Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2000 AMC 12 Problems/Problem 13"

m (→‎See also: hehe)
(84 intermediate revisions by 27 users not shown)
Line 1: Line 1:
 +
{{duplicate|[[2000 AMC 12 Problems|2000 AMC 12 #13]] and [[2000 AMC 10 Problems|2000 AMC 10 #22]]}}
 +
 
== Problem ==
 
== 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?
+
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?
  
 
<math>\text {(A)}\ 3 \qquad \text {(B)}\ 4 \qquad \text {(C)}\ 5 \qquad \text {(D)}\ 6 \qquad \text {(E)}\ 7</math>
 
<math>\text {(A)}\ 3 \qquad \text {(B)}\ 4 \qquad \text {(C)}\ 5 \qquad \text {(D)}\ 6 \qquad \text {(E)}\ 7</math>
  
−
== Solution ==
+
== Solution 1 ==
−
Let <math>a</math> be the total amount of coffee, <math>b</math> of milk, and <math>n</math> the number of people in the family. Then each person drinks the same amount of coffee and milk, so <math>\left(\frac{a}{6} + \frac{b}{4}\right)n = a + b</math>. Regrouping the terms, we get <math>2a(6-n)=3b(n-4)</math>. Since both <math>a,b</math> are positive, it follows that <math>6-n,n-4</math> are also positive, which is only possible when <math>n = 5\ \mathrm{(C)}</math>.  
+
 
 +
Let <math>c</math> be the total amount of coffee, <math>m</math> of milk, and <math>p</math> the number of people in the family. Then each person drinks the same total amount of coffee and milk (8 ounces), so  
 +
<cmath>\left(\frac{c}{6} + \frac{m}{4}\right)p = c + m</cmath>
 +
Regrouping, we get <math>2c(6-p)=3m(p-4)</math>. Since both <math>c,m</math> are positive, it follows that <math>6-p</math> and <math>p-4</math> are also positive, which is only possible when <math>p = 5\ \mathrm{(C)}</math>.
 +
 
 +
== 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 than her "fair share" of coffee in order to ensure that everyone has <math>8</math> ounces.  The "fair share" is <math>1/p.</math>  So,
 +
 
 +
<cmath>\frac{1}{6} < \frac{1}{p}<\frac{1}{4}</cmath>
 +
 
 +
Which requires that <math>p</math> be <math>p = 5\ \mathrm{(C)},</math> since <math>p</math> is a whole number.
 +
 
 +
== Solution 3 ==
 +
 
 +
Again, let <math>c,</math> <math>m,</math> and <math>p</math> 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 <math>8p,</math> and also <math>c+m.</math> Thus, <math>c+m = 8p,</math> so <math>m = 8p-c</math> and <math>c = 8p-m.</math>
 +
 
 +
We also know that the amount Angela drank, which is <math>\frac{c}{6} + \frac{m}{4},</math> is equal to <math>8</math> ounces, thus <math>\frac{c}{6} + \frac{m}{4} = 8.</math> Rearranging gives  <cmath>24p - c = 96.</cmath> Now notice that <math>c > 0</math> (by the problem statement). In addition, <math>m > 0,</math> so <math>c = 8p-m < 8p.</math> Therefore, <math>0 < c < 8p,</math> and so <math>24p > 24p-c > 16p.</math> We know that <math>24p-c = 96,</math> so <cmath>24p > 96 > 16p.</cmath> From the leftmost inequality, we get <math>p > 4,</math> and from the rightmost inequality, we get <math>p < 6.</math> The only possible value of <math>p</math> is <math>p = 5\ \mathrm{(C)}</math>.
 +
 
 +
== Solution 4 ==
 +
 
 +
Let <math>c,</math> <math>m,</math> and <math>p</math> be the total amount of coffee, total amount of milk, and number of people in the family, respectively. <math>c</math> and <math>m</math> obviously can't be <math>0</math>.
 +
We know <math>\frac{c}{6} + \frac{m}{4} = 8</math> or <math>2c + 3m = 96</math> and <math>c + m = 8p</math> or <math>2c + 2m = 16p</math>.
 +
Then, <cmath>(2c + 2m) + m = 16p + m = 96</cmath>Because <math>16p</math> and <math>96</math> are both divisible by <math>16</math>, <math>m</math> must also be divisible by <math>16</math>. Let <math>m = 16k</math>. Now, <cmath>3(16k) + 2c = 48k + 2c = 96</cmath> <math>k</math> can't be <math>0</math>, otherwise <math>m</math> is <math>0</math>, and <math>k</math> can't be <math>2</math>, otherwise <math>c</math> is <math>0</math>. Therefore <math>k</math> must be <math>1</math>, <math>m = 16</math> and <math>c = 24</math>. <math>c + m = 24 + 16 = 40 = 8p</math>. Therefore, <math>p = 5\ \mathrm{(C)}</math>.
 +
 
