Comparing Two Tutoring Center Bills
A student needs at least 8 hours of tutoring before a placement test. Center A charges a $25 registration fee plus $20 per hour, but it gives a 10% discount on the hourly charge for any package of 8 or more hours. Center B charges a $45 registration fee plus $16 per hour and offers a $10 coupon off the total bill. The student's family can spend at most $180. A friend says Center A is cheaper because its hourly rate is only $4 more and its registration fee is $20 lower. The student wants a comparison for 8 hours and for 10 hours, plus a recommendation that respects the budget. Both centers require a one-time registration before the first session; the discount at Center A applies only to the hourly portion, not to registration. The coupon at Center B cannot be combined with any other offer.
1. Calculate the total cost at each center for 8 hours and for 10 hours. Show the discount steps.
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For 8 hours: Center A charges $25 registration plus 8 × $20 = $160 in hourly charges. The 10% discount applies to the hourly portion only: 0.10 × $160 = $16, so the total is $25 + $160 − $16 = $169. Center B charges $45 + 8 × $16 = $45 + $128 = $173, then subtracts the $10 coupon: $163. Center B is cheaper by $6 at 8 hours. For 10 hours: Center A charges $25 + 10 × $20 = $225, then subtracts 10% of $200, which is $20, for a total of $205. Center B charges $45 + 10 × $16 = $205, then subtracts $10, for a total of $195. Center B is cheaper by $10 at 10 hours. The friend's quick comparison ignores the discounts and the coupon.
2. Find the number of hours at which the two centers' base prices (registration plus hourly rate, before any discounts or coupons) are equal. Explain what that result means in this situation.
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Set the base costs equal: 25 + 20h = 45 + 16h. Subtract 16h from both sides: 25 + 4h = 45. Subtract 25: 4h = 20. Divide by 4: h = 5. At 5 hours, both base prices are $125. This is the break-even point before discounts. But the student needs at least 8 hours, so the break-even point is below the required range. That means the comparison for this student must use the discounted totals, not just the base rates. The result also shows why the friend's reasoning is incomplete: at 5 hours the base prices are equal, but the coupon and the 10% discount change the totals at 8 and 10 hours.
3. The family can spend at most $180. Using the discounted totals, which center can provide at least 8 hours within budget, and what is the maximum whole number of hours the family can afford at that center?
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At 8 hours, both centers are within $180: Center A costs $169 and Center B costs $163. At 9 hours, Center A costs $25 + 9 × $20 = $205, minus 10% of $180, which is $18, for $187. That is over $180. Center B costs $45 + 9 × $16 = $189, minus $10, for $179. That is within $180. At 10 hours, Center B costs $195, which is over budget. So the best choice is Center B, and the family can afford a maximum of 9 whole hours there. The answer should not stop at 'both are under budget at 8 hours'; it must test the next hour to find the actual maximum.