450 $\div$ 9 = 50

50 = length $\times$ width, where length : width = 5 : 4

50 = 5 $\times$ 10

No closer factor pairs $\therefore$ no factor pairs in the ratio 5 : 4

1260 $\div$ 9 = 140

140 = 2 $\times$ 70

= 4 $\times$ 35

= 10 $\times$ 15

No closer factor pairs $\therefore$ no factor pairs in the ratio 5 : 4

1620 $\div$ 9 = 180

180 = 10 $\times$ 18

= 12 $\times$ 15 correct ratio

So 1620 is possible

The ratio length:breadth of the smaller rectangles is $5:4$. Let the length and breadth of these rectangles be $5x \; \text{cm}$ and $4x\; \text{cm}$ respectively. The area of the large rectangle, in cm$^2$, is $9 \times 20x^2 = 180x^2$ and the only one of the alternatives which is a product of $180$ and a perfect square is $1620$, which corresponds to $x = 3$.

This problem is taken from the UKMT Mathematical Challenges.

You can find more short problems, arranged by curriculum topic, in our short problems collection.