It costs `(s+t)` dollars for a box of brand A cat food, and `(q-r)` dollars for a box of brand B cat food. The difference between the cost of 15 boxes of brand A cat food and 7 boxes of brand B cat food is k dollars. Which of the following equations represents a relationship between s, t, q, r, and k?

Options

A. `105(s+t)(q-r)=k`
B. `|7(q-r)+15(s-t)|=k`
C. `|15(s+t)+7(q-r)|=k`
D. `|15(s+t)-7(q-r)|=k`

Answer

To solve the problem, we need to establish the relationship between the costs of the cat food brands and the given variables. Let's break it down step by step.

### Step 1: Identify the costs of the cat food brands
- The cost of one box of brand A cat food is given as \( s + t \) dollars.
- The cost of one box of brand B cat food is given as \( q - r \) dollars.

### Step 2: Calculate the total costs for 15 boxes of brand A and 7 boxes of brand B
- The cost for 15 boxes of brand A cat food is:
\[ 15 \times (s + t) = 15s + 15t \]
- The cost for 7 boxes of brand B cat food is:
\[ 7 \times (q - r) = 7q - 7r \]

### Step 3: Establish the difference in costs
According to the problem, the difference between the cost of 15 boxes of brand A and 7 boxes of brand B is \( k \) dollars. Therefore, we can write the equation:
\[ (15s + 15t) - (7q - 7r) = k \]

### Step 4: Rearranging the equation
We can rearrange the equation to isolate \( k \):
\[ 15s + 15t - 7q + 7r = k \]

### Step 5: Absolute value consideration
Since costs cannot be negative, we need to consider the absolute value of the difference:
\[ |15s + 15t - 7q + 7r| = k \]

### Final Equation
Thus, the relationship between \( s, t, q, r, \) and \( k \) can be expressed as:
\[ |15(s + t) - 7(q - r)| = k \]

### Conclusion
The correct equation representing the relationship between \( s, t, q, r, \) and \( k \) is:
\[ |15(s + t) - 7(q - r)| = k \]