# Ratios and Sharing

Here is a word problem that involves ratios and good will:

On a Halloween night, Brother and Sister went collecting the treats. On return home, they found that Brother collected only 3/4 of what Sister had. Being the older of the two, Sister decided to share with Brother and gave him half of her treats to Brother. After that noble action she had 12 treats fewer than Brother. How many of the treats did each of them collect on the Halloween night?

Solution

On a Halloween night, Brother and Sister went collecting the treats. On return home, they found that Brother collected only 3/4 of what Sister had. Being the older of the two, Sister decided to share with Brother and gave him half of her treats to Brother. After that noble action she had 12 treats fewer than Brother. How many of the treats did each of them collect on the Halloween night?

### Solution

The problem tells us that Brother had 3/4 of what Sister had. This means that for every 3 treats that Brother had, Sister had 4 treats. In other words, to every 3 parts of Brother's treats Sister had 4 parts. One half of 4,2,3,4,5,6 Sister's parts equals 2,2,3,4,5,6 parts.

So the situation is this: Brother has 3 parts, Sister has 4 parts. She gives 2 parts to Brother, after which Brother has 5,2,3,4,5,6 parts and Sister has 2,2,3,4,5,6 parts. So, at this point, Brother has 3,2,3,4,5,6 parts more than Sister. The problem tells us that these three parts count 12 treats, meaning that each part equals 4 treats.

Thus on return home Brother had 12 treats (3 parts each of which is 4 treats) and Sister had 16 treats.

Let's check. After Sister gave her Brother half of what she had, she was left with 8,4,5,6,7,8 while Brother got 20,18,19,20,21,22, which is exactly 12 more than the number of Sister's treats. We are done.

Doing this algebraically is not very much different. Let letter P stand for a "part" of which Brother had 3 and Sister 4. So it comes 3P for Brother and 4P,2P,3P,4P,6P,8P for Sister. After her giving Brother 2P she is left with 2P, while the boy gets 5P, which 3P more. We are getting an equation 3P = 12. Divide both sides by 3 and get P = 4. This makes 3P = 12 (for Brother) and 4P = 16 (for Sister).

Yet there is another way. Let B and S be the amounts of treats Brother and Sister brought home on the Halloween night. The first condition of the problem tells us that

B = 3/4 S, or S = 4/3 B.

After Sister gives half of S to Brother she is left with S/2,S/2,S/3,S/4,S/6, while Brother gets B + S/2. The difference

(B + S/2) - S/2 = B

is simply what Brother started with. This is equal to 12: B = 12, from which S = 4/3×12 = 16, as it should be.