Daniel, Michael, and Ethan share some money in the ratio four to five to eight. Twice Daniel’s share exceeds Michael’s share by 12 Egyptian pounds. How much money do Daniel, Michael, and Ethan each have?
Now we’re given the ratio in which the money is shared. This tells us how many parts each person gets relative to the other. In other words, Daniel gets four parts, whilst Michael gets five, and Ethan gets eight. Now, if we knew the total amount which they were sharing, we could work out how much each person gets. But we’re told a different bit of information. We’re told that twice Daniel’s share exceeds Michael’s share by 12 Egyptian pounds. So let’s see how that plays out in terms of the parts of the total amount they each get.
If Daniel gets four parts of the money, then twice the number of parts he gets is two times four, which is equal to eight. Twice Daniel’s share exceeds Michael’s share by 12 Egyptian pounds. So we’re going to work out the number of parts we get when twice Daniel’s parts exceeds Michael’s parts. In other words, eight minus five is equal to three. This means that three parts in our ratio represents 12 Egyptian pounds.
To work out the total amount of money that each person receives, we need to work out what one part represents. Since three divided by three is one, we’re going to divide through by three here. 12 divided by three is four. So that tells us that one part must be worth four Egyptian pounds.
Since Daniel receives four parts, he must get four times four Egyptian pounds, which is equal to 16. Similarly, Michael gets five parts, so that’s five times four, which is 20 Egyptian pounds. Finally, Ethan gets eight parts of the money, so he gets eight times four Egyptian pounds, which is 32 Egyptian pounds. So we can see that Daniel has 16 Egyptian pounds, Michael has 20, and Ethan has 32.
Now we can check this in a couple of ways. First, let’s double-check that twice Daniel’s share exceeds Michael’s. That’s two times 16 minus 20, or 32 minus 20, which is equal to 12. And we were expecting this to be equal to 12 Egyptian pounds. Next, let’s check that the ratio of the amounts of money they each receive simplifies to four to five to eight. It’s 16 to 20 to 32. And when we divide each part by four, we do indeed get four to five to eight, confirming we’ve answered the question correctly.