# Problem of the Week Problem D and Solution Mangoes and Oranges

## Problem

At POTW’s Supermarket, Livio stocks mangoes and Dhruv stocks oranges. One day they noticed that an equal number of mangoes and oranges were rotten. Also, $$\frac{2}{3}$$ of the mangoes were rotten and $$\frac{3}{4}$$ of the oranges were rotten. What fraction of the total number of mangoes and oranges was rotten?

## Solution

Solution 1:

Let the total number of mangoes be represented by $$a$$ and the total number of oranges be represented be $$b$$. Since there were an equal number of rotten mangoes and rotten oranges, then $$\frac{2}{3}a=\frac{3}{4}b$$, so $$b=\frac{4}{3}(\frac{2}{3}a)=\frac{8}{9}a$$.
Therefore, there were a total of $$a+b=a+\frac{8}{9}a=\frac{17}{9}a$$ mangoes and oranges.
Also, the total amount of rotten fruit was $$2(\frac{2}{3}a)=\frac{4}{3}a$$.
Therefore, $$\frac{\frac{4}{3}a}{\frac{17}{9}a}=\frac{4}{3} \left( \frac{9}{17}\right) = \frac{12}{17}$$ of the total number of mangoes and oranges was rotten.

Solution 2:

Since $$\frac{2}{3}$$ of the mangoes were rotten, $$\frac{3}{4}$$ of the oranges were rotten, and the number of rotten mangoes equaled the number of rotten oranges, suppose there were $$6$$ rotten mangoes. (We choose $$6$$ as it is a multiple of the numerator of each fraction.) Then the number of rotten oranges will also be $$6$$.
If there were 6 rotten mangoes, then there were a total of $$6 \div \frac{2}{3}= 6\left( \frac{3}{2}\right)=9$$ mangoes.
If there were 6 rotten oranges, then there were a total of $$6 \div \frac{3}{4}= 6\left( \frac{4}{3}\right)=8$$ oranges.
Therefore, there were $$9+8 =17$$ pieces if fruit in total, of which $$6+6 = 12$$ were rotten.
Thus, $$\frac{12}{17}$$ of the total number of mangoes and oranges was rotten.

Note: In Solution 2 we could have used any multiple $$6$$ for the number of rotten mangoes and thus the number of rotten oranges. The final fraction would always reduce to $$\frac{12}{17}$$. We will show this in general in Solution 3.

Solution 3:

According to the problem, there were an equal number of rotten mangoes and rotten oranges.
Let the number of rotten mangoes and rotten oranges each be $$6x$$, for some positive integer $$x$$.
The total number of mangoes was thus $$6x \div \frac{2}{3} = 9x$$.
The total number of oranges was thus $$6x \div \frac{3}{4} = 8x$$.
Therefore, the total number of mangoes and oranges was $$9x + 8x = 17x.$$
Also, the total number of rotten mangoes and rotten oranges was $$6x + 6x = 12x$$.
Therefore, $$\frac{12x}{17x}= \frac{12}{17}$$ of the total number of mangoes and oranges was rotten.