Question Video: Using Proportions to Find Unknowns

Video Transcript

If 10𝑥 is equal to 11𝑦 which is equal to 12𝑧, find the ratio of 𝑥 to 𝑦 to 𝑧.

There are lots of ways of solving this problem. One way would be to find values that solve different parts of the equation first. Let’s begin by considering 10𝑥 is equal to 11𝑦. Substituting in the values 𝑥 equals 11 and 𝑦 equals 10 would mean that this equation is true. 10 multiplied by 11 and 11 multiplied by 10 are both equal to 110. This means that the ratio of 𝑥 to 𝑦 could be written as 11 to 10. Let’s now consider the fact that 10𝑥 is also equal to 12𝑧. In this equation, 𝑥 equals 12 and 𝑧 equals 10 is a solution. 10 multiplied by 12 and 12 multiplied by 10 are both equal to 120. This means that the ratio of 𝑥 to 𝑧 is 12 to 10.

We now have two ratios, a ratio of 𝑥 to 𝑦 and a ratio of 𝑥 to 𝑧. In order to combine these ratios, we need to use equivalent ratios to ensure that the value for 𝑥 is the same. The lowest common multiple of 11 and 12 is 132. We can therefore multiply the top ratio by 12 and the bottom ratio by 11. The ratio 11 to 10 is equivalent to the ratio 132 to 120. Likewise, the ratio of 𝑥 to 𝑧 of 12 to 10 is equivalent to 132 to 110. As our value for 𝑥 is the same, we can now combine the ratios. The ratio of 𝑥 to 𝑦 to 𝑧 is 132 : 120 : 110.

This ratio can be simplified as all of our values are even and are therefore divisible by two. 132 divided by two is equal to 66. 120 divided by two is equal to 60. And 110 divided by two is equal to 55. The ratio of 𝑥 to 𝑦 to 𝑧 in its simplest form is 66 : 60 : 55 as these three numbers have no common factor apart from one.