180 pieces of chocolate are to be shared between five boys and 10 girls. Each girl’s share will be two-fifths of each boy’s share. How many pieces of chocolate will each boy and each girl get?
There are lots of different ways of solving this problem. We will use an algebraic method. We can let 𝑥 be the number of pieces of chocolate that each boy gets. And we will let 𝑦 be the number of pieces of chocolate that each girl gets. As there were five boys, the total number of pieces of chocolate that the boys get is five 𝑥. There were 10 girls. So, the total number of pieces of chocolate the girls get is 10 𝑦.
As there were 180 pieces of chocolate altogether, five 𝑥 plus 10𝑦 is equal to 180. We will call this equation one. We were also told that each girl’s share will be two-fifths of each boy’s share. As each girl’s share was equal to 𝑦 and each boy’s share was equal to 𝑥, then 𝑦 is equal to two-fifths of 𝑥. We will call this equation two.
We now have a pair of simultaneous equations, which we can solve. In this question, we will solve them by substitution. We will substitute equation two into equation one. This means that we will replace the 𝑦 in equation one with two-fifths of 𝑥. Equation one then becomes five 𝑥 plus 10 multiplied by two-fifths of 𝑥 is equal to 180.
10 multiplied by two-fifths is equal to four. As 10 multiplied by two is 20. And 20 divided by five is equal to four. The equation, therefore, simplifies to five 𝑥 plus four 𝑥 equals 180. Grouping, or collecting, the like terms on the left-hand side gives us nine 𝑥, as five plus four is equal to nine. We now have nine 𝑥 is equal to 180. Finally, we need to divide both sides of this equation by nine to calculate 𝑥.
Nine 𝑥 divided by nine is equal to 𝑥. And 180 divided by nine is equal to 20, as 18 divided by nine equals two. We can now substitute 𝑥 equals 20 back into equation two. This gives us 𝑦 is equal to two-fifths multiplied by 20, or two-fifths of 20. One-fifth of 20 is equal to four. Therefore, two-fifths of 20 is equal to eight. We have 𝑦 is equal to eight.
As 𝑥 represented the number of pieces of chocolate that each boy gets and 𝑦 represented the number of pieces that each girl gets, we can say that each boy gets 20 pieces of chocolate and each girl gets eight pieces of chocolate. We can check this is correct by multiplying five by 20, as there are five boys who got 20 pieces, and 10 by eight, as there were 10 girls who got eight pieces. Five multiplied by 20 is equal to 100. And 10 multiplied by eight equals 80. Adding these numbers gives us 180. As there were 180 pieces of chocolate that needed to be shared, this suggests that our answer is correct.