Question

Albert was challenged to a race by his friend Martin. Martin started running first, and Albert didn’t start running until Martin had finished a quarter mile lap. Albert overtook Martin exactly at their sixth lap. If both boys ran at a constant speed, with Albert running 2 miles an hour faster than Martin, find the speed of Martin.

Answer 1

Hint: Here, we need to find the speed of Martin. We will assume the speed of Martin to be \[x\] miles an hour. Using the speed distance time formula, we will find the time taken by Martin to complete 5 laps, and the time taken by Albert to complete 6 laps. Using this and the given information, we can form an equation. We will solve this equation to find the value to find the speed of Martin.

Formula Used:
We will use the formula of the time taken to cover a distance at a uniform speed is given by \[{\rm{Time}} = \dfrac{{{\rm{Distance}}}}{{{\rm{Speed}}}}\].

Complete step-by-step answer:
Let the speed of Martin be \[x\] miles an hour.
Thus, we get the speed of Albert as \[\left( {x + 2} \right)\] miles an hour.
It is given that each lap is a quarter mile in length. Albert and Martin crossed each other at the sixth lap. Albert started running after Martin completed 1 lap.
Therefore, the time taken by Albert to run 6 laps is equal to the time taken by Martin to run 5 laps.
We will use the speed distance time formula , \[{\rm{Time}} = \dfrac{{{\rm{Distance}}}}{{{\rm{Speed}}}}\], to find the time taken.
Distance travelled by Martin in 5 laps \[ = 5 \times 0.25 = 1.25\] miles
Speed of Martin \[ = x\] miles an hour
Therefore, we get
Time taken by Martin to complete 5 laps \[ = \dfrac{{1.25}}{x}\] hours
Similarly, distance travelled by Albert in 6 laps \[ = 6 \times 0.25 = 1.5\] miles
Speed of Albert \[ = \left( {x + 2} \right)\] miles an hour
Therefore, we get
Time taken by Albert to complete 6 laps \[ = \dfrac{{1.5}}{{x + 2}}\] hours
The time taken by Albert to run 6 laps is equal to the time taken by Martin to run 5 laps.
Therefore, we get the equation
\[ \Rightarrow \dfrac{{1.25}}{x} = \dfrac{{1.5}}{{x + 2}}\]
We need to solve this equation to get the value of \[x\] and hence, find the speed of Martin.
Simplifying the expression by cross-multiplying, we get
\[ \Rightarrow 1.25\left( {x + 2} \right) = 1.5x\]
Multiplying the terms in the above equation using distributive property, we get
\[ \Rightarrow 1.25x + 2.5 = 1.5x\]
Subtracting \[1.25x\] from both sides of the equation, we get
\[\begin{array}{l} \Rightarrow 1.25x + 2.5 - 1.25x = 1.5x - 1.25x\\ \Rightarrow 2.5 = 0.25x\end{array}\]
Dividing both sides by \[0.25\], we get
\[ \Rightarrow \dfrac{{2.5}}{{0.25}} = \dfrac{{0.25x}}{{0.25}}\]
Therefore, we get
\[ \Rightarrow x = 10\]
\[\therefore \] We get the speed of Martin as 10 miles an hour.

Note: We have formed a linear equation in one variable using the given information in this question. A linear equation in one variable is an equation that has only one variable with highest exponent of 1 and is of the form \[ax + b = 0\], where \[a\] and \[b\] are integers. A linear equation of the form \[ax + b = 0\] has only one solution.
We have used the distributive property in the solution. The distributive law of multiplication states that \[a\left( {b + c} \right) = a \cdot b + a \cdot c\].