Question

A bakery sells cupcakes in packs of 6 and packs of 9. If \( p \) packs of 6 and \( q \) packs of 9 are bought, and the total number of cupcakes is 90, what is the smallest possible value of \( p + q \)?

• A. \( 7 \)
• B. \( 8 \)
• C. \( 9 \)
• D. \( 10 \)
• E. \( 11 \)

Hint:

When solving problems involving linear Diophantine equations, test integer values for one variable and solve for the other.

Answer 1

The Correct Option is B

Step 1: The total number of cupcakes is given by \( 6p + 9q = 90 \), where \( p \) and \( q \) are the number of packs of 6 and 9 cupcakes, respectively. Step 2: Simplify the equation by dividing the entire equation by 3: \[ 2p + 3q = 30. \] Step 3: Now, we need to find integer values of \( p \) and \( q \) that satisfy this equation. We will test different values of \( q \) and solve for \( p \): - If \( q = 0 \), then \( 2p = 30 \) gives \( p = 15 \). - If \( q = 2 \), then \( 2p + 3(2) = 30 \) gives \( 2p + 6 = 30 \), so \( 2p = 24 \) and \( p = 12 \). - If \( q = 4 \), then \( 2p + 3(4) = 30 \) gives \( 2p + 12 = 30 \), so \( 2p = 18 \) and \( p = 9 \). - If \( q = 6 \), then \( 2p + 3(6) = 30 \) gives \( 2p + 18 = 30 \), so \( 2p = 12 \) and \( p = 6 \). - If \( q = 8 \), then \( 2p + 3(8) = 30 \) gives \( 2p + 24 = 30 \), so \( 2p = 6 \) and \( p = 3 \). - If \( q = 10 \), then \( 2p + 3(10) = 30 \) gives \( 2p + 30 = 30 \), so \( 2p = 0 \) and \( p = 0 \). Step 4: The smallest value of \( p + q \) occurs when \( p = 0 \) and \( q = 10 \), which gives \( p + q = 10 \), but the next smallest value is when \( p = 3 \) and \( q = 8 \), which gives \( p + q = 8 \). Thus, the smallest possible value of \( p + q \) is \( 8 \).