What is estimation?
To estimate means to find an answer that is close enough, quickly. We do not need the exact answer. We round the numbers first, so the sum is easy to do in our head.
Example: 48 × 21 is about 50 × 20 = 1000. The exact answer is 1008. The estimate is very close.
Why estimate?
- To check if a calculator answer makes sense.
- To plan when exact data is not known (food, money, time).
- To solve big questions, like the number of cars in a city.
Ways to estimate
1. Rounding
Round each number to 1 significant figure, then calculate. 389 + 612 ≈ 400 + 600 = 1000.
2. Sampling (count a part)
Count a small part and multiply. A box of 5 × 5 cubes in each layer and 4 layers has about 100 cubes.
3. Fermi problems
Break a big unknown into small guesses you can make. How many heartbeats in a day? About 70 per minute × 60 × 24 ≈ 100 000. This idea is named after the physicist Enrico Fermi.
4. Using a known reference
A door is about 2 m tall. A building as tall as 15 doors is about 30 m.
Order of magnitude
The order of magnitude of a number is the power of ten closest to it. 300 is closer to 10² than to 10³ in powers of ten, so its order is 2. (A common rule: write the number as a × 10ⁿ; if a is less than about 3.16, the order is n; otherwise it is n + 1.)
| Thing | Size | Order |
|---|---|---|
| Width of a hair | about 10⁻⁴ m | −4 |
| Height of a person | about 1.7 m | 0 |
| Height of Mount Everest | about 8.8 × 10³ m | 4 |
| Radius of Earth | about 6.4 × 10⁶ m | 7 |
Two numbers differ by one order of magnitude when one is about 10 times the other.
Choosing the right precision
An answer should not look more exact than the data. If you guessed 70 beats per minute, saying 100 800 beats a day looks too exact. Say about 100 000.
- For shopping, the nearest 10 rupees or 1 euro is fine.
- For medicine doses, we need high precision.
- For physics estimates, 1 significant figure or just the order of magnitude is enough.
Estimates also help us make a conjecture (a smart guess) and then test it. If your estimate and the exact answer are far apart, check your working.
Try it: estimate at home
Take a handful of rice. Count the grains in a small spoon. Count how many spoons fill a cup. Multiply. Now you know roughly how many grains are in a cup. Compare with a friend's estimate: are you in the same order of magnitude?
Key formulas and definitions
- Estimate = (count in a part) × (number of parts)
- Round to 1 significant figure, then calculate
- Order of magnitude of a × 10ⁿ: n (if a < 3.16) or n + 1
- One order of magnitude = a factor of 10
Worked examples
1. Estimate 52 × 39.
Round: 50 × 40 = 2000. (Exact: 2028.)
2. Estimate 7.8 × 412 ÷ 1.9.
Round: 8 × 400 ÷ 2 = 1600.
3. A stadium has 20 blocks. One block has 38 rows of 52 seats. Estimate the seats.
One block ≈ 40 × 50 = 2000. Total ≈ 20 × 2000 = 40 000 seats.
4. Estimate the number of seconds in a year and give its order of magnitude.
365 × 24 × 3600 = 31 536 000 ≈ 3 × 10⁷ s. Written as 3 × 10⁷, a = 3 < 3.16, so the order of magnitude is 7.
5. Estimate the mass of air in a classroom 10 m × 8 m × 3 m (air density ≈ 1.2 kg/m³).
Volume = 240 m³. Mass ≈ 240 × 1.2 ≈ 290 kg, about 300 kg. Order of magnitude 10².
6. Fermi problem: how many litres of water does a city of 1 million people use for drinking each day?
Each person drinks about 2 L a day. 10⁶ × 2 = 2 × 10⁶ L a day (about 2 million litres).
Common mistakes
- Giving too many digits: an estimate of 103 412 should be written as about 100 000.
- Rounding one number up a lot and another down a lot, so the estimate drifts far from the real value.
- Forgetting units when comparing sizes (mm and m differ by 3 orders of magnitude).
- Thinking an estimate is a wild guess: a good estimate always uses reasons and known facts.