In a bakery, donuts and cupcakes are sold only in boxes. A box of 8 donuts costs $15 and a box of 5 cupcakes costs$7. Alice and Ben were given a sum of money to buy donuts and cupcakes from the bakery. Alice needed 20 donuts and 3 cupcakes for her party. After buying the donuts and cupcakes from the bakery, Alice passed the rest of the money to Ben. Ben spent $\frac{4}{7}$ of the rest of the money to buy donuts and cupcakes for his party and had $\frac{1}{3}$ of the sum of money left.

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 =$2 \times 5 = 10$.