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Difference between revisions of "1996 AHSME Problems/Problem 13"

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==Problem==
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Sunny runs at a steady rate, and Moonbeam runs <math>m</math> times as fast, where <math>m</math> is a number greater than 1.  If Moonbeam gives Sunny a head start of <math>h</math> meters, how many meters must Moonbeam run to overtake Sunny?
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<math> \text{(A)}\ hm\qquad\text{(B)}\ \frac{h}{h+m}\qquad\text{(C)}\ \frac{h}{m-1}\qquad\text{(D)}\ \frac{hm}{m-1}\qquad\text{(E)}\ \frac{h+m}{m-1} </math>
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==Solution ==
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If Sunny runs at a rate of <math>s</math> for <math>x</math> meters in <math>t</math> minutes, then <math>s = \frac{x}{t}</math>.
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In that case, Moonbeam's rate is <math>ms</math>, and Moonbeam's distance is <math>x + h</math>, and the amount of time <math>t</math> is the same.  Thus, <math>ms = \frac{x + h}{t}</math>
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Solving each equation for <math>t</math>, we have <math>t = \frac{x}{s} = \frac{x + h}{ms}</math>
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Cross multiplying, we get <math>msx = xs + hs</math>
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Solving for <math>x</math>, we get <math>msx - xs = hs</math>, which leads to <math>x = \frac{h}{m - 1}</math>.
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Note that <math>x</math> is the distance that Sunny ran.  Moonbeam ran <math>h</math> meters more, for a total of <math>h + \frac{h}{m-1} = \frac{h(m-1) + h}{m-1} = \frac{hm}{m-1}</math>.  This is answer <math>\boxed{D}</math>. 
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==Solution 2==
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Note that <math>h</math> is a length, while <math>m</math> is a dimensionless constant.  Thus, <math>h</math> and <math>m</math> cannot be added, and <math>B</math> and <math>E</math> are not proper answers, since they both contain <math>h+m</math>.
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Thus, we only concern ourselves with answers <math>A, C, D</math>.
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If <math>m</math> is a very, very large number, then Moonbeam will have to run just over <math>h</math> meters to reach Sunny.  Or, in the language of limits:
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<math>\lim_{x\rightarrow \infty} d(m) = h</math>, where <math>d(m)</math> is the distance Moonbeam needs to catch Sunny at the given rate ratio of <math>m</math>.
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In option <math>A</math>, when <math>m</math> gets large, the distance gets large.  Thus, <math>A</math> is not a valid answer.
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In option <math>C</math>, when <math>m</math> gets large, the distance approaches <math>0</math>, not <math>h</math> as desired.  This is not a valid answer.  (In fact, this is the distance Sunny runs, which does approach <math>0</math> as Moonbeam gets faster and faster.)
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In option <math>D</math>, when <math>m</math> gets large, the ratio <math>\frac{m}{m-1}</math> gets very close to, but remains just a tiny bit over, the number <math>1</math>.  Thus, when you multiply it by <math>h</math>, the ratio in option <math>D</math> gets very close to, but remains just a tiny bit over, <math>h</math>.  Thus, the best option out of all the choices is <math>\boxed{D}</math>.
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==See also==
 
==See also==
 
{{AHSME box|year=1996|num-b=12|num-a=14}}
 
{{AHSME box|year=1996|num-b=12|num-a=14}}

Revision as of 11:51, 19 August 2011

Problem

Sunny runs at a steady rate, and Moonbeam runs $m$ times as fast, where $m$ is a number greater than 1. If Moonbeam gives Sunny a head start of $h$ meters, how many meters must Moonbeam run to overtake Sunny?

$\text{(A)}\ hm\qquad\text{(B)}\ \frac{h}{h+m}\qquad\text{(C)}\ \frac{h}{m-1}\qquad\text{(D)}\ \frac{hm}{m-1}\qquad\text{(E)}\ \frac{h+m}{m-1}$

Solution

If Sunny runs at a rate of $s$ for $x$ meters in $t$ minutes, then $s = \frac{x}{t}$.

In that case, Moonbeam's rate is $ms$, and Moonbeam's distance is $x + h$, and the amount of time $t$ is the same. Thus, $ms = \frac{x + h}{t}$

Solving each equation for $t$, we have $t = \frac{x}{s} = \frac{x + h}{ms}$

Cross multiplying, we get $msx = xs + hs$

Solving for $x$, we get $msx - xs = hs$, which leads to $x = \frac{h}{m - 1}$.

Note that $x$ is the distance that Sunny ran. Moonbeam ran $h$ meters more, for a total of $h + \frac{h}{m-1} = \frac{h(m-1) + h}{m-1} = \frac{hm}{m-1}$. This is answer $\boxed{D}$.

Solution 2

Note that $h$ is a length, while $m$ is a dimensionless constant. Thus, $h$ and $m$ cannot be added, and $B$ and $E$ are not proper answers, since they both contain $h+m$.

Thus, we only concern ourselves with answers $A, C, D$.

If $m$ is a very, very large number, then Moonbeam will have to run just over $h$ meters to reach Sunny. Or, in the language of limits:

$\lim_{x\rightarrow \infty} d(m) = h$, where $d(m)$ is the distance Moonbeam needs to catch Sunny at the given rate ratio of $m$.

In option $A$, when $m$ gets large, the distance gets large. Thus, $A$ is not a valid answer.

In option $C$, when $m$ gets large, the distance approaches $0$, not $h$ as desired. This is not a valid answer. (In fact, this is the distance Sunny runs, which does approach $0$ as Moonbeam gets faster and faster.)

In option $D$, when $m$ gets large, the ratio $\frac{m}{m-1}$ gets very close to, but remains just a tiny bit over, the number $1$. Thus, when you multiply it by $h$, the ratio in option $D$ gets very close to, but remains just a tiny bit over, $h$. Thus, the best option out of all the choices is $\boxed{D}$.

See also

1996 AHSME (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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions
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