Rounding & Estimation Masterclass

The Golden Rule of Rounding

Look at the next digit (the "decider").

If it is 5 or more (5, 6, 7, 8, 9) → Round UP.
If it is 4 or less (0, 1, 2, 3, 4) → Stay DOWN (Keep the digit the same).

1. Decimal Places (d.p.)

We count digits after the decimal point.

Round 4.538 to 2 decimal places (2 d.p.)
1. Count 2 digits after the point: \(4.53\underline{8}\)
2. Look at the 3rd digit (the decider). It is 8.
3. Since 8 \(\ge\) 5, we round up the previous digit (3 becomes 4).
Answer: 4.54

2. Significant Figures (s.f.)

This is about the importance of the digits. We start counting from the first non-zero digit.

Identifying Significant Figures:

\(3405\) (4 s.f.) - Zeros in the middle count.
\(0.0052\) (2 s.f.) - Leading zeros never count.
\(5200\) (2 s.f.) - Trailing zeros in whole numbers are usually placeholders (unless specified).
\(5.20\) (3 s.f.) - Trailing zeros after a decimal count (they show precision).

Round 0.004592 to 2 significant figures
1. Ignore the first zeros. Start at 4.
2. Count 2 digits: \(4, 5\). The decider is the next one (9).
3. 9 rounds up, so 5 becomes 6.
Answer: 0.0046

3. Estimation Strategies

When asked to "Estimate", do not work out the exact answer. Round every number to 1 Significant Figure first.

Estimating Cylinder Volumes

Formula: \(V = \pi r^2 h\).
Strategy: Round \(r\) and \(h\) to 1 s.f. and approximate \(\pi \approx 3\).

Estimate the volume of a cylinder with radius 4.8cm and height 10.2cm.
\(\pi \approx 3\)
\(r = 4.8 \approx 5\)
\(h = 10.2 \approx 10\)
Calculate: \(V \approx 3 \times 5^2 \times 10 = 3 \times 25 \times 10 = 750\)
Estimate: 750 cm³

Estimating Square Roots

Find the two square numbers your number sits between.

Estimate \(\sqrt{53}\)
Square numbers nearby: \(\sqrt{49} = 7\) and \(\sqrt{64} = 8\).
53 is much closer to 49 than 64.
Estimate: 7.3

Practice Questions (Click to Reveal)

Q1: Round 12.846 to 1 decimal place.

Look at the 1st decimal place: 8.
The next digit (decider) is 4.
4 is low, so the 8 stays the same.
Answer: 12.8

Q2: Round 0.007692 to 2 significant figures.

Start counting at the first non-zero: 7.
The 2nd sig fig is 6.
The decider is 9 (round up).
The 6 becomes a 7.
Answer: 0.0077

Q3: Round 89,520 to 1 significant figure.

The 1st sig fig is 8.
The decider is 9 (round up).
8 becomes 9. Replace the rest with placeholder zeros.
Answer: 90,000

Q4: Estimate the value of \( \frac{21.4 \times 3.9}{0.48} \)

Round everything to 1 s.f.
\(21.4 \approx 20\)
\(3.9 \approx 4\)
\(0.48 \approx 0.5\)
Calculation: \( \frac{20 \times 4}{0.5} = \frac{80}{0.5} \)
Dividing by 0.5 is doubling.
Answer: 160

Q5: Estimate the square root of 90.

Lower square: \(\sqrt{81} = 9\)
Upper square: \(\sqrt{100} = 10\)
90 is roughly halfway between 81 and 100.
Answer: 9.5 (Accept 9.4 - 9.6)

Q6: Round 19.996 to 2 decimal places.

Look at 2nd d.p: 9.
Decider is 6 (Round up).
9 becomes 10, so carry the 1...
The next 9 becomes 10, carry the 1...
19 becomes 20.
Answer: 20.00

Q7: Estimate the volume of a cylinder with \(r=2.9\)cm and \(h=21.1\)cm.

Round to 1 s.f:
\(\pi \approx 3\), \(r \approx 3\), \(h \approx 20\).
\(V = \pi r^2 h \approx 3 \times 3^2 \times 20\)
\(V \approx 3 \times 9 \times 20\)
\(V \approx 27 \times 20\)
Answer: 540 cm³

Q8: Write 504.92 to 2 significant figures.

1st s.f is 5.
2nd s.f is 0.
Decider is 4 (Stay down).
Keep the 5 and 0, add a placeholder zero for the units column.
Answer: 500

Q9: Estimate \( \sqrt{5.2 \times 18.9} \)

Round inside the root first:
\(5.2 \approx 5\)
\(18.9 \approx 20\)
Calculation: \(\sqrt{5 \times 20} = \sqrt{100}\)
Answer: 10

Q10: Tricky Zero - Round 0.05060 to 3 significant figures.

Start at first non-zero: 5.
Next is 0.
Next is 6 (This is the 3rd s.f).
Decider is 0 (Stay down).
We must include the trailing zero to show it was measured.
Answer: 0.0506