Difference between revisions of "1971 AHSME Problems/Problem 2"
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<math>\textbf{(A) }fb^2\qquad \textbf{(B) }b/f^2\qquad \textbf{(C) }f^2/b\qquad \textbf{(D) }b^2/f\qquad \textbf{(E) }f/b^2</math> | <math>\textbf{(A) }fb^2\qquad \textbf{(B) }b/f^2\qquad \textbf{(C) }f^2/b\qquad \textbf{(D) }b^2/f\qquad \textbf{(E) }f/b^2</math> | ||
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== Solution == | == Solution == | ||
− | We can use a modified version of the equation <math>\text{Distance} = \text{Rate} \times {\text{Time}}</math>, which is <math>\text{Work Done} = \text{Rate of Work} \times{ \text{Time Worked}}</math>. In this case, the work done is the number of bricks laid, the rate of work is the number of men working, and the time worked is the number of days. With these definitions, we see from the first equation that <math>f = b \cdot c</math>. If we let <math>X</math> be the number of days it will take <math>c</math> men working at the same rate to <math>b</math> bricks, then we have the equation <math>b = c \cdot x</math>. So, <math>x = \frac{b}{c}</math>. The first equation says that <math>\frac{f}{c^2} = \frac{b}{c}</math>, which leads to <math>x = \frac{f}{c^2}</math>. This doesn't match any of our answer choices though, so we have to fiddle around a bit before we realize that <math>c = \frac{f}{b}</math>, a substitution we can make to see that <math>x = \frac{b^2}{f}</math>. Thus, the answer is \boxed{\ | + | We can use a modified version of the equation <math>\text{Distance} = \text{Rate} \times {\text{Time}}</math>, which is <math>\text{Work Done} = \text{Rate of Work} \times{ \text{Time Worked}}</math>. In this case, the work done is the number of bricks laid, the rate of work is the number of men working, and the time worked is the number of days. With these definitions, we see from the first equation that <math>f = b \cdot c</math>. If we let <math>X</math> be the number of days it will take <math>c</math> men working at the same rate to <math>b</math> bricks, then we have the equation <math>b = c \cdot x</math>. So, <math>x = \frac{b}{c}</math>. The first equation says that <math>\frac{f}{c^2} = \frac{b}{c}</math>, which leads to <math>x = \frac{f}{c^2}</math>. This doesn't match any of our answer choices though, so we have to fiddle around a bit before we realize that <math>c = \frac{f}{b}</math>, a substitution we can make to see that <math>x = \frac{b^2}{f}</math>. Thus, the answer is <math>\boxed{\textbf{(D) }b^2/f}</math>. |
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+ | == See Also == | ||
+ | {{AHSME 35p box|year=1971|num-b=1|num-a=3}} | ||
+ | {{MAA Notice}} | ||
+ | [[Category:Introductory Algebra Problems]] |
Latest revision as of 09:30, 1 August 2024
Problem
If men take
days to lay
bricks, then the number of days it will take
men working at the same rate to lay
bricks, is
Solution
We can use a modified version of the equation , which is
. In this case, the work done is the number of bricks laid, the rate of work is the number of men working, and the time worked is the number of days. With these definitions, we see from the first equation that
. If we let
be the number of days it will take
men working at the same rate to
bricks, then we have the equation
. So,
. The first equation says that
, which leads to
. This doesn't match any of our answer choices though, so we have to fiddle around a bit before we realize that
, a substitution we can make to see that
. Thus, the answer is
.
See Also
1971 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.