A man travels from Town $A$ to Town $B$ at an average speed of $4 \mathrm{~km} / \mathrm{h}$ and from Town B to Town A at an average speed of $6 \mathrm{~km} / \mathrm{h}$ . If he takes $45$ minutes to complete the entire journey, find $h_{\text {is }}$ total distance travelled.

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Q. A man travels from Town

A

to Town

B

at an average speed of

4 \mathrm{~km} / \mathrm{h}

and from Town B to Town A at an average speed of

6 \mathrm{~km} / \mathrm{h}

. If he takes

45

minutes to complete the entire journey, find

h_{\text {is }}

total distance travelled.

Convert to Hours: First, let's convert the time taken for the entire journey from minutes to hours, since the speeds are given in kilometers per hour (km/h). $\newline$ $45$ minutes is equal to $\frac{45}{60}$ hours. $\newline$ $\frac{45}{60} = \frac{3}{4}$ hours.
Distance Calculation: Now, let's denote the distance from Town A to Town B as $D$ kilometers. The man travels this distance twice, once from $A$ to $B$ and once from $B$ to $A$ . The total time taken for the journey is the sum of the time taken to travel from $A$ to $B$ and the time taken to travel from $B$ to $A$ .
Total Time Equation: The time taken to travel from A to B at $4\,\text{km/h}$ is $\frac{D}{4}$ hours, and the time taken to travel from B to A at $6\,\text{km/h}$ is $\frac{D}{6}$ hours. $\newline$ The total time for the journey is therefore $\frac{D}{4} + \frac{D}{6}$ hours.
Solve for Distance: We know the total time for the journey is $\frac{3}{4}$ hours, so we can set up the equation: $\newline$ $\frac{D}{4} + \frac{D}{6} = \frac{3}{4}$ $\newline$ To solve for $D$ , we need to find a common denominator for the fractions on the left side of the equation.
Calculate Total Distance: The least common multiple of $4$ and $6$ is $12$ . We multiply each term by $12$ to clear the denominators: $\newline$ $12 \times (D/4) + 12 \times (D/6) = 12 \times (3/4)$ $\newline$ This simplifies to: $\newline$ $3D + 2D = 9$
Calculate Total Distance: The least common multiple of $4$ and $6$ is $12$ . We multiply each term by $12$ to clear the denominators: $\newline$ $12 \times (D/4) + 12 \times (D/6) = 12 \times (3/4)$ $\newline$ This simplifies to: $\newline$ $3D + 2D = 9$ Combining like terms, we get: $\newline$ $5D = 9$ $\newline$ Now, we solve for $D$ by dividing both sides of the equation by $5$ : $\newline$ $D = 9/5$ $\newline$ $6$ $0$ kilometers
Calculate Total Distance: The least common multiple of $4$ and $6$ is $12$ . We multiply each term by $12$ to clear the denominators: $\newline$ $12 \times \left(\frac{D}{4}\right) + 12 \times \left(\frac{D}{6}\right) = 12 \times \left(\frac{3}{4}\right)$ $\newline$ This simplifies to: $\newline$ $3D + 2D = 9$ Combining like terms, we get: $\newline$ $5D = 9$ $\newline$ Now, we solve for $D$ by dividing both sides of the equation by $5$ : $\newline$ $D = \frac{9}{5}$ $\newline$ $D = 1.8 \text{ kilometers}$ Since the man travels the distance $D$ from Town A to Town B and then the same distance back from Town B to Town A, the total distance traveled is $2D$ . $\newline$ $2D = 2 \times 1.8 = 3.6 \text{ kilometers}$