GCSE Maths Practice: estimation

Question 10 of 10

Use front-end estimation, compatible numbers, and error bounds to get a fast, reliable subtraction estimate before doing exact arithmetic.

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\( \begin{array}{l}\textbf{Estimate:}\\1586.4 - 938.7\end{array} \)

Choose one option:

Choose a rounding place that keeps the calculation simple and the estimate tight — hundreds are often best for four-digit numbers.

Front-End Estimation for Subtraction

Front-end estimation focuses on the leading digits to get a rapid, reliable sense of size. For subtraction, compare the thousands and hundreds first, then make a small adjustment if needed. This approach is quick for mental maths and excellent for checking that a detailed calculation is sensible in GCSE Maths.

Method 1: Front-End First, Adjust Later

  1. Look at the leading parts (thousands/hundreds). Ignore the rest for the first pass.
  2. Subtract the big chunks to get an initial estimate.
  3. Adjust only if the dropped digits are unusually large or small.

Example (new numbers): 2,478 − 1,935 → front-end 2,400 − 1,900 = 500. The exact is 543, so the front-end estimate is close and very fast.

Method 2: Compatible Numbers

Sometimes you can nudge each number to a nearby value that works neatly with the other. For subtraction, choose round hundreds or thousands that keep the difference stable.

  • Example: 3,197 − 1,804 → use 3,200 − 1,800 = 1,400. Easy boundaries, clean mental arithmetic.
  • This works best when one number is already close to a round figure.

Method 3: Error Bounds (Upper–Lower Box)

Instead of one estimate, place the result in a small interval to prove reasonableness. Round one number up and the other down to get an upper bound, then reverse for a lower bound. The true value should lie between them.

  • Example (new numbers): 5,062.8 − 2,471.2.
    Upper bound: 5,100 − 2,400 = 2,700.
    Lower bound: 5,000 − 2,500 = 2,500.
    So the exact result should be between 2,500 and 2,700.

Choosing the Rounding Place

Pick the place value that keeps mental work simple and keeps the estimate tight. For four-digit numbers, the hundreds are often ideal. Rounding to tens may be fussier; rounding to thousands may be too coarse.

Worked Examples (varied)

  • Example A: 6,842.9 − 1,973.4 → 6,800 − 2,000 = 4,800 (front-end).
  • Example B: 1,249 − 612 → 1,200 − 600 = 600 (compatible numbers).
  • Example C: 9,504 − 8,978 → 9,500 − 9,000 = 500 (error bound check shows ~500).

Common Mistakes

  • Over-rounding: Jumping to thousands when hundreds would keep accuracy high.
  • Sign errors: Mixing up which number is larger when subtracting mentally.
  • Ignoring direction: If you round one number up and the other down, remember the estimate may shift slightly.

Real-World Uses

Front-end subtraction is used for budgeting (“About how much is left this month?”), stock checks (“Roughly how many items remain?”), and time planning (“How many minutes are left in the lesson?”). In all cases, you want a quick, defendable figure before confirming details.

FAQ

  • How accurate should an estimate be? Within about 5–10% is typical for quick checks.
  • Is it better to round both up or both down? Consistency matters more than direction; just be aware of the bias (both up slightly inflates, both down slightly deflates).
  • When should I switch to exact arithmetic? When decisions depend on small differences or when marks require a precise value.

Study Tip

Say the front-end step out loud: “Sixteen hundred minus nine hundred is seven hundred.” Verbalising the big-chunk difference keeps your working clear and prevents small-digit distractions.