 +
== Solution 5 ==
 +
 
 +
Let <math>m</math>, <math>c</math> be the total amounts of milk and coffee, respectively. In order to know the number of people, we first need to find the total amount of mixture <math>t = m + c</math>. We are given that
 +
<cmath>\frac{m}{4} + \frac{c}{6} = 8</cmath>
 +
Multiplying the equation by <math>4</math> yields
 +
<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>. Now multiplying the equation by <math>6</math> yields
 +
<cmath>\frac{3}{2}m + c = (m+c) + \frac{1}{2}m = t + \frac{1}{2}m = 48</cmath>
 +
Since <math>\frac{1}{2}m > 0</math>, we have <math>t < 48</math>. Thus, <math>32 < t < 48</math>.
 +
 
 +
Since <math>t</math> is a multiple of <math>8</math>, the only possible value for <math>t</math> in that range is <math>40</math>. Therefore, there are <math>\frac{40}{8} = 5</math> people in Angela's family. <math>\mathrm{(C)}</math>.
 +
 
 +
~ Nafer
 +
 
 +
== Solution 6 (Constraints) ==
 +
 
 +
If there were 4 people in the family, and each of them drank exactly the same amount of coffee and milk as Angela, there would be too much coffee. If there were 6 people in the family, and each of them drank exactly the same amount of coffee and milk as Angela, there would be not enough milk. Thus, it has to be <math>\boxed{5}</math>.
 +
 
 +
== Solution 7 (Guess and check) ==
 +
The number of ounces that Angela's family drank has to be a multiple of 8, so we can find the right answer by guessing random values for the number of ounces of coffee and milk Angela drank. With Angela drinking 4 ounces of milk and 4 ounces of coffee, we get 40 total ounces Angela's family drank. Dividing that by 8(each person drank 8 ounces) we get 5 members who drank the coffee with milk, or <math>\boxed{C}</math>  ~ Mathyguy88
 +
 
 +
==Video Solution 1==
 +
https://youtu.be/UGcqyhh03LQ?si=WKAVn2VitpmeCvSd ~ Pi Academy
 +
 
 +
==Video Solution 2==
 +
https://youtu.be/k6G5BjjILGY
 +
https://www.youtube.com/watch?v=OT42J21ZNC8&feature=youtu.be
 +
 
 +
https://www.youtube.com/watch?v=7QU8OlnljHw    ~David
 +
 
 +
==Sidenote==
 +
If we now solve for <math>c</math> and <math>m</math>, we find that <math>m=16</math> and <math>c=24</math>. Thus in total the family drank <math>16</math> ounces of milk and <math>24</math> ounces of coffee. Angela drank exactly <math>4</math> ounces of milk and <math>4</math> ounces of coffee.
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2000|num-b=12|num-a=14}}
 
{{AMC12 box|year=2000|num-b=12|num-a=14}}
 +
{{AMC10 box|year=2000|num-b=21|num-a=23}}
  
 
[[Category:Introductory Number Theory Problems]]
 
[[Category:Introductory Number Theory Problems]]
 +
{{MAA Notice}}

Revision as of 20:59, 5 October 2024

The following problem is from both the 2000 AMC 12 #13 and 2000 AMC 10 #22, so both problems redirect to this page.

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?

$\text {(A)}\ 3 \qquad \text {(B)}\ 4 \qquad \text {(C)}\ 5 \qquad \text {(D)}\ 6 \qquad \text {(E)}\ 7$

Solution 1

Let $c$ be the total amount of coffee, $m$ of milk, and $p$ the number of people in the family. Then each person drinks the same total amount of coffee and milk (8 ounces), so \[\left(\frac{c}{6} + \frac{m}{4}\right)p = c + m\] Regrouping, we get $2c(6-p)=3m(p-4)$. Since both $c,m$ are positive, it follows that $6-p$ and $p-4$ are also positive, which is only possible when $p = 5\ \mathrm{(C)}$.

