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Tim and Robert have entered a race, the rules of which stipulate that

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Tim and Robert have entered a race, the rules of which stipulate that  [#permalink]

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New post 19 Jun 2018, 23:39
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Tim and Robert have entered a race, the rules of which stipulate that each runner must run for at least 4 hours and no runner can run for more than 6 hours. Together, they must run a total of 50 miles. If it takes Tim 15 minutes to run a mile and Robert 12 minutes to run a mile, what is the minimum number of miles Robert must run if both Tim and Robert must individually run a whole number of miles?

A. 18
B. 20
C. 22
D. 24
E. 26

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Re: Tim and Robert have entered a race, the rules of which stipulate that  [#permalink]

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New post 19 Jun 2018, 23:52
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Bunuel wrote:
Tim and Robert have entered a race, the rules of which stipulate that each runner must run for at least 4 hours and no runner can run for more than 6 hours. Together, they must run a total of 50 miles. If it takes Tim 15 minutes to run a mile and Robert 12 minutes to run a mile, what is the minimum number of miles Robert must run if both Tim and Robert must individually run a whole number of miles?

A. 18
B. 20
C. 22
D. 24
E. 26


As we're given a range of possible values, we'll look at its extremes.
This is a Logical approach

Tim runs at 15 minutes per mile, which is 4 miles per hour..
This is between 4*4=16 and 6*4 = 24 miles
Robert runs at 12 minutes per mile, which is 5 miles per hour.
This is between 4*5=20 and 6*5 = 30 miles.

So, the maximum Tim can run is 24 miles meaning the minimum Robert can run is 50 - 24 = 26 miles.

(E) is our answer.
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Re: Tim and Robert have entered a race, the rules of which stipulate that  [#permalink]

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New post 20 Jun 2018, 04:29
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Solution



Given:

    • Tim and Robert have to race for 50 miles combined.
    • Tim runs 1 mile in 15 min and Robert runs 1 mile in 12 min.
    • Each runner must run for at least 4 hours and no more 6 hours.

To find:

    • We need to find the minimum number of miles Robert must run if Tim and Robert both run a whole number of time.

Approach and Working:

We have a constraint in the question that both the runner should run for at least 4 hours.
    • So, Tim covers= 1 mile* (\(\frac{240}{15}\)) = 16 miles in 4 hours.
    • And, Robert covers= 1 mile* (\(\frac{240}{12}\)) = 20 miles in 4 hours.
Hence, both of them run 36 miles in 4 hours.

Now, for Robert to run the minimum number of miles, time must cover the maximum number of miles.
    • Hence, Tim should run for remaining 2 hours also.
    • Hence, Tim will run= 1 mile* (120/15) = 8 miles in 2 hour.
Now, both of them have covered 36+8= 44 miles.

Hence, remaining 6 miles Robert should cover.
Thus, Robert runs for minimum 26 miles.

Hence, option E is the correct answer.
Answer: E
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Re: Tim and Robert have entered a race, the rules of which stipulate that  [#permalink]

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New post 22 Jun 2018, 13:11
For Robert to cover minimum distance, Tim needs to cover maximum distance.
But maximum time Tim can run is 6 hours, therefore he can cover maximum distance of = 4*6 = 24 miles.
Hence, the rest is the minimum distance that needs to be covered by Robert = 50-24 = 26 miles.
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Tim and Robert have entered a race, the rules of which stipulate that  [#permalink]

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New post 23 Jun 2018, 12:09
Bunuel wrote:
Tim and Robert have entered a race, the rules of which stipulate that each runner must run for at least 4 hours and no runner can run for more than 6 hours. Together, they must run a total of 50 miles. If it takes Tim 15 minutes to run a mile and Robert 12 minutes to run a mile, what is the minimum number of miles Robert must run if both Tim and Robert must individually run a whole number of miles?

A. 18
B. 20
C. 22
D. 24
E. 26

Maximize Tim's number of miles to minimize Robert's number if miles.

(One part of this question might seem tricky, but that part is irrelevant. The number of miles has to be a whole number. The time each person takes to run those miles does not have to be a whole number.)

T's rate: \((\frac{1mile}{15mins}*\frac{4}{4})=\frac{4mi}{60mins}=\frac{4mi}{1hr}\)

R's rate: \((\frac{1mile}{12mins}*\frac{5}{5})=\frac{5mi}{60mins}=\frac{5mi}{1hr}\)

Both runners must run at least 4 hours and at most 6 hours.

Maximize Tim's # of miles by making him run the maximum number of 6 hours.
Tim can run (6 hours * 4 mph) = 24 miles

Tim is done. Robert must finish the 50-mile total, so Robert runs (50 - 24) = 26 miles

(And while TIM'S time is a whole number, Robert's time is not. Does not matter. Only # of miles run must be a whole number.)

