2015 AMC 8 Problems/Problem 20
Ralph went to the store and bought 12 pairs of socks for a total of $24. Some of the socks he bought cost $1 a pair, some of the socks he bought cost $3 a pair, and some of the socks he bought cost $4 a pair. If he bought at least one pair of each type, how many pairs of $1 socks did Ralph buy?
Solution
So let there be pairs of
dollar socks,
pairs of
dollar socks,
pairs of
dollar socks.
We have ,
, and
.
Now we subtract to find , and
.
It follows that
is a multiple of
and
is a multiple of
, so since
, we must have
.
Therefore, , and it follows that
. Now
, as desired.
See Also
2015 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
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