### Video Transcript

If seven over ๐ฅ equals 42 over 78, find the value of ๐ฅ.

Weโve been given an equation that describes the proportion between different values. Weโre told that the proportion of seven to ๐ฅ 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 ๐ฅ the subject. Alternatively, weโll think about what it means when things are in proportion to one another.

If seven is in the same proportion to ๐ฅ as 42 is to 78, then we could say that seven times some constant ๐ gives us the value of ๐ฅ, where that constant of ๐ also holds in the equation 42 times ๐ equals 78. With our second equation, if we divide through by 42, we get ๐ equals 78 divided by 42. Replacing ๐ in our first equation with 78 over 42, and we get seven times 78 over 42 equals ๐ฅ.

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 ๐ฅ. In other words, ๐ฅ is 78 divided by six. Then weโll calculate the value of ๐ฅ by using the bus stop method. We begin by asking how many sixes make seven. Well, itโs seven with a remainder of one. Next, how many sixes make 18? Itโs three. So 78 divided by six is 13, meaning ๐ฅ must be equal to 13.

Remember though, we said this wasnโt the only way to solve the problem. We could have solved just like solving any normal equation. Letโs take the equation seven over ๐ฅ 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, thatโs ๐ฅ over seven. And on the right-hand side, itโs 78 over 42. Then to make ๐ฅ 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 ๐ฅ is 13.