在一家面包店里，甜甜圈和纸杯蛋糕仅按箱销售。一箱 8 个甜甜圈的价格为 15 美元，一箱 5 个纸杯蛋糕的价格为 7 美元。爱丽丝（Alice）和本（Ben）得到了一笔钱，去这家面包店买甜甜圈和纸杯蛋糕。 爱丽丝的派对需要 20 个甜甜圈和 3 个纸杯蛋糕。从面包店买完甜甜圈和纸杯蛋糕后，爱丽丝把剩下的钱交给了本。 本用剩下钱的 $\frac{4}{7}$ 为他的派对购买了甜甜圈和纸杯蛋糕，并且还剩下这笔总钱数的 $\frac{1}{3}$。

1. Find the number of boxes of donuts Alice needs to buy and their cost.

Alice needs 20 donuts. Since they are sold in boxes of 8:
$20 \div 8 = 2$ with a remainder of $4$.
She must buy $3$ boxes to have enough donuts.
Cost of donuts = $3 \times \char36{}15 = \char36{}45$

2. Find the number of boxes of cupcakes Alice needs to buy and their cost.

Alice needs 3 cupcakes. Since they are sold in boxes of 5:
She must buy $1$ box to have enough cupcakes.
Cost of cupcakes = $1 \times \char36{}7 = \char36{}7$

3. Calculate the total least amount of money spent.

Total amount spent = $\char36{}45 + \char36{}7 = \char36{}52$

1. Find the fraction of the remaining money that Ben had left.

Ben spent $\frac{4}{7}$ of the rest of the money, so the fraction of the rest he had left is:
$1 - \frac{4}{7} = \frac{3}{7}$

2. Express the rest of the money as a fraction of the total sum.

We are given that Ben had $\frac{1}{3}$ of the total sum of money left. This means $\frac{3}{7}$ of the rest of the money is equal to $\frac{1}{3}$ of the total sum.
$\frac{3}{7} \times \text{Rest} = \frac{1}{3} \times \text{Total}$
$\text{Rest} = \frac{1}{3} \div \frac{3}{7} = \frac{1}{3} \times \frac{7}{3} = \frac{7}{9} \text{ of the Total}$

3. Calculate the fraction of the total sum that Alice spent.

Since the rest of the money is $\frac{7}{9}$ of the total sum, the fraction Alice spent is:
$1 - \frac{7}{9} = \frac{2}{9}$

1. Determine the total sum of money using the fraction left

Let the rest of the money (after Alice's purchase) be $R$.
Ben spent $\frac{4}{7}$ of $R$, so he had $1 - \frac{4}{7} = \frac{3}{7}$ of $R$ left.
We are given that this amount left is $\frac{1}{3}$ of the total sum of money ($T$).
So, $\frac{3}{7} R = \frac{1}{3} T \implies R = \frac{7}{9} T$.
This means Alice spent $T - R = T - \frac{7}{9} T = \frac{2}{9} T$.

2. Calculate the total sum of money and the amount Ben spent

From part (a), Alice spent $52. So,$\frac{2}{9} T = 52 \implies T = 52 \times \frac{9}{2} = 234$. The rest of the money$R = 234 - 52 = 182$. Ben spent$\frac{4}{7}$of$R$, which is$\frac{4}{7} \times 182 = 104$.

3. Determine the number of donuts and cupcakes Ben bought

Ben spent $104 on donuts and cupcakes. Let$d$be the number of boxes of donuts and$c$be the number of boxes of cupcakes.$15d + 7c = 104$. Since$d$and$c$must be non-negative integers, we can test values for$d$: If$d = 1$,$15(1) + 7c = 104 \implies 7c = 89$(not divisible by 7). If$d = 2$,$15(2) + 7c = 104 \implies 7c = 74$(no). If$d = 3$,$15(3) + 7c = 104 \implies 7c = 59$(no). If$d = 4$,$15(4) + 7c = 104 \implies 7c = 44$(no). If$d = 5$,$15(5) + 7c = 104 \implies 7c = 29$(no). If$d = 6$,$15(6) + 7c = 104 \implies 7c = 14 \implies c = 2$. So Ben bought 6 boxes of donuts and 2 boxes of cupcakes. Number of donuts =$6 \times 8 = 48$. Number of cupcakes =$2 \times 5 = 10$.