Difference between revisions of "2000 AMC 12 Problems/Problem 13"
(→Solution 5) |
(→Solution 5) |
||
Line 43: | Line 43: | ||
Multiplying the equation by 4 to get <cmath>m + \frac{2}{3}c = (m + c) - \frac{1}{3}c = t - \frac{1}{3}c = 32</cmath> | Multiplying the equation by 4 to get <cmath>m + \frac{2}{3}c = (m + c) - \frac{1}{3}c = t - \frac{1}{3}c = 32</cmath> | ||
Since <math>\frac{1}{3}c > 0</math>, we have <math>t > 32</math>. | Since <math>\frac{1}{3}c > 0</math>, we have <math>t > 32</math>. | ||
+ | |||
Now multiplying the equation by 6 to get <cmath>\frac{3}{2}m + c = (m+c) + \frac{1}{3}m = t + \frac{1}{3}m = 48</cmath> | Now multiplying the equation by 6 to get <cmath>\frac{3}{2}m + c = (m+c) + \frac{1}{3}m = t + \frac{1}{3}m = 48</cmath> | ||
Since <math>\frac{1}{3}m > 0</math>, we have <math>t < 48</math> | Since <math>\frac{1}{3}m > 0</math>, we have <math>t < 48</math> |
Revision as of 13:57, 29 June 2019
- The following problem is from both the 2000 AMC 12 #13 and 2000 AMC 10 #22, so both problems redirect to this page.
Contents
Problem
One morning each member of Angela’s family drank an 8-ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?
Solution
Solution 1
Let be the total amount of coffee,
of milk, and
the number of people in the family. Then each person drinks the same total amount of coffee and milk (8 ounces), so
Regrouping, we get
. Since both
are positive, it follows that
and
are also positive, which is only possible when
.
Solution 2 (less rigorous)
One could notice that (since there are only two components to the mixture) Angela must have more than her "fair share" of milk and less then her "fair share" of coffee in order to ensure that everyone has ounces. The "fair share" is
So,
Which requires that be
since
is a whole number.
Solution 3
Again, let
and
be the total amount of coffee, total amount of milk, and number of people in the family, respectively. Then the total amount that is drunk is
and also
Thus,
so
and
We also know that the amount Angela drank, which is is equal to
ounces, thus
Rearranging gives
Now notice that
(by the problem statement). In addition,
so
Therefore,
and so
We know that
so
From the leftmost inequality, we get
and from the rightmost inequality, we get
The only possible value of
is
.
Solution 4
Let
and
be the total amount of coffee, total amount of milk, and number of people in the family, respectively.
and
obviously can't be
.
We know
or
and
or
.
Then,
Because
and
are both divisible by
,
must also be divisible by
. Let
. Now,
can't be
, otherwise
is
, and
can't be
, otherwise
is
. Therefore
must be
,
and
.
. Therefore,
.
~grtuit
Solution 5
Let ,
be the total amounts of milk and coffee, respectively. In order to know the number of people, we need to first find the total amount of mixture
. We are given that
Multiplying the equation by 4 to get
Since
, we have
.
Now multiplying the equation by 6 to get
Since
, we have
Thus, .
Since is a multiple of 8, the only possible value for
in that range is 40. Therefore, there are
people in Angela's family. (C)
~ Nafer
Sidenote
If we now solve for and
, we find that
and
. Thus in total the family drank
ounces of milk and
ounces of coffee. Angela drank exactly
ounces of milk and
ounces of coffee.
See also
2000 AMC 12 (Problems • Answer Key • Resources) | |
Preceded by Problem 12 |
Followed by Problem 14 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
2000 AMC 10 (Problems • Answer Key • Resources) | ||
Preceded by Problem 21 |
Followed by Problem 23 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.