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 than her "fair share" of coffee in order to ensure that everyone has $8$ ounces. The "fair share" is $1/p.$ So,

\[\frac{1}{6} < \frac{1}{p}<\frac{1}{4}\]

Which requires that $p$ be $p = 5\ \mathrm{(C)},$ since $p$ is a whole number.

Solution 3

Again, let $c,$ $m,$ and $p$ 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 $8p,$ and also $c+m.$ Thus, $c+m = 8p,$ so $m = 8p-c$ and $c = 8p-m.$

We also know that the amount Angela drank, which is $\frac{c}{6} + \frac{m}{4},$ is equal to $8$ ounces, thus $\frac{c}{6} + \frac{m}{4} = 8.$ Rearranging gives \[24p - c = 96.\] Now notice that $c > 0$ (by the problem statement). In addition, $m > 0,$ so $c = 8p-m < 8p.$ Therefore, $0 < c < 8p,$ and so $24p > 24p-c > 16p.$ We know that $24p-c = 96,$ so \[24p > 96 > 16p.\] From the leftmost inequality, we get $p > 4,$ and from the rightmost inequality, we get $p < 6.$ The only possible value of $p$ is $p = 5\ \mathrm{(C)}$.

Solution 4

Let $c,$ $m,$ and $p$ be the total amount of coffee, total amount of milk, and number of people in the family, respectively. $c$ and $m$ obviously can't be $0$. We know $\frac{c}{6} + \frac{m}{4} = 8$ or $2c + 3m = 96$ and $c + m = 8p$ or $2c + 2m = 16p$. Then, \[(2c + 2m) + m = 16p + m = 96\]Because $16p$ and $96$ are both divisible by $16$, $m$ must also be divisible by $16$. Let $m = 16k$. Now, \[3(16k) + 2c = 48k + 2c = 96\] $k$ can't be $0$, otherwise $m$ is $0$, and $k$ can't be $2$, otherwise $c$ is $0$. Therefore $k$ must be $1$, $m = 16$ and $c = 24$. $c + m = 24 + 16 = 40 = 8p$. Therefore, $p = 5\ \mathrm{(C)}$.

Solution 5

Let $m$, $c$ be the total amounts of milk and coffee, respectively. In order to know the number of people, we first need to find the total amount of mixture $t = m + c$. We are given that \[\frac{m}{4} + \frac{c}{6} = 8\] Multiplying the equation by $4$ yields \[m + \frac{2}{3}c = (m + c) - \frac{1}{3}c = t - \frac{1}{3}c = 32\] Since $\frac{1}{3}c > 0$, we have $t > 32$. Now multiplying the equation by $6$ yields \[\frac{3}{2}m + c = (m+c) + \frac{1}{2}m = t + \frac{1}{2}m = 48\] Since $\frac{1}{2}m > 0$, we have $t < 48$. Thus, $32 < t < 48$.

Since $t$ is a multiple of $8$, the only possible value for $t$ in that range is $40$. Therefore, there are $\frac{40}{8} = 5$ people in Angela's family. $\mathrm{(C)}$.

~ Nafer

Solution 6 (Constraints)

If there were 4 people in the family, and each of them drank exactly the same amount of coffee and milk as Angela, there would be too much coffee. If there were 6 people in the family, and each of them drank exactly the same amount of coffee and milk as Angela, there would be not enough milk. Thus, it has to be $\boxed{5}$.

Solution 7 (Guess and check)

The number of ounces that Angela's family drank has to be a multiple of 8, so we can find the right answer by guessing random values for the number of ounces of coffee and milk Angela drank. With Angela drinking 4 ounces of milk and 4 ounces of coffee, we get 40 total ounces Angela's family drank. Dividing that by 8(each person drank 8 ounces) we get 5 members who drank the coffee with milk, or $\boxed{C}$ ~ Mathyguy88

Video Solution 1

https://youtu.be/UGcqyhh03LQ?si=WKAVn2VitpmeCvSd ~ Pi Academy

Video Solution 2

https://youtu.be/k6G5BjjILGY https://www.youtube.com/watch?v=OT42J21ZNC8&feature=youtu.be

https://www.youtube.com/watch?v=7QU8OlnljHw ~David

Sidenote

If we now solve for $c$ and $m$, we find that $m=16$ and $c=24$. Thus in total the family drank $16$ ounces of milk and $24$ ounces of coffee. Angela drank exactly $4$ ounces of milk and $4$ 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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2000_AMC_12_Problems/Problem_13&oldid=229061"