Difference between revisions of "2014 AMC 10A Problems/Problem 6"
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+ | {{duplicate|[[2014 AMC 12A Problems|2014 AMC 12A #4]] and [[2014 AMC 10A Problems|2014 AMC 10A #6]]}} | ||
+ | |||
==Problem== | ==Problem== | ||
Suppose that <math>a</math> cows give <math>b</math> gallons of milk in <math>c</math> days. At this rate, how many gallons of milk will <math>d</math> cows give in <math>e</math> days? | Suppose that <math>a</math> cows give <math>b</math> gallons of milk in <math>c</math> days. At this rate, how many gallons of milk will <math>d</math> cows give in <math>e</math> days? | ||
− | <math> \textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D) | + | <math>\textbf{(A)}\ \frac{bde}{ac}\qquad\textbf{(B)}\ \frac{ac}{bde}\qquad\textbf{(C)}\ \frac{abde}{c}\qquad\textbf{(D)}\ \frac{bcde}{a}\qquad\textbf{(E)}\ \frac{abc}{de}</math> |
− | ==Solution== | + | ==Solution 1== |
− | We see that the the | + | |
+ | We need to multiply <math>b</math> by <math>\frac{d}{a}</math> for the new cows and <math>\frac{e}{c}</math> for the new time, so the answer is <math>b\cdot \frac{d}{a}\cdot \frac{e}{c}=\frac{bde}{ac}</math>, or <math>\boxed{\textbf{(A) } \frac{bde}{ac}}</math>. | ||
+ | |||
+ | ==Solution 2== | ||
+ | |||
+ | |||
+ | We plug in <math>a=2</math>, <math>b=3</math>, <math>c=4</math>, <math>d=5</math>, and <math>e=6</math>. Hence the question becomes "2 cows give 3 gallons of milk in 4 days. How many gallons of milk do 5 cows give in 6 days?" | ||
+ | |||
+ | If 2 cows give 3 gallons of milk in 4 days, then 2 cows give <math>\frac{3}{4}</math> gallons of milk in 1 day, so 1 cow gives <math>\frac{3}{4\cdot2}</math> gallons in 1 day. This means that 5 cows give <math>\frac{5\cdot3}{4\cdot2}</math> gallons of milk in 1 day. Finally, we see that 5 cows give <math>\frac{5\cdot3\cdot6}{4\cdot2}</math> gallons of milk in 6 days. Substituting our values for the variables, this becomes <math>\frac{dbe}{ac}</math>, which is <math>\boxed{\textbf{(A)}\ \frac{bde}{ac}}</math>. | ||
+ | |||
+ | ==Solution 3== | ||
+ | We see that the number of cows is inversely proportional to the number of days and directly proportional to the gallons of milk. So our constant is <math>\dfrac{ac}{b}</math>. | ||
Let <math>g</math> be the answer to the question. We have <math>\dfrac{de}{g}=\dfrac{ac}{b}\implies gac=bde\implies g=\dfrac{bde}{ac}\implies\boxed{ \textbf{(A)}\ \frac{bde}{ac}}</math> | Let <math>g</math> be the answer to the question. We have <math>\dfrac{de}{g}=\dfrac{ac}{b}\implies gac=bde\implies g=\dfrac{bde}{ac}\implies\boxed{ \textbf{(A)}\ \frac{bde}{ac}}</math> | ||
+ | |||
+ | ==Solution 4== | ||
+ | The problem specifics "rate," so it would be wise to first find the rate at which cows produce milk. We can find the rate of work/production by finding the gallons produced by a single cow in a single day. To do this, we divide the amount produced by the number of cows and number of days <cmath>\implies\text{rate}=\dfrac{b}{ac}</cmath> | ||
+ | |||
+ | Now that we have the gallons produced by a single cow in a single day, we simply multiply that rate by the number of cows and the number of days <cmath>\implies\boxed{\textbf{(A) } \dfrac{bde}{ac}}</cmath> | ||
+ | ==Solution 5== | ||
+ | If <math>a</math> cows give <math>b</math> gallons of milk in <math>c</math> days, that means that one cow will give <math>\frac{b}{a}</math> gallons of milk in <math>c</math> days. Also, we want to find the number of gallons of milk will <math>d</math> cows give in <math>e</math> days, so in <math>\frac{e}{c}</math> days <math>d</math> cows give <math>\frac{bd}{a}</math> gallons of milk. Multiplying with the formula <math>d=rt</math>, we get <math>\boxed{\textbf{(A) } \dfrac{bde}{ac}}</math> | ||
+ | |||
+ | ==Solution 6== | ||
+ | |||
