2015 AMC 10A Problems/Problem 13
Problem 13
Ashraful has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Ashraful have?
Solution 1
Let Claudia have 5-cent coins and
10-cent coins. It is easily observed that any multiple of
between
and
inclusive can be obtained by a combination of coins. Thus,
combinations can be made, so
. But the answer is not
because we are asked for the number of 10-cent coins, which is
Solution 2
Since the coins are 5-cent and 10-cent, all possible values that can be made will be multiples of To have exactly
different multiples of
we will need to make up to
cents. If all twelve coins were 5-cent coins, we will have
cents possible. Each trade of a 5-cent coin for a 10-cent coin will gain
cents, and as we need to gain
cents, the answer is
For small scale substitutions of 5 cent for 10 cent, like in this problem, it is quite fast, however, if the problem is a little bit more complex, knowing that you need 85 cents, explained above, it is also possible to use a system of equations. It would be
Solving this
(5-cent coins)
and
(10-cent coins)
, so again the answer is
Video Solution
~savannahsolver
See Also
2015 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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All AMC 10 Problems and Solutions |
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