# Simple but strange ratio problem, I understand the general case but not for this specific question

There is this ratio question that I understand the general case, but not for this specific case.

Problem: A rectangle measures 32 cm by 24 cm. Given that its measurements are increased in the ratio 5 : 4 to obtain a second rectangle, find the ratio of their areas.

First, I just want to confirm the meaning of "its measurements are increased in the ratio 5 : 4", does that mean $32+5x$ and $24+4x$ for some $x$? or it means something else?

The solution is 16 : 25.

Looking from the solution, the sentence might mean the ratio of the old to new measurement is 4 : 5.

Then there is a general rule that the ratio of old area to new area is just $old^2 : new^2$ given old is the old measurement and new is the new measurement. Am I correct?

But how can we show it for this specific rectangle in this question. Does that mean the information of length 32 cm and width 24 cm are useless?

In other words, how can we get the solution of 16 : 25 for this problem?

Many many thanks!

This is a simple problem indeed! Given a 5:4 ratio means making a bigger rectangle, you would multiply the dimensions by 5/4 to get the dimensions of the second rectangle. Find both areas and solve, and that should give you the answer.

I believe your intuition about the old squared new squared rule is correct, but it does seem weird to me that the ratio is flipped. I can't explain that.

You are wrong about the x and y part, because a ratio means a scalar multiple, not an added number. In other words, it's 5/4(dimension) as I said earlier.

Let me solve for you.

(5/4)(32)=40, and (5/4)(24)=30, the scaled rectangle being the larger dimensions of 40cm by 30cm.

(24)(32)=768cm gives us the area of the first rectangle. (40)(30)=1200cm, the area of the second rectangle. The problem is not clear about which rectangle is on the bottom or top of the ratio, so I'll assume smaller first. That gives us an area ratio of 768:1200, which reduces to 16:25.

• wow, thanks so much! yes, this is very easy indeed! :) and you are right, we cannot add the numbers. – user71346 Jan 29 '14 at 5:04

To add on to the answer: yes, the actual lengths of the rectangle are irrelevant. We can prove this much more general rule.

Let's say you have a rectangle with dimensions x, y. Then it's area is x*y.

If you increase both lengths by a ratio r, Then the dimensions are rx, ry. And the area is rx*ry = r^2*x*y.

So the ratio of old area to new area is x*y:r^2*x*y. This reduces to 1:r^2.

In this case, r = 5/4, So the ratio is 1:(5/4)^2 This simplifies to 16:25. It didn't matter what x or y were.