+ | In the right formula, plugging in <math>d=a</math> and <math>e=c</math> should simplify to <math>b</math>, as if it doesn't, we'd essentially be saying "<math>a</math> cows give <math>b</math> gallons of milk in <math>c</math> days, but <math>a</math> cows don't give <math>b</math> gallons of milk in <math>c</math> days." The only one of the answer choices that simplifies like this is <math>\boxed{\textbf{(A)}}</math> | ||
+ | |||
+ | ==Solution 7== | ||
+ | |||
+ | Call the rate at which a single cow produces milk <math>r</math>. Using the given information we get that <math>rac=b</math>, and we want to find <math>rde</math>. We solve for <math>r</math> in the first equation to get <math>r=b \cdot \tfrac{1}{ac}</math>. Plugging in this value for <math>r</math> into the expression we want, we get that <math>rde=\frac{bde}{ac}</math>, or <math>\boxed{\textbf{(A)}}</math>. | ||
+ | |||
+ | ==Video Solution (CREATIVE THINKING)== | ||
+ | https://youtu.be/4THtSkiSGMg | ||
+ | |||
+ | ~Education, the Study of Everything | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/OW4rHUTPgPA | ||
+ | |||
+ | ~savannahsolver | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2014|ab=A|num-b=5|num-a=7}} | {{AMC10 box|year=2014|ab=A|num-b=5|num-a=7}} | ||
+ | {{AMC12 box|year=2014|ab=A|num-b=3|num-a=5}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
+ | |||
+ | [[Category: Introductory Algebra Problems]] |
Latest revision as of 00:09, 27 June 2023
- The following problem is from both the 2014 AMC 12A #4 and 2014 AMC 10A #6, so both problems redirect to this page.
Contents
Problem
Suppose that cows give
gallons of milk in
days. At this rate, how many gallons of milk will
cows give in
days?
Solution 1
We need to multiply by
for the new cows and
for the new time, so the answer is
, or
.
Solution 2
We plug in ,
,
,
, and
. Hence the question becomes "2 cows give 3 gallons of milk in 4 days. How many gallons of milk do 5 cows give in 6 days?"
If 2 cows give 3 gallons of milk in 4 days, then 2 cows give gallons of milk in 1 day, so 1 cow gives
gallons in 1 day. This means that 5 cows give
gallons of milk in 1 day. Finally, we see that 5 cows give
gallons of milk in 6 days. Substituting our values for the variables, this becomes
, which is
.
Solution 3
We see that the number of cows is inversely proportional to the number of days and directly proportional to the gallons of milk. So our constant is .
Let be the answer to the question. We have
Solution 4
The problem specifics "rate," so it would be wise to first find the rate at which cows produce milk. We can find the rate of work/production by finding the gallons produced by a single cow in a single day. To do this, we divide the amount produced by the number of cows and number of days
Now that we have the gallons produced by a single cow in a single day, we simply multiply that rate by the number of cows and the number of days
Solution 5
If cows give
gallons of milk in
days, that means that one cow will give
gallons of milk in
days. Also, we want to find the number of gallons of milk will
cows give in
days, so in
days
cows give
gallons of milk. Multiplying with the formula
, we get
Solution 6
In the right formula, plugging in and
should simplify to
, as if it doesn't, we'd essentially be saying "
cows give
gallons of milk in
days, but
cows don't give
gallons of milk in
days." The only one of the answer choices that simplifies like this is
Solution 7
Call the rate at which a single cow produces milk . Using the given information we get that
, and we want to find
. We solve for
in the first equation to get
. Plugging in this value for
into the expression we want, we get that
, or
.
Video Solution (CREATIVE THINKING)
https://youtu.be/4THtSkiSGMg
~Education, the Study of Everything
Video Solution
~savannahsolver
See Also
2014 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
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 |
2014 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 3 |
Followed by Problem 5 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.