 # Problem of the Week Problem D and Solution Boxes of Doughnuts

## Problem

A bakery is famous for its specialty doughnuts. One weekend they had three flavours available, each packaged in boxes. Peach caramel doughnuts were sold in small boxes with $$3$$ doughnuts per box, chocolate fudge doughnuts were sold in medium boxes with $$4$$ doughnuts per box, and rainbow doughnuts were sold in large boxes with $$8$$ doughnuts per box.

At the end of the weekend, the owner calculated that they sold $$68$$ boxes of doughnuts in total. Also an equal number of doughnuts of each flavour were sold. How many doughnuts did they sell in total? ## Solution

Solution 1

Let $$n$$ represent the number of doughnuts of each flavour sold. Since there were $$3$$ peach caramel doughnuts per box, then $$n$$ must be divisible by $$3$$. Since there were $$4$$ chocolate fudge doughnuts per box, then $$n$$ must be divisible by $$4$$. Since there were $$8$$ rainbow doughnuts per box, then $$n$$ must be divisible by $$8$$. Therefore, $$n$$ must be divisible by $$3,~4,$$ and $$8$$. The smallest number divisible by $$3,~4,$$ and $$8$$ is $$24$$. (This number is called the least common multiple or LCM).

If there were $$24$$ doughnuts of each flavour sold, there would have been $$24\div 3=8$$ boxes of peach caramel doughnuts, $$24\div 4=6$$ boxes of chocolate fudge doughnuts, and $$24\div 8=3$$ boxes of rainbow doughnuts. This would mean $$8+6+3=17$$ boxes of doughnuts would have been sold in total. However, we know that $$68$$ boxes of doughnuts were sold, and since $$68=17\times 4$$, it follows that $$4$$ times as many doughnuts were sold. Therefore, $$24\times 4=96$$ doughnuts of each flavour were sold. Thus, the total number of doughnuts sold was $$96\times 3=288$$.

We can check the correctness of this solution. Since there were $$3$$ peach caramel doughnuts per box, then there were $$96\div 3=32$$ boxes of peach doughnuts sold. Since there were $$4$$ chocolate fudge doughnuts per box, then there were $$96\div 4=24$$ boxes of chocolate fudge doughnuts sold. Since there were $$8$$ rainbow doughnuts per box, then there were $$96\div 8=12$$ boxes of rainbow doughnuts sold. The total number of boxes sold was therefore $$32+24+12=68$$, as expected.

Solution 2

This solution uses algebra and equation solving. Let $$n$$ represent the number of doughnuts of each flavour sold. Since there were $$3$$ peach caramel doughnuts per box, then $$\frac{n}{3}$$ boxes of peach caramel doughnuts were sold. Since there were $$4$$ chocolate fudge doughnuts per box, then $$\frac{n}{4}$$ boxes of chocolate fudge doughnuts were sold. Since there were $$8$$ rainbow doughnuts per box, then $$\frac{n}{8}$$ boxes of rainbow doughnuts were sold. Since $$68$$ boxes of doughnuts were sold in total, \begin{aligned} \frac{n}{3} + \frac{n}{4} + \frac{n}{8} &= 68\\ \frac{8n}{24} + \frac{6n}{24} + \frac{3n}{24} &= 68\\ \frac{17n}{24} &= 68\\ 17n &= 68 \times 24\\ 17n &= 1632\\ n &= \frac{1632}{17} = 96 \end{aligned} Therefore, $$96$$ doughnuts of each flavour were sold. Thus, the total number of doughnuts sold was $$96\times 3=288$$.

Solution 3

This solution uses ratios. Let $$n$$ represent the number of doughnuts of each flavour sold. The ratio of the number of boxes of rainbow doughnuts to peach caramel doughnuts sold is $\frac{n}{8} : \frac{n}{3} = \frac{3n}{24} : \frac{8n}{24} = 3n : 8n = 3:8$ Similarly, the ratio of the number of boxes of peach caramel doughnuts to chocolate fudge doughnuts sold is $$4:3 = 8:6$$. So the ratio of the number of boxes of rainbow doughnuts to peach caramel doughnuts to chocolate fudge doughnuts sold is $$3:8:6$$. Let the number of boxes of rainbow doughnuts be $$3k$$, the number of boxes of peach caramel doughnuts be $$8k$$, and the number of boxes of chocolate fudge doughnuts be $$6k$$. Since $$68$$ boxes of doughnuts were sold in total, \begin{aligned} 3k+8k+6k &= 68\\ 17k &= 68\\ k &= \frac{68}{17}=4 \end{aligned} It follows that the number of boxes of rainbow doughnuts sold was $$3 \times 4 =12$$, so the number of rainbow doughnuts sold was $$12 \times 8 = 96$$. Therefore $$n=96$$, so $$96$$ doughnuts of each flavour were sold. Thus, the total number of doughnuts sold was $$96\times 3=288$$.