# 45 arithmetic/7-11.p

## Description

This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

# 45 arithmetic/7-11.p

A customer at a 7-11 store selected four 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 of the four
individual items. The customer protested that the four prices
should have been ADDED, not MULTIPLIED. The clerk said that that
was OK with him, but, the result was still the same: exactly \$7.11.

What were the prices of the four items?

arithmetic/7-11.s

The prices are: \$1.20, \$1.25, \$1.50, and \$3.16

\$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.
Note that \$7.11 has a single, unique solution.

```       1 -  \$6.44      1 -  \$7.83      2 -  \$9.20      3 - \$10.89
1 -  \$6.51      1 -  \$7.86      1 -  \$9.23      1 - \$10.95
1 -  \$6.60      3 -  \$7.92      1 -  \$9.24      2 - \$11.00
1 -  \$6.63      1 -  \$8.00      1 -  \$9.27      1 - \$11.07
1 -  \$6.65      1 -  \$8.01      2 -  \$9.35      1 - \$11.13
1 -  \$6.72      1 -  \$8.03      3 -  \$9.36      1 - \$11.16
2 -  \$6.75      5 -  \$8.10      1 -  \$9.38      1 - \$11.22
1 -  \$6.78      1 -  \$8.12      5 -  \$9.45      2 - \$11.25
1 -  \$6.80      1 -  \$8.16      2 -  \$9.48      2 - \$11.27
2 -  \$6.84      2 -  \$8.19      1 -  \$9.54      1 - \$11.30
1 -  \$6.86      1 -  \$8.22      1 -  \$9.57      1 - \$11.36
1 -  \$6.89      1 -  \$8.25      1 -  \$9.59      1 - \$11.40
2 -  \$6.93      3 -  \$8.28      2 -  \$9.60      2 - \$11.43
1 -  \$7.02      3 -  \$8.33      1 -  \$9.62      2 - \$11.52
1 -  \$7.05      1 -  \$8.36      2 -  \$9.63      2 - \$11.55
2 -  \$7.07      1 -  \$8.37      1 -  \$9.66      2 - \$11.61
1 -  \$7.08      2 -  \$8.40      1 -  \$9.68      1 - \$11.69
1 -  \$7.11      1 -  \$8.45      2 -  \$9.69      1 - \$11.70
1 -  \$7.13      2 -  \$8.46      1 -  \$9.78      1 - \$11.88
2 -  \$7.14      1 -  \$8.52      2 -  \$9.80      1 - \$11.90
3 -  \$7.20      5 -  \$8.55      1 -  \$9.81      1 - \$11.99
1 -  \$7.25      1 -  \$8.60      1 -  \$9.87      1 - \$12.06
1 -  \$7.26      4 -  \$8.64      4 -  \$9.90      1 - \$12.15
2 -  \$7.28      1 -  \$8.67      1 -  \$9.92      1 - \$12.18
1 -  \$7.29      1 -  \$8.69      2 -  \$9.99      1 - \$12.24
3 -  \$7.35      1 -  \$8.73      1 - \$10.01      1 - \$12.30
1 -  \$7.37      2 -  \$8.75      1 - \$10.05      1 - \$12.32
1 -  \$7.47      1 -  \$8.76      2 - \$10.08      1 - \$12.35
1 -  \$7.50      1 -  \$8.78      1 - \$10.17      2 - \$12.42
1 -  \$7.52      5 -  \$8.82      1 - \$10.20      1 - \$12.51
4 -  \$7.56      1 -  \$8.85      2 - \$10.26      1 - \$12.65
1 -  \$7.62      1 -  \$8.88      3 - \$10.29      2 - \$12.69
4 -  \$7.65      2 -  \$8.91      3 - \$10.35      1 - \$12.75
1 -  \$7.67      1 -  \$8.94      2 - \$10.44      1 - \$12.92
2 -  \$7.70      1 -  \$8.96      1 - \$10.53      1 - \$12.96
3 -  \$7.74      3 -  \$9.00      1 - \$10.56      1 - \$13.23
1 -  \$7.77      1 -  \$9.02      1 - \$10.64      1 - \$13.41
1 -  \$7.79      2 -  \$9.03      2 - \$10.71      1 - \$13.56
2 -  \$7.80      1 -  \$9.12      3 - \$10.80      1 - \$14.49
1 -  \$7.82      2 -  \$9.18      1 - \$10.85      1 - \$15.18
```

There are plenty of solutions for five summands. Here are a few:

```         \$8.28  -- at least two solutions
\$8.47  -- at least two solutions
\$8.82  -- at least two solutions
```

--
Mark Johnson mark@microunity.com (408) 734-8100

There may be many approximate solutions, for example: \$1.01, \$1.15, \$2.41,
and \$2.54. These sum to \$7.11 but the product is 7.1100061.

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