Question

Deb's Stationery sells thank-you cards in packs of 11 and birthday cards in packages of 8. Surprisingly, the store sold the same number of each type of card this month. What is the minimum number of each card type that the store must have sold?

Answer 1

Let the number of packs of thank-you cards sold be denoted by \( x \) and the number of packs of birthday cards sold be denoted by \( y \).

Since each pack of thank-you cards contains 11 cards, the total number of thank-you cards sold is given by:
\[
11x
\]
Similarly, as each pack of birthday cards contains 8 cards, the total number of birthday cards sold is given by:
\[
8y
\]
According to the problem, the total number of thank-you cards sold is equal to the total number of birthday cards sold:
\[
11x = 8y
\]

To find a solution, we can rewrite this equation:
\[
\frac{x}{y} = \frac{8}{11}
\]
This implies that \( x \) is proportional to 8 and \( y \) is proportional to 11. We can express \( x \) and \( y \) in terms of a common variable \( k \):
\[
x = 8k \quad \text{and} \quad y = 11k
\]
where \( k \) is a positive integer (the common factor).

Now, we are looking to find the minimum number of thank-you cards and birthday cards sold. The total cards sold can be calculated as:
\[
11x = 11(8k) = 88k \quad \text{(thank-you cards)}
\]
\[
8y = 8(11k) = 88k \quad \text{(birthday cards)}
\]

To minimize the number of cards sold, we select the smallest positive integer value for \( k \), which is \( k = 1 \). Plugging this value into our equations gives:
\[
x = 8(1) = 8 \quad \text{(packs of thank-you cards)}
\]
\[
y = 11(1) = 11 \quad \text{(packs of birthday cards)}
\]

Thus, the minimum number of thank-you cards sold is:
\[
11x = 11 \times 8 = 88
\]
And the minimum number of birthday cards sold is:
\[
8y = 8 \times 11 = 88
\]

Hence, the minimum number of each card type that the store must have sold is:
\[
\boxed{88}
\]