Estimation and Approximation

Estimation is arguably the single most powerful strategy for the UCAT Quantitative Reasoning subtest. On many questions, you do not need to calculate the exact answer — you need to calculate an answer that is close enough to distinguish between the four options. This lesson teaches you when and how to estimate effectively.

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Why Estimation Works in UCAT QR

The QR subtest is multiple choice with four options. The question designers must make the wrong options plausible, but they cannot make all four options extremely close together (otherwise the question would be unfairly difficult). In practice, the options are usually spread across a meaningful range.

Example options: A) 145 B) 290 C) 435 D) 580

These options are roughly 145 apart. An estimate accurate to within ±50 would identify the correct answer with certainty.

Example options: A) 12.3% B) 18.7% C) 24.1% D) 31.5%

These differ by about 6 percentage points. An estimate accurate to within ±3% would suffice.

Key Principle: You do not need the exact answer. You need an answer accurate enough to be closer to the correct option than to any other option.

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Technique 1: Rounding to Convenient Numbers

The most common estimation technique. Replace awkward numbers with nearby round numbers that are easy to work with.

Rules for Rounding

1. Round to the nearest convenient number — this might be the nearest 10, 100, or 1,000 depending on the magnitude.
2. Try to balance your rounding — if you round one number up, try to round another down. This keeps errors small.
3. Round more aggressively when options are far apart, more conservatively when they are close.

Worked Example

Question: A school orders 487 textbooks at £23.50 each. What is the total cost?

Options: A) £8,452 B) £11,445 C) £14,230 D) £17,108

Estimation:

• 487 ≈ 500 (rounded up by 13)
• £23.50 ≈ £24 (rounded up by £0.50)
• 500 × 24 = 12,000

Wait — 12,000 is not close to any option. Let us refine:

• 500 × £23.50 = £11,750
• But we rounded 487 up to 500, so the actual answer is a bit less than £11,750
• £11,445 is the closest option

Answer: B) £11,445 ✓ (Exact answer: 487 × 23.50 = £11,444.50)

Time: ~8 seconds with estimation vs ~12 seconds with calculator.

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Technique 2: Order-of-Magnitude Checking

Before calculating, determine the approximate size (order of magnitude) of the answer. This instantly eliminates wildly wrong options.

Method

Round each number to one significant figure, then multiply/divide mentally.

Example: 3,847 × 0.042

• ≈ 4,000 × 0.04 = 160

Options: A) 16.2 B) 38.5 C) 161.6 D) 384.7

Only C is near 160. Answer: C) 161.6 ✓

Example: 12,500 ÷ 32

• ≈ 12,000 ÷ 30 = 400

Options: A) 39.1 B) 156.3 C) 390.6 D) 625.0

Only C is near 400. Answer: C) 390.6 ✓

When this is enough: If only one option is in the right order of magnitude (hundreds vs thousands, tens vs hundreds), the order-of-magnitude check alone identifies the answer.

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Technique 3: Compatible Numbers

Replace the actual numbers with "compatible" numbers that divide or multiply neatly.

Method

Find numbers close to the originals that have a clean mathematical relationship.

Example: 623 ÷ 19

• 623 ≈ 620, and 620 ÷ 20 = 31
• Or: 19 ≈ 20, and 620 ÷ 20 = 31
• Actual answer: 32.8

Options: A) 24.6 B) 32.8 C) 41.2 D) 53.7

The estimate of 31 clearly points to B.

Example: 17% of 5,200

• 17% ≈ 1/6 (16.7%), and 5,200 ÷ 6 ≈ 867
• Or: 17% ≈ 20% − 3%, and 20% of 5,200 = 1,040, minus 3% of 5,200 = 156, giving ~884
• Actual answer: 884

Both estimation methods get close enough.

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Technique 4: Percentage Estimation Using Benchmarks

For percentage calculations, use benchmark percentages you know instantly:

Benchmark	Equivalent
10%	÷10
25%	÷4
33%	÷3
50%	÷2
75%	÷4 × 3

Build other percentages from these:

Example: 37% of 840

• 37% ≈ 33% + 4%
• 33% of 840 = 280 (÷3)
• 4% of 840 ≈ 34 (1% = 8.4, ×4 = 33.6)
• Estimate: 280 + 34 = 314
• Actual: 310.8

Or more roughly:

• 37% ≈ 40% − 3%
• 40% of 840 = 336
• 3% of 840 ≈ 25
• Estimate: 336 − 25 = 311

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Technique 5: The Sanity Check

After any calculation (mental or with the calculator), quickly check whether the answer "makes sense."

Common Sanity Checks