Estimating (AQA GCSE Maths): Revision Notes

Worked Example: Estimating a Complex Fraction

Estimate the value of $\frac{127.8 + 41.9}{56.5 \times 3.2}$

Step 1: Round all numbers to easier values

• 127.8 becomes 130
• 41.9 becomes 40
• 56.5 becomes 60
• 3.2 becomes 3

Step 2: Substitute the rounded values $\frac{130 + 40}{60 \times 3} = \frac{170}{180}$

Step 3: Simplify further if needed Round 170 to 180, giving you $\frac{180}{180} = 1$

Answer: The estimate is 1

Worked Example: Volume of a Cylinder

A cylindrical glass has a height of 18 cm and radius of 3 cm. Estimate its volume using $V = \pi r^2 h$.

Step 1: Round values to 1 significant figure

• $\pi \approx 3$
• height = 18 cm $\approx$ 20 cm
• radius = 3 cm (already simple)

Step 2: Substitute into the formula $V \approx 3 \times 3^2 \times 20 = 3 \times 9 \times 20 = 540 \text{ cm}^3$

Step 3: Apply to practical problem If filling glasses from a 2.5 litre bottle:

• Convert: 2.5 litres = 2500 cm³
• Round 540 to 500 for easier division
• Number of glasses: $2500 \div 500 = 5$ glasses

Answer: The glass volume is approximately 540 cm³

Worked Example: Estimating $\sqrt{87}$

Estimate $\sqrt{87}$ to 1 decimal place.

Step 1: Find the perfect squares on either side of 87

• $81 = 9^2$
• $100 = 10^2$

So 87 falls between 81 and 100, meaning $\sqrt{87}$ is between 9 and 10.

Step 2: Determine which it's closer to Since 87 is closer to 81 than to 100, $\sqrt{87}$ will be closer to 9 than to 10.

Step 3: Make a sensible estimate $\sqrt{87} \approx 9.3$

Verification: The actual value of $\sqrt{87}$ is approximately 9.32737, so our estimate is very reasonable.