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TchrWill

USA
80 Posts

 Posted - 05/17/2013 :  12:04:55 An oldie that you might have come across before.A customer at a 7-11 store selected 4 items to buy, and was told that the cost was \$7.11. He was curious that the cost was the same as the store name, so he inquired as to how the figure was derived. The clerk said that he had simply multiplied the prices for the 4 induvidual items.The customer protested that the 4 prices should have been ADDED, not MULTIPLIED. The clerk said that that was okay with him, but the result was still the same, exactly \$7.11 What were the prices of the four items?

Ultraglide

299 Posts

 Posted - 05/18/2013 :  16:23:28 We have to look at the factors of 711 i.e. 3, 3, and 79, so one solution would be 1,3,3,79. Other possibilities are 1,1,9,79 or 1,1,1,711.

TchrWill

USA
80 Posts

 Posted - 05/19/2013 :  10:40:52 quote:Originally posted by UltraglideWe have to look at the factors of 711 i.e. 3, 3, and 79, so one solution would be 1,3,3,79. Other possibilities are 1,1,9,79 or 1,1,1,711.Remember that a + b + c + d = a(b)c(d) = \$7.11

Ultraglide

299 Posts

 Posted - 05/19/2013 :  18:52:57 Oops, helps if you read the question.

the_hill1962

USA
1468 Posts

 Posted - 05/21/2013 :  15:59:45 You need to be looking at factors of 71100000000, not 711 because when you multiply the four items, the "cents" make an 8 place decimal.i.e. "1" means 1 cent === 0.01.01*.01*.01*.01=.00000001, not 1.Again, the problem got the best of me and I had to look up the solution.Sure enough, it involves solvinga+b+c+d=711abcd=711000000Nota+b+c+d=711abcd=711Note, I had originally thought 71100000000, not 711000000.I gather that it is 711000000 because the two ones count as part of the 8 place decimal. Edited by - the_hill1962 on 05/21/2013 16:00:19

someguy

143 Posts

 Posted - 05/21/2013 :  21:50:18 Hi thehill, my take on where the equations come from is as follows.If you let a, b, c, and d be the cost in cents, then a/100, b/100, c/100, and d/100 are the costs of each item in dollars.We are told the sum of the prices equals \$7.11.(a/100) + (b/100) + (c/100) + (d/100) = 7.11Multiplying through by 100 gives a+b+c+d=711.We are told the product of the prices is \$7.11.(a/100)*(b/100)*(c/100)*(d/100)=711/100Multiplying both sides by (100)^4 to clear the denominator gives abcd=711*(100)^3=711000000I haven't figured out a way to do it by hand yet, but did write a small bit of code to find the solution.I need to find a way to cut down the number of ways to group the primes together to make the problem small enough to handle by hand.

TchrWill

USA
80 Posts

the_hill1962

USA
1468 Posts

 Posted - 05/23/2013 :  12:49:47 Yes, I like your explanation.The solution is, indeed \$1.20, \$1.25, \$1.50 and \$3.16Now, how was this problem conceived?Can anyone create a problem such as this? Edited by - the_hill1962 on 05/23/2013 14:27:32

TchrWill

USA
80 Posts

 Posted - 05/24/2013 :  15:47:23 quote:Originally posted by the_hill1962Yes, I like your explanation.The solution is, indeed \$1.20, \$1.25, \$1.50 and \$3.16Now, how was this problem conceived?Can anyone create a problem such as this?If you enjoy these puzzles, there are other numbers that work like this. \$7.11 is not the only number which works. Here are the first 160 such numbers, preceded by a count of distinct solutions for that price. While \$7.11 has a single, unique solution, some of the other numbers have several distinct answers. I found these many years ago but have lost the site address.1 - \$6.44 1 - \$7.83 2 - \$9.20 3 - \$10.891 - \$6.51 1 - \$7.86 1 - \$9.23 1 - \$10.951 - \$6.60 3 - \$7.92 1 - \$9.24 2 - \$11.001 - \$6.63 1 - \$8.00 1 - \$9.27 1 - \$11.071 - \$6.65 1 - \$8.01 2 - \$9.35 1 - \$11.131 - \$6.72 1 - \$8.03 3 - \$9.36 1 - \$11.162 - \$6.75 5 - \$8.10 1 - \$9.38 1 - \$11.221 - \$6.78 1 - \$8.12 5 - \$9.45 2 - \$11.251 - \$6.80 1 - \$8.16 2 - \$9.48 2 - \$11.272 - \$6.84 2 - \$8.19 1 - \$9.54 1 - \$11.301 - \$6.86 1 - \$8.22 1 - \$9.57 1 - \$11.361 - \$6.89 1 - \$8.25 1 - \$9.59 1 - \$11.402 - \$6.93 3 - \$8.28 2 - \$9.60 2 - \$11.431 - \$7.02 3 - \$8.33 1 - \$9.62 2 - \$11.521 - \$7.05 1 - \$8.36 2 - \$9.63 2 - \$11.552 - \$7.07 1 - \$8.37 1 - \$9.66 2 - \$11.611 - \$7.08 2 - \$8.40 1 - \$9.68 1 - \$11.691 - \$7.11 1 - \$8.45 2 - \$9.69 1 - \$11.701 - \$7.13 2 - \$8.46 1 - \$9.78 1 - \$11.882 - \$7.14 1 - \$8.52 2 - \$9.80 1 - \$11.903 - \$7.20 5 - \$8.55 1 - \$9.81 1 - \$11.991 - \$7.25 1 - \$8.60 1 - \$9.87 1 - \$12.061 - \$7.26 4 - \$8.64 4 - \$9.90 1 - \$12.152 - \$7.28 1 - \$8.67 1 - \$9.92 1 - \$12.181 - \$7.29 1 - \$8.69 2 - \$9.99 1 - \$12.243 - \$7.35 1 - \$8.73 1 - \$10.01 1 - \$12.301 - \$7.37 2 - \$8.75 1 - \$10.05 1 - \$12.321 - \$7.47 1 - \$8.76 2 - \$10.08 1 - \$12.351 - \$7.50 1 - \$8.78 1 - \$10.17 2 - \$12.421 - \$7.52 5 - \$8.82 1 - \$10.20 1 - \$12.514 - \$7.56 1 - \$8.85 2 - \$10.26 1 - \$12.651 - \$7.62 1 - \$8.88 3 - \$10.29 2 - \$12.694 - \$7.65 2 - \$8.91 3 - \$10.35 1 - \$12.751 - \$7.67 1 - \$8.94 2 - \$10.44 1 - \$12.922 - \$7.70 1 - \$8.96 1 - \$10.53 1 - \$12.963 - \$7.74 3 - \$9.00 1 - \$10.56 1 - \$13.231 - \$7.77 1 - \$9.02 1 - \$10.64 1 - \$13.411 - \$7.79 2 - \$9.03 2 - \$10.71 1 - \$13.562 - \$7.80 1 - \$9.12 3 - \$10.80 1 - \$14.491 - \$7.82 2 - \$9.18 1 - \$10.85 1 - \$15.18
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