If seven over 𝑥 equals 42 over 78, find the value of 𝑥.
We’ve been given an equation that describes the proportion between different values. We’re told that the proportion of seven to 𝑥 is equal to the proportion of 42 to 78. And so there are a number of ways we can answer this question. The first is just to rearrange and make 𝑥 the subject. Alternatively, we’ll think about what it means when things are in proportion to one another.
If seven is in the same proportion to 𝑥 as 42 is to 78, then we could say that seven times some constant 𝑐 gives us the value of 𝑥, where that constant of 𝑐 also holds in the equation 42 times 𝑐 equals 78. With our second equation, if we divide through by 42, we get 𝑐 equals 78 divided by 42. Replacing 𝑐 in our first equation with 78 over 42, and we get seven times 78 over 42 equals 𝑥.
Next, we can simplify this equation. We notice both seven and 42 have a factor of seven. Seven divided by seven is one. And 42 divided by seven is six. So our equation becomes one times 78 over six equals 𝑥. In other words, 𝑥 is 78 divided by six. Then we’ll calculate the value of 𝑥 by using the bus stop method. We begin by asking how many sixes make seven. Well, it’s seven with a remainder of one. Next, how many sixes make 18? It’s three. So 78 divided by six is 13, meaning 𝑥 must be equal to 13.
Remember though, we said this wasn’t the only way to solve the problem. We could have solved just like solving any normal equation. Let’s take the equation seven over 𝑥 equals 42 over 78. Since we have purely fractional expressions on both sides, we can find the reciprocal of both sides of this equation. So on the left-hand side, that’s 𝑥 over seven. And on the right-hand side, it’s 78 over 42. Then to make 𝑥 the subject, we multiply through by seven. And At this stage, we notice we have the exact same expression as we did earlier. Either way, we found the value of 𝑥 is 13.