21) Watch the video and then solve the problem given below. Click here to watch the video. The employees from maintenance go for coffee together every day at 9 a m . On Monday, Hector paid $$ \$ 6.00 $$ for three cartons of milk, nine for four milks, seven coffees, and eight doughnuts. On Wednesday, Anna paid $$ \$ 4.05 $$ for two milks, five coffees, and six doughnuts. What is 13 The price of a carton of milk is $$ \square $$. (Type an integer or a decimal.) The price of a coffee is The price of a doughnut is $$ \square $$. (Type an integer or a decimal.)

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UpStudy AI · Accepted Answer

To solve the problem, we need to set up a system of equations based on the information given about the purchases made by Hector and Anna.

Let:
- $$ m $$ = price of a carton of milk
- $$ c $$ = price of a coffee
- $$ d $$ = price of a doughnut

From Hector's purchases on Monday, we have the equation:
$$
3m + 7c + 8d = 6.00 \quad \text{(1)}
$$

From Anna's purchases on Wednesday, we have the equation:
$$
2m + 5c + 6d = 4.05 \quad \text{(2)}
$$

Now we have two equations with three unknowns. To solve for the prices, we can express one variable in terms of the others. Let's solve for $$ m $$ in terms of $$ c $$ and $$ d $$ using equation (1):

1. Rearranging equation (1):
$$
3m = 6.00 - 7c - 8d
$$
$$
m = \frac{6.00 - 7c - 8d}{3} \quad \text{(3)}
$$

Now, we can substitute equation (3) into equation (2):

2. Substitute $$ m $$ into equation (2):
$$
2\left(\frac{6.00 - 7c - 8d}{3}\right) + 5c + 6d = 4.05
$$

3. Multiply through by 3 to eliminate the fraction:
$$
2(6.00 - 7c - 8d) + 15c + 18d = 12.15
$$
$$
12.00 - 14c - 16d + 15c + 18d = 12.15
$$

4. Combine like terms:
$$
12.00 + c + 2d = 12.15
$$
$$
c + 2d = 0.15 \quad \text{(4)}
$$

Now we have a new equation (4) relating $$ c $$ and $$ d $$. We can express $$ c $$ in terms of $$ d $$:
$$
c = 0.15 - 2d \quad \text{(5)}
$$

Next, we can substitute equation (5) back into equation (3) to find $$ m $$:
$$
m = \frac{6.00 - 7(0.15 - 2d) - 8d}{3}
$$
$$
m = \frac{6.00 - 1.05 + 14d - 8d}{3}
$$
$$
m = \frac{4.95 + 6d}{3}
$$

Now we have expressions for $$ m $$ and $$ c $$ in terms of $$ d $$. We can substitute $$ d $$ with a reasonable value to find integer or decimal prices.

Let's assume $$ d = 0.05 $$ (5 cents):
$$
c = 0.15 - 2(0.05) = 0.15 - 0.10 = 0.05
$$
$$
m = \frac{4.95 + 6(0.05)}{3} = \frac{4.95 + 0.30}{3} = \frac{5.25}{3} = 1.75
$$

Thus, we have:
- Price of a carton of milk $$ m = 1.75 $$
- Price of a coffee $$ c = 0.05 $$
- Price of a doughnut $$ d = 0.05 $$

Final answers:
- The price of a carton of milk is $$ 1.75 $$.
- The price of a coffee is $$ 0.05 $$.
- The price of a doughnut is $$ 0.05 $$.
The price of a carton of milk is $$ 1.75 $$. The price of a coffee is $$ 0.05 $$. The price of a doughnut is $$ 0.05 $$.

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