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?
Kate bought two CDs.
By defining the number of CDs Paul and Kate bought, we can set up a system of equations based on the total cost and the total number of CDs. Solving these equations reveals that Kate purchased two CDs for $15 each.
This choice is correct; if Kate bought 2 CDs at $15 each, she spent $30. Consequently, Paul would have purchased 4 CDs at $12 each, totaling $48. Together, their spending sums to $78, confirming the solution.
If Kate bought 3 CDs at $15 each, her expenditure would be $45. This would leave only $33 for Paul to spend, allowing him to purchase just 2 CDs at $12 each, which totals $24. The combined total would then be $69, which does not match the given total of $78.
Should Kate have bought 4 CDs at $15 each, her total cost would be $60. This would leave Paul needing to spend $18, which would only allow him to buy 1 CD at $12. The total then would be $72, which again falls short of $78.
If Kate purchased 5 CDs at $15 each, her total would be $75. This would leave Paul with only $3 to spend, which is insufficient to buy even one CD at $12. Hence, this scenario is impossible within the given constraints.
Through logical deduction and solving the equations, we determine that Kate indeed bought 2 CDs. The calculations show that this choice satisfies both the total number of CDs and the total amount spent. The other options fail to meet the required conditions, emphasizing the uniqueness of the correct solution.
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