The height of tree A is 𝑥 centimeters. The height of tree B is 𝑦 centimeters. The height pf tree C is 𝑧 over two centimeters. The height of tree B is 20 percent more than the height of tree A. The height of tree B is 40 percent less than the height of tree C. Work out the ratio 𝑥 to 𝑦 to 𝑧. Write the ratio in its simplest form.
To find a ratio of 𝑥 to 𝑦 to 𝑧, we should begin by forming expressions for the height of the tree in one variable. Currently, we have three variables: they are 𝑥, 𝑦, and 𝑧. We are to hold the relationship of the heights of trees A and C compared to B. So let’s form some equations for 𝑥 and 𝑧 in terms of 𝑦.
We are told that the height of tree B — remember that was 𝑦 centimeters — is 20 percent more than the height of tree A that was 𝑥 centimeters. Remember the original value is always 100 percent. So to increase by 20 percent, we can add 120. 100 plus 20 is 120 percent. So an increase of 20 percent is the same as finding 120 percent of that same number.
We can then find the decimal multiplier that corresponds to an increase of 20 percent by dividing this value by 100. When we divide by 100, we move the digits to the right two places. So 120 percent is equal to 1.2. To increase by 20 percent then, we multiply by 1.2. That means that 𝑦 is equal to 1.2 multiplied by 𝑥.
Remember we wanted to form an equation for 𝑥 in terms of 𝑦. To achieve this, we’re going to divide both sides of this equation by 1.2. And in doing so, we get that 𝑦 over 1.2 is equal to 𝑥. In fact, if we type one divided by 1.2 into our calculator, we get five-sixths. So this tells us that five-sixths of 𝑦 is equal to 𝑥 or 𝑥 is equal to five-sixths of 𝑦.
Let’s repeat this process with the information that the height of tree B is 40 percent less than the height of tree C. This time the height of tree B is 𝑦. But tree C is 𝑧 over two. And of course, this time we’re reducing. So we’re going to subtract 40 percent from 100. That’s 60.
So to find a 40 percent reduction, we find 60 percent of that number. Dividing by 100 gives us a decimal multiplier of 0.6. This time we can say that 𝑦 is equal to 0.6 lots of 𝑧 over two. 0.6 divided by two is 0.3. So 𝑦 is equal to 0.3𝑧.
We want an expression for 𝑧 in terms of 𝑦. So we’re going to divide both sides of this equation by 0.3. And we get 𝑦 divided by 0.3 is equal to 𝑧. This time one divided by 0.3 is ten-thirds. And we can, therefore, see that 𝑧 is equal to ten-thirds of 𝑦.
Now that we have expressions for 𝑥 and 𝑧 in terms of 𝑦, we can substitute all of these into the original ratio. 𝑥 is equal to five-sixths of 𝑦, 𝑦 is still 𝑦, and 𝑧 was equal to ten-thirds of 𝑦. Remember we’re looking to write our ratio in its simplest form. And we can see that there is a common factor throughout of 𝑦. We can, therefore, divide everything through by 𝑦. We get five-sixths to one to ten-thirds.
Finally, to write a ratio in its simplest form, we need these values to be integers — that’s whole numbers. The quickest way to achieve this is to multiply through by the lowest common multiple of each of the denominators. The lowest common multiple of six and three is six. So we’re going to multiply each part of our ratio by six.
Five-sixths multiplied by six is five, one multiplied by six is six, and ten-thirds multiplied by six is 20. Now, we are allowed to use a calculator here. But it is useful to know how to multiply a fraction by a whole number. Should it arise on a non-calculator paper, we give the whole number a denominator and its denominator is one.
So ten-thirds multiplied by six is the same as ten-thirds multiplied by six over one. We could cross cancel or we could simply multiply the numerators together — 10 multiplied by six is 60 — and then the denominators — three multiplied by one is three. We can then see that 60 divided by three is 20, as we showed.
In its simplest form then, the ratio of 𝑥 to 𝑦 to 𝑧 is five to six to 20.