. (250 + 260) ÷ 2 = 255. Numbers below 255 round to 250; numbers at or above 255 round to 260.
3. Compare both numbers to the midpoint. 253.79 < 255, so it rounds down. 257.11 ≥ 255, so it rounds up.
4. Explain using place-value language. The ones digit decides because it shows whether the number is before or after the midpoint. 3 is before 5, while 7 is after.

Worked Examples

Example 1. Compare 184 and 188 when rounding to the nearest ten.

• 184 → 180 (ones 4 < 5)
• 188 → 190 (ones 8 ≥ 5)
• They round differently because 185 is the midpoint between 180 and 190.

Example 2. Compare 372.4 and 374.9.

• 372.4 → 370
• 374.9 → 370 (still below midpoint 375)
• Both round the same this time because both are below 375.

Example 3. 253.79 vs 257.11.

• 253.79 < 255 → 250
• 257.11 ≥ 255 → 260
• Different tens because they fall on opposite sides of the midpoint.

Common Mistakes

• Looking at the wrong digit: Students sometimes check the tenths (after the decimal) instead of the ones digit when rounding to tens.
• Thinking decimals automatically round up: Decimals do not always increase the value—what matters is the digit immediately after the place you’re rounding to.
• Forgetting the midpoint concept: Rounding is based on proximity, not arbitrary cutoff. Visualising a number line helps avoid errors.

Real-Life Applications

Explaining why values round differently is not just theoretical. In finance, reporting £253.79 as £250 and £257.11 as £260 helps communicate that small differences can change an estimate category. In engineering, small measurement differences affect which specification a component fits into. In computing, data is often grouped by tens or hundreds; understanding rounding boundaries ensures consistency in summaries or database grouping.

FAQ

Q1: What exactly causes two numbers so close together to round differently?
A: Each is on a different side of the rounding midpoint. 255 divides the interval between 250 and 260; numbers below it round down, numbers above it round up.

Q2: If a number is exactly 255, which way should I round?
A: By convention, numbers ending exactly in 5 round up, so 255 → 260.

Q3: Does the decimal part affect rounding to tens?
A: Only if the decimal changes the ones digit after rounding. The rule still focuses on the ones digit.

Study Tip

When asked to explain a rounding result, always refer to the midpoint and relative distance. Sketch a quick number line: mark the two endpoints and the midpoint. Then show which side each number lies on. This visual reasoning earns full marks in “Explain why…” questions and prepares you for bounds and estimation topics later in the GCSE Maths course.