To find the minimum number of each type of card that Jill's Stationery must have sold, we are looking for the least common multiple (LCM) of the pack sizes for the thank-you cards and birthday cards.
The thank-you cards are sold in packs of 11 and the birthday cards in packs of 8. We can denote the number of packs of thank-you cards sold as \( x \) and the number of packs of birthday cards sold as \( y \).
To have the same number of thank-you cards and birthday cards sold, we can express this mathematically as follows: \[ 11x = 8y \] To find integer values of \( x \) and \( y \) that satisfy this equation, we need to find the LCM of 11 and 8.
Since 11 is a prime number and does not divide 8, the LCM can be calculated directly as: \[ \text{LCM}(11, 8) = 11 \times 8 = 88 \]
Next, we use the LCM to find the minimum number of each type of card sold. We find the number of packs required for this minimum total:
-
The number of thank-you cards sold is given by: \[ \text{Number of thank-you cards} = \frac{88}{11} = 8 \text{ packs} \]
-
The number of birthday cards sold is given by: \[ \text{Number of birthday cards} = \frac{88}{8} = 11 \text{ packs} \]
Now, we calculate the total number of cards sold for each type:
- Total number of thank-you cards sold: \[ 8 \text{ packs} \times 11 \text{ cards/pack} = 88 \text{ thank-you cards} \]
- Total number of birthday cards sold: \[ 11 \text{ packs} \times 8 \text{ cards/pack} = 88 \text{ birthday cards} \]
Thus, the minimum number of each card type that the store must have sold is: \[ \boxed{88} \]