Estimation

Worked example

1. 1

   $\frac{6.87\times 213}{0.487}$ — The question. These numbers are too awkward to work with in your head.

2. 2

   $\approx \frac{7\times 200}{0.5}$ — Round each number to 1 s.f.: 6.87 → 7, 213 → 200, 0.487 → 0.5. Write them down, because they earn a mark.

3. 3

   $=\frac{1400}{0.5}$ — Work out the top first: 7 × 200 = 1400.

4. 4

   $=2800$ — Dividing by 0.5 doubles, because there are 2 halves in every 1. So 1400 × 2 = 2800.

Worked example

1. 1

   $\frac{58.4+21.7}{6.83-2.1}$ — The question. A plus on top and a minus below.

2. 2

   $\approx \frac{60+20}{7-2}$ — Round all four numbers to 1 s.f.: 58.4 → 60, 21.7 → 20, 6.83 → 7, 2.1 → 2. Write this line down: it earns a mark.

3. 3

   $=\frac{80}{7-2}$ — Finish the top: 60 + 20 = 80. The fraction line works like brackets, so the top is done on its own.

4. 4

   $=\frac{80}{5}$ — Finish the bottom: 7 − 2 = 5.

5. 5

   $=16$ — Now divide: 80 ÷ 5 = 16. This is the estimate.

Worked example

1. 1

   $\frac{2.93\times \sqrt{385}}{0.512^2}$ — The question. A root on top and a square below.

2. 2

   $\approx \frac{3\times \sqrt{400}}{{0.5}^2}$ — Round each number to 1 s.f., including the ones under the √ and under the square: 2.93 → 3, 385 → 400, 0.512 → 0.5.

3. 3

   $=\frac{3\times 20}{0.5^2}$ — √400 = 20, because 20 × 20 = 400.

4. 4

   $=\frac{3\times 20}{0.25}$ — 0.5² = 0.5 × 0.5 = 0.25. Squaring a number less than 1 makes it smaller.

5. 5

   $=\frac{60}{0.25}$ — Work out the top: 3 × 20 = 60.

6. 6

   $=240$ — 0.25 is a quarter, and there are 4 quarters in every 1. So dividing by 0.25 multiplies by 4: 60 × 4 = 240.

Worked example

1. 1

   $\frac{9.78\times 312}{2.07}$ — Mia's calculation. She got 147.4. Estimate it yourself to test her answer.

2. 2

   $\approx \frac{10\times 300}{2}$ — Round each number to 1 s.f.: 9.78 → 10, 312 → 300, 2.07 → 2. Note 9.78 rounds up to 10, not 9.

3. 3

   $=\frac{3000}{2}$ — Work out the top first: 10 × 300 = 3000.

4. 4

   $=1500$ — 3000 ÷ 2 = 1500. This is the estimate.

5. 5

   $147.4\text{ vs }1500$ — Compare. Her answer is about ten times too small, so a decimal point has probably slipped.

Worked example

1. 1

   $38\times 19.60$ — Total cost = number of tickets × price of one ticket.

2. 2

   $\approx 40\times 20$ — Round each number to 1 s.f.: 38 → 40, 19.60 → 20.

3. 3

   $=800$ — 40 × 20 = 800. So the total is about $800.

4. 4

   $40>38,20>19.60$ — Is the estimate too big or too small? Both numbers were rounded up, so $800 is more than the exact cost.