The ratios 12 to four to eight and 𝑥 to five to 𝑦 are equivalent. Find 𝑥 and find 𝑦.
We’re given two sets of ratios, each of them involving three values. In order to solve for our missing values, we’ll want our middle terms to be equal. Our first ratio has a middle term of four. And our second ratio has a middle term of five. We could convert these ratios so that they both have a middle term of 20. We know that four times five equals 20. And since we’re dealing with ratios, we would then need to multiply both the first and the third term by five as well to keep the proportion correct. 12 times five equals 60. And eight times five equals 40. And now we have a ratio of 60 to 20 to 40.
To go from five to 20, we need to multiply by four. And that means we need to multiply 𝑥 and 𝑦 by four as well. 𝑥 times four is four 𝑥. And 𝑦 times four is four 𝑦. So our second ratio is four 𝑥 to 20 to four 𝑦. If we line these two ratios up. Since we have two equal middle terms, then 60 must be equal to four 𝑥. And 40 must be equal to four 𝑦. At this point, we’ll need to solve for 𝑥 and 𝑦. On the left, we need to divide both sides by four. 60 divided by four is 15. Four 𝑥 divided by four equals 𝑥. And so 𝑥 equals 15. On the right, we’ll solve for 𝑦 by dividing both sides by four. 40 divided by four equals 10. Four 𝑦 divided by four equals 𝑦. And so we can say that 𝑦 equals 10. Let’s take these values and plug them back into our original ratio. We can say that 12 to four to eight is equivalent to 15 to five to 10.
Now that we found that 𝑥 equals 15 and 𝑦 equals 10, I’m going to clear some space and show you a second way to solve this problem. Back to the beginning, we were given 12 to four to eight. And that is equivalent to 𝑥 to five to 𝑦. We can think about what we see in this first ratio. We know that eight is equal to four times two. And that 12 is equal to four times three. And if we call 𝑚 our middle term, then the third term will be equal to two times 𝑚. And our first term will be equal to three times 𝑚. If we use this logic for our equivalent ratio, then we need to multiply five times two to find 𝑦. And we need to multiply five times three to find 𝑥. Five times three is 15. Five times two equals 10, which confirms what we found for 𝑥 and 𝑦 using the first method. The first method that I showed you works even when you can’t recognize a pattern like this.
Either way, we confirm that 𝑥 equals 15 and 𝑦 equals 10.