Overview

Estimation is a powerful meta-skill on the AMC 8. When you can quickly bound an answer between two nice numbers, you can eliminate three or four answer choices before doing any precise computation. This module trains that instinct.

Key Ideas

• Round factors to one significant figure before multiplying to get a quick estimate.
• Use upper and lower bounds: if $a < x < b$, then $f(a) < f(x) < f(b)$ for increasing functions.
• On multiple-choice tests, estimate first, then only compute precisely if two choices are close together.
• Squeezing: find a simple number above and below the true answer.

Core Skills

Rough Multiplication

Replace $197 \times 43$ with $200 \times 40 = 8000$ to narrow the answer range.

Eliminating Answer Choices

If you estimate 8000 and the choices are 3000, 5000, 8451, 12000, 20000 — only 8451 is close.

Bounding Sums

For large sums, compare to a known formula or integral bound.

Worked Example

Estimate $\frac{2021 \times 49}{98}$ without a calculator.

$\frac{2021 \times 49}{98} = 2021 \times \frac{1}{2} \approx 1010.5.$

The exact value is $1010.5$.

Practice Problems

Practice Problems

Problem 1

Estimate $317 \times 52$ without a calculator.

Solution: Round $317 \approx 300$ and $52 \approx 50$. Then:

$317 \times 52 \approx 300 \times 50 = 15000.$

The exact value is $16484$. If the answer choices were $500$, $2000$, $16484$, $50000$, $200000$ — only $16484$ is near our estimate.

Problem 2

Estimate $\dfrac{4987 \times 6}{1003}$ without a calculator.

Solution: Round $4987 \approx 5000$ and $1003 \approx 1000$. Then:

$\frac{4987 \times 6}{1003} \approx \frac{5000 \times 6}{1000} = \frac{30000}{1000} = 30.$

The exact value is $\approx 29.84$. Our estimate of $30$ immediately eliminates any answer choices far from $30$.

Problem 3

The answer choices for a problem are $12$, $120$, $1200$, $12000$, $120000$. You estimate that the answer is near $198 \times 63$. Which choice is correct?

Solution: Round $198 \approx 200$ and $63 \approx 60$. Then:

$198 \times 63 \approx 200 \times 60 = 12000.$

The exact value is $12474$. The closest answer choice is $12000$.

Problem 4

Estimate $\dfrac{999 \times 999}{1001}$ without a calculator.

Solution: Round $999 \approx 1000$ and $1001 \approx 1000$. Then:

$\frac{999 \times 999}{1001} \approx \frac{1000 \times 1000}{1000} = 1000.$

The exact value is $\approx 998$. Since $999 \times 999 \approx 1000^2$ and $1001 \approx 1000$, a quick squeeze confirms the answer is just under $1000$.

Problem 5

Bound the sum $1 + 2 + 3 + \cdots + 99$ using estimation.

Solution: Use the formula bound: the sum of the first $n$ integers is $\dfrac{n(n+1)}{2}$. Estimating:

$\frac{99 \times 100}{2} \approx \frac{100 \times 100}{2} = 5000.$

The exact value is $4950$. You can also bound directly: there are $99$ terms each between $1$ and $99$, so a midpoint estimate gives $50 \times 99 = 4950$.

Problem 6

Estimate $\sqrt{620}$ to the nearest integer.

Solution: Squeeze between perfect squares: $24^2 = 576$ and $25^2 = 625$. Since $576 < 620 < 625$:

$24 < \sqrt{620} < 25.$

Since $620$ is very close to $625$, we know $\sqrt{620}$ is very close to $25$. The exact value is $\approx 24.9$.

Problem 7

Estimate $\dfrac{2998 \times 15}{497}$ without a calculator.

Solution: Round $2998 \approx 3000$ and $497 \approx 500$. Then:

$\frac{2998 \times 15}{497} \approx \frac{3000 \times 15}{500} = \frac{45000}{500} = 90.$

The exact value is $\approx 90.54$. Our estimate of $90$ is essentially exact.

Problem 8

A rectangle has side lengths $203$ and $397$. Estimate its area.

Solution: Round $203 \approx 200$ and $397 \approx 400$. Then:

$203 \times 397 \approx 200 \times 400 = 80000.$

The exact value is $80591$. If answer choices are $8000$, $40000$, $80591$, $160000$, $400000$ — only $80591$ matches.

Problem 9

Estimate $\dfrac{1}{0.0204}$ without a calculator.

Solution: Round $0.0204 \approx 0.02 = \dfrac{1}{50}$. Then:

$\frac{1}{0.0204} \approx \frac{1}{0.02} = 50.$

The exact value is $\approx 49.02$. Rounding the denominator up slightly means our estimate is slightly high, so the true answer is just under $50$.

Problem 10

Estimate $31 \times 29$ using the difference-of-squares identity as a bounding tool.

Solution: Notice that $31 \times 29 = (30+1)(30-1) = 30^2 - 1^2 = 900 - 1 = 899.$

Even without this identity, round both to $30$:

$31 \times 29 \approx 30 \times 30 = 900.$

Since one factor is slightly above $30$ and one slightly below, the product is just under $900$. The exact value is $899$ — a perfect squeeze.