Answer E
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Re: Tim and Robert have entered a race, the rules of which stipulate that  [#permalink]

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New post 23 Jun 2018, 21:32
Bunuel wrote:
Tim and Robert have entered a race, the rules of which stipulate that each runner must run for at least 4 hours and no runner can run for more than 6 hours. Together, they must run a total of 50 miles. If it takes Tim 15 minutes to run a mile and Robert 12 minutes to run a mile, what is the minimum number of miles Robert must run if both Tim and Robert must individually run a whole number of miles?

A. 18
B. 20
C. 22
D. 24
E. 26



Answer

Please make a note of rules very carefully. Atleast 4 hours each has to run , and no once can run for more than 6 hours.

How calculate the speed of Tim and Robert in miles per hour
Tim Speed = 60/15 = 4 miles /hour
robert speed = 60/12 = 5 miles/hour

Now , both have to run for at least 4 hours, , that implies in four hours , distance travelled by both individually will be,
Tim = 16 miles
Robert = 20 miles.

In Question stem it is mentioned , together they traveled 50 miles. 36 miles have already been covered (16+20 ), total of 14 miles left to be covered.

The Q stem asks us to find the minimum miles Robert must run, for that Tim must run for the complete 6 hours.
Therefore Tim distance would be 4miles/hour*6hours = 24 miles

Remaining distance = 50-24 = 26 miles, is the minimum distance traveled by Robert .

Thus E is the correct ans.
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Tim and Robert have entered a race, the rules of which stipulate that  [#permalink]

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New post 28 Sep 2018, 08:09
Bunuel wrote:
Tim and Robert have entered a race, the rules of which stipulate that each runner must run for at least 4 hours and no runner can run for more than 6 hours. Together, they must run a total of 50 miles. If it takes Tim 15 minutes to run a mile and Robert 12 minutes to run a mile, what is the minimum number of miles Robert must run if both Tim and Robert must individually run a whole number of miles?

A. 18
B. 20
C. 22
D. 24
E. 26

Let T and R be the number of minutes Tim and Robert run, respectively. Hence:

DATA:

\(4 \cdot 60\,\,\, \le \,\,\,\,\,T,R\,\,\,\, \le \,\,\,6 \cdot 60\,\,\,\,\,\,\,\left( {\rm{I}} \right)\)

\(\left. \matrix{
{\mathop{\rm int}} \,\,\, = \,\,\,T\,\,\min \,\,\,\left( {{{1\,\,{\rm{mile}}} \over {15\,\,\min }}} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,T\,\,{\rm{divisible}}\,\,{\rm{by}}\,\,15\,\,\,\,\, \hfill \cr
{\mathop{\rm int}} \,\,\,\mathop = \limits^{\left( * \right)} \,\,\,R\,\,\min \,\,\,\left( {{{1\,\,{\rm{mile}}} \over {12\,\,\min }}} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,R\,\,{\rm{divisible}}\,\,{\rm{by}}\,\,12 \hfill \cr} \right\}\,\,\,\,\,\,\,\left( {{\rm{II}}} \right)\)

\(\boxed4\,T\,\,\min \,\,\,\left( {\frac{{1\,\,{\text{mile}}}}{{3 \cdot 5 \cdot \boxed4\,\,\min }}} \right)\,\,\, + \,\,\boxed5\,R\,\,\min \,\,\,\left( {\frac{{1\,\,{\text{mile}}}}{{3 \cdot 4 \cdot \boxed5\,\,\min }}} \right) = \frac{{50 \cdot \boxed{3 \cdot 4 \cdot 5}}}{{\boxed{3 \cdot 4 \cdot 5}}}\,\,{\text{miles}}\)

\(4T + 5R = 50 \cdot 3 \cdot 4 \cdot 5\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,4T = 5\left( {3 \cdot 4 \cdot 5 \cdot 10 - R} \right)\,\,\,\,\mathop \leqslant \limits^{\left( {\text{I}} \right)} \,\,\,\,4 \cdot 6 \cdot 60\)

\(R\,\,\, \geqslant \,\,\,3 \cdot 4 \cdot 5 \cdot 10 - 4 \cdot 6 \cdot 12 = 4 \cdot 6 \cdot \left( {25 - 12} \right) = 4 \cdot 6 \cdot 13\,\,\,\,\,\,\,\left( {{\text{III}}} \right)\)


FOCUS:

\(?\,\,\,:\,\,\,\min \,\,\,\frac{R}{{12}}\,\,\,\,\,\left( * \right)\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\min \,\,\,R\,\,\,\,\,\,\,\,\,\,\,\,\left[ {\,? = {{\left( {\frac{R}{{12}}} \right)}_{\,\min }}\,} \right]\,\,\,\,\,\)


DATA-FOCUS CONNECTION:

