Question

Dear Mitch,

From your previous posts, I think I now understand the ratios. However, I am a little stuck on a particular question and wanted your opinion. Here it is:

Two numbers are in the ratio 5 : 3.

If the smaller number is increased by 11, the

ratio becomes 4 : 3. What are the numbers?

The part that I am not getting is increasing the smaller number by 11. The answer according to the answer sheet I have is -55 and -33. Even if I were to multiply 5 x 11 and 3 x 11, I understand the 55 and 33, but I do not understand how that comes up with negative 55 and negative 33. Can you explain it in simple terms?

Thank you!

Dina K.

Answer

Dear Dina,

Yes, I think I can explain this one in 'simple terms'.

In fact, since you have actually supplied the answer with your question and would like it "explained" – and since I am always interested in trying novel approaches to standard topics, I'll explain this one by working backwards, i.e., from the answer back to the question, which is an approach I don't think I've used on this website before.

First, though, I want you to know something: This question is trickier than most ratio questions because it has a 'twist'. And that twist is, as I see you surmised, the oddity of having negative numbers for ratios. In my previous posts on ratios, which you alluded to, I did not cover these because they are not part of the basics of ratios that one encounters in real life and on most standard and non standard tests. Still, here we go...

When you look at the -55 and the -33, I think it takes most people a moment to recall/realize that the "smaller number" is the -55. (This we determine by looking at an imaginary number line in our mind to see which of the two integers appears further to the left. And, on the negative side of zero, the left side of zero, the negative 55 is indeed further out on the left; it is the smaller. (You can also think of finances: Which is a smaller amount of 'wealth' – to have a balance in your bank of negative 55 or a balance of negative 33?..)

So, when you increase the "smaller number" (the -55) by 11, you get -44. That happens because you increase a negative number by moving it to the right for the number of units of increase. (Here, 11 units).

BEFORE THE CHANGE, your ratio was -55 : -33 (which of course the question simplified to 5:3 by dividing both sides by the common factor of -11.

So, from -55 : -33,

you add 11 units to the -55 and get:

Your ratio is now:

-44 : -33

And once again, reducing both sides by the common factor of -11, you get 4 : 3.

Hope that helps!

Good Luck,

Mitch