21. At a music store, the CDs Paul bought were $12 each, and the CDs Kate bought were $15 each. If together they paid a total of $78 for 6 CDs, how many CDs did Kate buy?
Answer: A
Kate bought two CDs.
To determine how many CDs Kate bought, we can set up a system of equations based on the prices and total number of CDs purchased. Let \( x \) be the number of CDs Paul bought and \( y \) be the number of CDs Kate bought. The equations are \( x + y = 6 \) and \( 12x + 15y = 78 \). Solving these equations shows that Kate bought 2 CDs.
A) Two
This option is correct. By substituting \( y = 2 \) into the first equation, we find that \( x = 4 \). Checking the total cost: \( 12(4) + 15(2) = 48 + 30 = 78 \), which matches the total amount spent.
B) Three
This option is incorrect. If Kate bought 3 CDs, then substituting \( y = 3 \) into the first equation gives \( x = 3 \). The total cost would then be \( 12(3) + 15(3) = 36 + 45 = 81 \), which exceeds the total spent of $78.
C) Four
This option is incorrect. If Kate bought 4 CDs, substituting \( y = 4 \) leads to \( x = 2 \). The total cost would be \( 12(2) + 15(4) = 24 + 60 = 84 \), which again exceeds the total amount of $78.
D) Five
This option is incorrect. If Kate bought 5 CDs, substituting \( y = 5 \) results in \( x = 1 \). The total cost would be \( 12(1) + 15(5) = 12 + 75 = 87 \), which is also higher than the total of $78.
Conclusion
The analysis confirms that the only viable solution is that Kate bought 2 CDs, as this correctly satisfies both the total number of CDs and the total cost. All other options lead to sums that exceed the given total, reinforcing the correctness of option A.