Question

A man reaches his office 30 minutes too late if he walks from his home at 3 km an hour and reaches 40 minutes too early if he walks at 4 km an hour. How far is his office from his house?
a. 7 km
b. 14 km
c. 5 km
d. 3 km

Answer 1

Hint: We will use the formula, $\text{time=}\dfrac{\text{distance}}{\text{speed}}$ to solve this question. We will find the relation between the two given conditions using $\text{time=}\dfrac{\text{distance}}{\text{speed}}$ as $30=\dfrac{x}{3}$ and $40=\dfrac{x}{4}$. In the last step, we will subtract the obtained relations to find the distance, x.

Complete step-by-step answer:
It is given in the question that, a man reaches his office 30 minutes too late if he walks from his home at 3 km an hour and reaches 40 minutes too early if he walks at 4 km an hour. And we have been asked to find how far his office is from his house.
We know that, $\text{speed=}\dfrac{\text{distance}}{\text{time}}$. From this, we get, $\text{time=}\dfrac{\text{distance}}{\text{speed}}$. Let us assume that the distance between his office and home is x km.
So, the time taken by the man to reach the office if he walks at a speed of 3 kmph, can be written as,
$\text{-30=}\dfrac{x}{3}.........\left( i \right)$
We have taken the time as -30 because at a speed of 3kmph, he reaches 30 minutes late.
Similarly, the time taken by the man to reach the office if he walks at a speed of 4 kmph, can be written as,
$\text{40=}\dfrac{x}{4}.........\left( ii \right)$
Now, we will subtract equation (ii) from equation (i). So, we get,
\[\begin{align}
  & \dfrac{x}{3}-\dfrac{x}{4}=40-\left( -30 \right)\min \\
 & \dfrac{x}{3}-\dfrac{x}{4}=70\min \\
\end{align}\]
But, here we are given the time in minutes, so we will convert it to hours. We know that 1 hour = 60 minutes. Therefore, we can write, 70 minutes = $\dfrac{70}{60}$ hours.
So, we will get our equation as,
 \[\begin{align}
  & \dfrac{x}{3}-\dfrac{x}{4}=\dfrac{70}{60} \\
 & \dfrac{4x-3x}{12}=\dfrac{70}{60} \\
 & \dfrac{x}{12}=\dfrac{70}{60} \\
 & x=\dfrac{70\times 12}{60} \\
 & x=14km \\
\end{align}\]
Therefore, the distance between the office and the house is 14 km.
Hence, option (b) is the correct answer.

Note: The possible mistake that the students can make in this question is by not considering the negative sign for 30 minutes and it will result in the wrong answer. The negative sign shows that the man is 30 minutes late. So, the negative sign is required in the first case. Also, it is to be noted that the unit conversions must be made wherever required. Hence, the students must solve this question carefully.