The ratio between two integers is two to five. If you subtract 33 from each, then the ratio becomes three to two. What are the two integers?
Now, there are two ways to answer this problem. The first is an algebraic method. The second is a little bit more like trial and error. Let’s consider both methods. We’ll begin with the algebraic method. We’re given that the ratio between two integers is two to five. So those integers could be two, five; four, 10; six, 15; and so on. In other words, we can rewrite the ratio as two 𝑎 to five 𝑎, where 𝑎 itself is an integer.
So what happens when we subtract 33 from each? We get two 𝑎 minus 33 equals five 𝑎 minus 33. And we’re told that the ratio now is equivalent to three to two. So two 𝑎 minus 33 to five 𝑎 minus 33 equals three to two. And this is great because we can now divide one part of our ratio by the other to create an equation in terms of 𝑎. In both cases, we’ll divide the left-hand side of our ratio by the right. When we do, we get two 𝑎 minus 33 over five 𝑎 minus 33 equals three over two. To solve this equation, we’re going to get rid of these nasty fractions. And to achieve this, we’ll multiply both sides by five 𝑎 minus 33 and by two.
On the left-hand side, that leaves us with two times two 𝑎 minus 33, which distributes to make four 𝑎 minus 66. And on the right, we get three times five 𝑎 minus 33, which is 15𝑎 minus 99. To solve for 𝑎, we’re going to subtract four 𝑎 from both sides to get negative 66 equals 11𝑎 minus 99. Then, we’ll add 99 to both sides to get 11𝑎 equals 33. Then, our final step is to divide through by 11, giving us 𝑎 equals three. Once we have the value of 𝑎, we can substitute this back into our earlier ratio if 𝑎 is three, two 𝑎 is six, and five 𝑎 is 15. And so, our two integers are six and 15.
So this was one method. But we said that there were more than one. The second method is a numeric method. Consider the ratio two to five. By multiplying both sides of this ratio, we see it’s equivalent to four to 10, six to 15, eight to 20. And we can continue in this manner to form at least the first seven possible ratios. We know, though, that we’re going to subtract 33 from each part. If we do that to our first ratio, we get negative 31 to negative 28. For our second, we get negative 29 to negative 23. And our third, we get negative 27 to negative 18.
We can continue in this manner, but actually we don’t need to. Negative 31 and negative 28 have no common factors other than negative one or one. And so negative 31 to negative 28 will not simplify to three to two. Similarly, nor will negative 29 to negative 23. But if we divide negative 27 and negative 18 by negative nine, we get the ratio three to two. So this means this pair of values is correct and our original ratio must have been six to 15. And of course, either method is perfectly valid. The former is more efficient but possibly can be a little bit confusing, whereas the second is less efficient but certainly more simplistic. Either way, the two integers are six and 15.