Question Video: Using Proportions to Form and Solve Linear Equations Mathematics

Video Transcript

Julie starts with the numbers five and 12. If she adds a number 𝑘 to each of her numbers, the ratio of the resulting numbers is 23 to 24. Find the value of 𝑘.

Let’s begin by considering the ratio of the numbers at the start of this problem. To begin with, the numbers are in a ratio five to 12. Remember, order here is important. Then we’re told that she adds some number 𝑘 to each of her numbers. So her first number five becomes five plus 𝑘, and her second number 12 becomes 12 plus 𝑘. This means the new ratio of these numbers can be represented as five plus 𝑘 to 12 plus 𝑘.

But remember, the question told us that the ratio of these resulting numbers is in fact 23 to 24. So five plus 𝑘 to 12 plus 𝑘 is in fact equal to 23 to 24. And whilst this might look quite nasty, this now means we can create a single equation in terms of 𝑘. Since these ratios are equal, if we divide the left-hand side of the first ratio by the right-hand side of the first ratio, this will be equal to dividing the left-hand side of the second ratio by the right-hand side of the second.

In other words, five plus 𝑘 over 12 plus 𝑘 is equal to 23 over 24. We’re going to now need to solve this equation for 𝑘. And to do so, we need to get rid of these fractions. So we’re going to multiply both sides of the equation by 12 plus 𝑘 and by 24. When we do, on the left-hand side, we get 24 times five plus 𝑘, and on the right, 23 times 12 plus 𝑘.

Our next job will be to distribute these parentheses. 24 times five plus 𝑘 is 120 plus 24𝑘. And 23 times 12 plus 𝑘 is 276 plus 23𝑘. Since we’re solving for 𝑘, we need to find a way to make it the subject. Let’s subtract 23𝑘 from both sides. That gives us 120 plus 𝑘 equals 276. And we can now see that there’s just one more step to help us find the value of 𝑘; we need to subtract 120 from both sides. 276 minus 120 is 156. And so we have the value 𝑘 that Julie added to each of her numbers. It was 156.