Exercise solution 2, page 311 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) When a measurement is given correct to 3 significant figures, it means the value has been rounded. To find the exact range of possible values, you need the lower bound and the upper bound.

If a length is written with 2 decimal places, then it has been rounded to the nearest \(0.01\). Half of \(0.01\) is \(0.005\), so:

\[ \text{lower bound} = \text{stated value} - 0.005 \qquad \text{upper bound} = \text{stated value} + 0.005 \]

For example, if a length is \(4.28\text{ cm}\) correct to 3 significant figures, then the actual length satisfies

\[ 4.275 \leq \text{length} < 4.285 \]

B) The lower bound is included, but the upper bound is not included. This is why we write inequalities in the form

\[ a \leq x < b \]

This means the value can be equal to the lower bound, but it must stay strictly less than the upper bound.

This happens because a value exactly at the upper bound would round up to the next number.

C) To find the bounds of the area of a rectangle, use the fact that

\[ \text{Area} = \text{length} \times \text{width} \]

If both measurements are positive, then:

lowest possible area \(=\) lower bound of length \(\times\) lower bound of width
highest possible area \(=\) upper bound of length \(\times\) upper bound of width

For example, suppose a rectangle has sides \(5.42\text{ cm}\) and \(1.86\text{ cm}\), each correct to 3 significant figures. Then:

\[ 5.415 \leq L < 5.425 \qquad 1.855 \leq W < 1.865 \]

So the area bounds are found by multiplying the matching bounds:

\[\begin{aligned} \text{lower bound of area} &= 5.415 \times 1.855 \\ &= 10.046825 \end{aligned}\]

\[\begin{aligned} \text{upper bound of area} &= 5.425 \times 1.865 \\ &= 10.116625 \end{aligned}\]

D) Be careful not to mix up decimal places and significant figures.

Decimal places count digits after the decimal point.
Significant figures start from the first non-zero digit.

For example, \(7.34\) is correct to 3 significant figures, and it is also correct to 2 decimal places. But \(0.0476\) correct to 3 significant figures is not the same idea as 3 decimal places.

E) After finding exact bounds, you may be asked to give the answers correct to 3 significant figures. This means you round the bound itself to 3 significant figures.

For example:

\[ 12.4837 \text{ to 3 significant figures is } 12.5 \]

\[ 0.068249 \text{ to 3 significant figures is } 0.0682 \]

Always count significant figures starting from the first non-zero digit.

F) A good method for this type of question is:

1. Identify the rounding unit of each measurement.
2. Find each lower bound and upper bound.
3. Multiply the two lower bounds to get the minimum area.
4. Multiply the two upper bounds to get the maximum area.
5. Round the final bounds if the question asks for a given number of significant figures.

Working in this order helps avoid mistakes and makes your reasoning clear.