A rectangular garden is drawn on a plan using a scale of \(1:200\). On the plan the garden measures \(6.5\) cm by \(4.2\) cm. Calculate the actual area of t...

A scale of \(1:200\) means one unit on the plan represents 200 of the same units in reality, so every plan length is multiplied by 200 to get the real length. The trap in this question is the area: the scale factor applies to lengths, so areas scale by \(200^{2}\), not by 200.

The safest route is to convert both lengths first and then find the area.

\[6.5 \times 200 = 1300 \text{ cm} = 13 \text{ m}\]

\[4.2 \times 200 = 840 \text{ cm} = 8.4 \text{ m}\]

Obtaining both real lengths scores [M1]. Recall that \(100\) cm \(= 1\) m, so dividing the centimetre values by 100 gives metres.

The actual area is then

\[13 \times 8.4 = 109.2 \text{ m}^{2}\]

[A1] cao.

As a check by the area factor: the plan area is \(6.5 \times 4.2 = 27.3\) cm\(^2\), and \(27.3 \times 200^{2} = 27.3 \times 40000 = 1\,092\,000\) cm\(^2\). Since \(1\) m\(^2 = 10\,000\) cm\(^2\), this is \(1\,092\,000 \div 10\,000 = 109.2\) m\(^2\), which agrees.

The common error is to multiply the plan area by 200 once and then convert, giving a hundredfold error. Converting the lengths first avoids the issue entirely.