Question

The area of lllinois on the smaller map is $15$ cm${}^2.$ If the distance between Chicago and Rockford is $2$ cm on the smaller map and it is $6$ cm on the larger map, then what is the area of llinois on the larger map?

• A. $115$ cm${}^2$
• B. $120$ cm${}^2$
• C. $125$ cm${}^2$
• D. $135$ cm${}^2$

Answer 1

The correct option is D $135$ cm${}^2$


The area of Illinois on the smaller map $=15$ cm${}^2$
Now let's consider the distance between Chicago and Rockford

2 cm distance on the smaller map corresponds to 6 cm on the larger map.

Small map larger map
2 cm $\iff$ 6 cm
$\text{Ratio}=\frac{6}{2}=3$

We know, Area = Base $\times$ height
If length is increased by a multiple of 3, then Area is increased by a multiple of: length $\times$ length = 3 $\times$ 3 = 9

Small map larger map
2 cm $\iff$ 6 cm
1 cm $\iff$ 3 cm
Area: 1 $\iff$ $3^2$ cm${}^2$ area

So, $1{\text{cm}}^2$ corresponds to $3^2$ cm${}^2$ area
$\therefore$ 15 cm${}^2$ will corresponds to $15\times 3^2$ cm${}^2=135$ cm${}^2$ area

So, it's not just a length ratio that we are looking for but if we change the units and look at the area then we have to change the ratio accordingly.