Question

A market sells 6 cans of food for every 7 boxes of food. The market sold a total of 26 cans and boxes today. How many of each kind did the market sell?
6 cans and 20 boxes
12 cans and 14 boxes
14 cans and 12 boxes
20 cans and 6 boxes

Answer 1

Let the number of cans sold be \( c \) and the number of boxes sold be \( b \). According to the problem, we have two key pieces of information:

1. The total number of cans and boxes sold is 26:
\[
c + b = 26
\]

2. The ratio of cans to boxes is 6 to 7. We can express this as:
\[
\frac{c}{b} = \frac{6}{7}
\]
This can be rearranged to:
\[
7c = 6b
\]

Now, we can solve these two equations step by step. Let's start with the second equation by expressing \( b \) in terms of \( c \):
\[
b = \frac{7}{6}c
\]

Now substitute \( b \) in the first equation:
\[
c + \frac{7}{6}c = 26
\]

To combine the terms, convert \( c \) into a fraction:
\[
\frac{6}{6}c + \frac{7}{6}c = 26
\]
\[
\frac{13}{6}c = 26
\]

Now, solve for \( c \):
\[
13c = 156
\]
\[
c = 12
\]

Now that we have \( c \), substitute it back into the equation for \( b \):
\[
b = 26 - c = 26 - 12 = 14
\]

So, the market sold 12 cans and 14 boxes.

Now let's verify the ratio to ensure correctness:
- Cans: 12
- Boxes: 14

The ratio is:
\[
\frac{12}{14} = \frac{6}{7}
\]

This confirms that the sales adhere to the given ratio.

Therefore, the market sold **12 cans and 14 boxes**.