\(\left. \matrix{
\left( {\rm{I}} \right)\,\,\, \Rightarrow \,\,\,R \ge 4 \cdot 60 = 4 \cdot 3 \cdot 20\,\,\, \hfill \cr
\left( {{\rm{II}}} \right)\,\,\, \Rightarrow \,\,\,R = 3 \cdot 4 \cdot {\mathop{\rm int}} \hfill \cr
\left( {{\rm{III}}} \right)\,\,\, \Rightarrow \,\,\,R \ge 2 \cdot 3 \cdot 4 \cdot 13 \hfill \cr} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,? = {\mathop{\rm int}} = 26\)


\(\left( \begin{gathered}
{R_{\,\min }} = \underline {3 \cdot 4 \cdot 26} \,\,\,\, \Rightarrow \,\,\,4T = 5\left( {3 \cdot 4 \cdot 5 \cdot 10 - \underline {3 \cdot 4 \cdot 26} } \right) = \underleftrightarrow {60\left( {50 - 26} \right)} = 60 \cdot 24 \hfill \\
T = 6 \cdot 60\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\left\{ \begin{gathered}
\left( {\text{I}} \right)\,\,\,{\text{ok}} \hfill \\
\left( {{\text{II}}} \right)\,\,\,{\text{ok}} \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,{R_{\,\min }} = 3 \cdot 4 \cdot 26\,\,\,{\text{viable}}\,\,\,\, \hfill \\
\end{gathered} \right)\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: Tim and Robert have entered a race, the rules of which stipulate that  [#permalink]

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New post 28 Sep 2018, 14:41
Bunuel wrote:
Tim and Robert have entered a race, the rules of which stipulate that each runner must run for at least 4 hours and no runner can run for more than 6 hours. Together, they must run a total of 50 miles. If it takes Tim 15 minutes to run a mile and Robert 12 minutes to run a mile, what is the minimum number of miles Robert must run if both Tim and Robert must individually run a whole number of miles?

A. 18
B. 20
C. 22
D. 24
E. 26

Alternate approach: (Risky but many-times-"awarded", as in this case!)

Let (again) T and R be the number of minutes Tim and Robert run, respectively.

To minimize Roberts mileage (our FOCUS), we must minimize R, therefore (by duality) we must maximize T.

The maximum POTENTIAL value of T is 6*60 (minutes), hence let´s check whether this number - and R obtained from it - are viable:

\({\rm{Tim}}\,\,\left( {6\,\,{\rm{h}}} \right)\,\,:\,\,\,\,6 \cdot 60\,\,\min \,\,\,\left( {{{1\,\,\,{\rm{mile}}} \over {15\,\,\,\min }}\,\matrix{
\nearrow \cr
\nearrow \cr

} } \right)\,\,\, = \,\,24\,\,{\rm{miles}}\)

\({?_{{\rm{potencial}}\,\left( {{\rm{Robert}}} \right)}}\,\,\,\mathop \Rightarrow \limits^{\sum {\, = \,50\,\,{\rm{miles}}} } \,\,\,\,26\,\,{\rm{miles}}\,\,\,\left( {{{12\,\,\,\min } \over {1\,\,\,{\rm{mile}}}}\,\matrix{
\nearrow \cr
\nearrow \cr

} } \right)\,\,\, = {2^3} \cdot 3 \cdot 13\,\,\, > \,\,\,\underbrace {{2^3} \cdot 3 \cdot 10}_{4\,\, \cdot \,60}\)

They are! The answer is therefore 26 (miles).

Regards,
Fabio.
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Re: Tim and Robert have entered a race, the rules of which stipulate that  [#permalink]

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New post 13 Oct 2018, 18:59
Bunuel wrote:
Tim and Robert have entered a race, the rules of which stipulate that each runner must run for at least 4 hours and no runner can run for more than 6 hours. Together, they must run a total of 50 miles. If it takes Tim 15 minutes to run a mile and Robert 12 minutes to run a mile, what is the minimum number of miles Robert must run if both Tim and Robert must individually run a whole number of miles?

A. 18
B. 20
C. 22
D. 24
E. 26



Since it takes Tim 15 minutes to run a mile, he runs 4 miles per hour. Similarly, it takes Robert 12 minutes to run a mile, he runs 5 miles per hour.

Since each runner must run for at least 4 hours and no runner can run for more than 6 hours, Tim runs at least 4 x 4 = 16 miles and at most 6 x 4 = 24 miles. Similarly, Robert runs at least 4 x 5 = 20 miles and at most 6 x 5 = 30 miles.

Since we want to determine the minimum number of miles Robert runs, we can assume Tim runs the the greatest number of miles he possibly can, which is 24. So Robert runs at least 50 - 24 = 26 miles.

Answer: E
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Re: Tim and Robert have entered a race, the rules of which stipulate that   [#permalink] 13 Oct 2018, 18:59
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