MATHCOUNTS® Problem of the Week Archive Congratulations Jennie! – June 24, 2013 Problems & Solutions As a graduation gift, Nancy purchased a watch for her daughter and planned to have it engraved with the following text: Congrats Jennie!
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Nancy received a quote on engraving from a gift shop in the mall. At this gift shop, all engraving orders are subject to a service charge, which is the same for every order. Then there is a charge to engrave the message ‘Congrats Jennie!’ and a separate charge for the date enscription. For this engraving, Nancy was quoted a total amount, before sales tax, of $15. When Nancy reviewed the break‐down of the quote, she noticed that the charge to engrave her message happened to be equal to the service charge, which was twice the charge for the date enscription. What was the amount of the service charge? The total quoted charge for the engraving is the sum of the service charge, s, the charge for engraving the
message, m and the charge for the date enscription, d. We are told that the service charge and the
charge for engraving the message were the same, and that this amount was twice charge for the date
enscription. That means s = m = 2d. Since the total charge is s + m + d = 15, we can substitute to get
s + s + (1/2)s = 15  (5/2)s = 15. Solving for s, we see that the service charge was $6.
Instead of the previous message, Nancy decided she preferred the following text: Congratulations Jennie!
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She was informed by the gift shop manager that the service charge and the charge for the date enscription would be the same as previously quoted, and the only variable cost was the charge for the engraved message. Not including spaces, the previous message ‘Congrats Jennie!’ has a total of 15 characters. If the charge per character remained the same, how much more than the previously quoted total amount would Nancy expect to be the charge for having her preferred text engraved on the watch? From the previous problem we know that the charge for engraving the message was the same as the
service charge, $6. Therefore, the rate per letter was 6  15 = $0.40. At a rate of 40 cents per letter, the new
message, which contains 22 letters would cost 22  0.40 = $8.80. The charge for the date enscription, which
remained unchanged, was 1/2 the service charge, or $3. The total for the new message would be
6 + 3 + 8.80 = $17.80. That is 17.80 − 15 = $2.80 more than the previous quote.
Nancy then requested a quote for engraving from the jewelry store where she purchased the watch. The charge to have the above preferred text engraved would normally be $20, before sales tax. However, since the engraving was for an item purchased at jewelry store, she was entitled to a 25% discount. How much would Nancy save by having the engraving done at the jewelry store instead of the gift shop? A 25% discount on the normal charge of $20 means Nancy would be charged 20  0.75 = $15. This is the
amount she was orignally quoted from the gift shop. Therefore, she would get her new preferred text
engraved at a savings of $2.80 over the gift shop’s cost.