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A man travels from Town A to Town B at an average speed of 4km//h and from Town B to Town A at an average speed of 6km//h. If he takes 45 minutes to complete the entire journey, find h_("is ") total distance travelled.

A man travels from Town A A to Town B B at an average speed of 4 km/h 4 \mathrm{~km} / \mathrm{h} and from Town B to Town A at an average speed of 6 km/h 6 \mathrm{~km} / \mathrm{h} . If he takes 4545 minutes to complete the entire journey, find his  h_{\text {is }} total distance travelled.

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Q. A man travels from Town A A to Town B B at an average speed of 4 km/h 4 \mathrm{~km} / \mathrm{h} and from Town B to Town A at an average speed of 6 km/h 6 \mathrm{~km} / \mathrm{h} . If he takes 4545 minutes to complete the entire journey, find his  h_{\text {is }} total distance travelled.
  1. 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).\newline4545 minutes is equal to 4560\frac{45}{60} hours.\newline4560=34\frac{45}{60} = \frac{3}{4} hours.
  2. Distance Calculation: Now, let's denote the distance from Town A to Town B as DD kilometers. The man travels this distance twice, once from AA to BB and once from BB to AA. The total time taken for the journey is the sum of the time taken to travel from AA to BB and the time taken to travel from BB to AA.
  3. Total Time Equation: The time taken to travel from A to B at 4 km/h4\,\text{km/h} is D4\frac{D}{4} hours, and the time taken to travel from B to A at 6 km/h6\,\text{km/h} is D6\frac{D}{6} hours.\newlineThe total time for the journey is therefore D4+D6\frac{D}{4} + \frac{D}{6} hours.
  4. Solve for Distance: We know the total time for the journey is 34\frac{3}{4} hours, so we can set up the equation:\newlineD4+D6=34\frac{D}{4} + \frac{D}{6} = \frac{3}{4}\newlineTo solve for DD, we need to find a common denominator for the fractions on the left side of the equation.
  5. Calculate Total Distance: The least common multiple of 44 and 66 is 1212. We multiply each term by 1212 to clear the denominators:\newline12×(D/4)+12×(D/6)=12×(3/4)12 \times (D/4) + 12 \times (D/6) = 12 \times (3/4)\newlineThis simplifies to:\newline3D+2D=93D + 2D = 9
  6. Calculate Total Distance: The least common multiple of 44 and 66 is 1212. We multiply each term by 1212 to clear the denominators:\newline12×(D/4)+12×(D/6)=12×(3/4)12 \times (D/4) + 12 \times (D/6) = 12 \times (3/4)\newlineThis simplifies to:\newline3D+2D=93D + 2D = 9 Combining like terms, we get:\newline5D=95D = 9\newlineNow, we solve for DD by dividing both sides of the equation by 55:\newlineD=9/5D = 9/5\newline6600 kilometers
  7. Calculate Total Distance: The least common multiple of 44 and 66 is 1212. We multiply each term by 1212 to clear the denominators:\newline12×(D4)+12×(D6)=12×(34)12 \times \left(\frac{D}{4}\right) + 12 \times \left(\frac{D}{6}\right) = 12 \times \left(\frac{3}{4}\right)\newlineThis simplifies to:\newline3D+2D=93D + 2D = 9Combining like terms, we get:\newline5D=95D = 9\newlineNow, we solve for DD by dividing both sides of the equation by 55:\newlineD=95D = \frac{9}{5}\newlineD=1.8 kilometersD = 1.8 \text{ kilometers}Since the man travels the distance DD from Town A to Town B and then the same distance back from Town B to Town A, the total distance traveled is 2D2D.\newline2D=2×1.8=3.6 kilometers2D = 2 \times 1.8 = 3.6 \text{ kilometers}

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