Rounding, Estimation & BoundsEdexcel IGCSE Maths: Revision notes

Decimal places (d.p.) count digits after the decimal point. Significant figures (s.f.) count from the first non-zero digit, reading left to right.

• To round, look at the digit after the one you're keeping. If it is 5 or more, round up; otherwise round down.
• Leading zeros (e.g. in $0.00456$) are never significant. Trailing zeros in a whole number (e.g. $2400$) may or may not be significant — context/rounding tells you.

Estimation rounds each number to 1 significant figure before calculating, giving a quick check that an exact answer is sensible.

Steps:

1. Round every value in the calculation to 1 s.f.
2. Carry out the simplified calculation.
3. Compare the estimate to your exact answer — if they're wildly different, you likely made an arithmetic slip.

Example: estimate $\dfrac{58.7 \times 3.92}{0.51}$. Round: $\dfrac{60 \times 4}{0.5} = \dfrac{240}{0.5} = 480$.

When a measurement is rounded, the true value lies within a range. The upper bound (UB) and lower bound (LB) define that range.

For a value rounded to the nearest unit $u$: add/subtract half of $u$.

$\text{LB} = x - \dfrac{u}{2}, \qquad \text{UB} = x + \dfrac{u}{2}$

Example: a length of $7.4\text{ cm}$ measured to the nearest $0.1\text{ cm}$ has $u = 0.1$, so: $\text{LB} = 7.4 - 0.05 = 7.35\text{ cm}, \qquad \text{UB} = 7.4 + 0.05 = 7.45\text{ cm}$

This is written as $7.35 \leq x < 7.45$. Note the upper bound uses a strict inequality ($<$) because $7.45$ would itself round up to $7.5$.

When combining measurements, work out which combination of bounds gives the extreme result you want.

• Maximum of $a + b$: use $\text{UB}_a + \text{UB}_b$
• Minimum of $a + b$: use $\text{LB}_a + \text{LB}_b$
• Maximum of $a - b$: use $\text{UB}_a - \text{LB}_b$
• Minimum of $a - b$: use $\text{LB}_a - \text{UB}_b$
• Maximum of $a \div b$: use $\text{UB}_a \div \text{LB}_b$
• Minimum of $a \div b$: use $\text{LB}_a \div \text{UB}_b$

Multiplication follows the same pattern as division: maximum uses $\text{UB}_a \times \text{UB}_b$, minimum uses $\text{LB}_a \times \text{LB}_b$.

Bounds questions often disguise themselves as area, volume or speed calculations.

• Area of a rectangle $= \text{length} \times \text{width}$: maximum area uses $\text{UB}_{\text{length}} \times \text{UB}_{\text{width}}$.
• Speed $= \dfrac{\text{distance}}{\text{time}}$: maximum speed uses $\text{UB}_{\text{distance}} \div \text{LB}_{\text{time}}$ — travelling the furthest possible distance in the shortest possible time.

Always identify the formula first, then decide whether each variable needs its upper or lower bound for the extreme you're asked for.