A new housing estate is represented by an area of 200 cm^2 on a Map A drawn to a scale of 1 : n. Given that the actual area of the estate is 32 km^2, find the value of n. Answer n = ___ The scale of another map, Map B is 1 : 65 000.
The length of a road on Map B is 50 cm.
Find the length of the road on Map A. Answer ___ cm

1. Convert the actual area from km² to cm².

Since $1\text{ km}=100000\text{ cm}$,

$1\text{ km}^2=(100000)^2\text{ cm}^2=10^{10}\text{ cm}^2$

$32\text{ km}^2=32\times 10^{10}=320000000000\text{ cm}^2$

2. Use the area scale to find the linear scale denominator $n$.

Map area : actual area

$200:320000000000=1:1600000000$

For a scale of $1:n$, the area ratio is

$1^2:n^2=1:n^2$

So

$n^2=1600000000$

$n=\sqrt{1600000000}=40000$

1. Find the actual length of the road using Map B's scale.

Map B has scale $1:65000$.

Actual length of road
$=50\times 65000=3250000\text{ cm}.$

2. Convert the actual road length to its length on Map A using the scale from part (a).

From part (a), Map A has scale $1:40000$.

Length on Map A
$=\frac{3250000}{40000}=81.25\text{ cm}.$