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Fatima leaves her house to go to her friend Nabil’s place. Her speed is 4.5 km/h4.5\,\text{km/h}. When she returns home, her speed is slower, 3.5 km/h3.5\,\text{km/h}. Her total walking time to get to Nabil’s and back was 4040 minutes. - how long does it take Fatima to get to Nabil’s house?

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Q. Fatima leaves her house to go to her friend Nabil’s place. Her speed is 4.5 km/h4.5\,\text{km/h}. When she returns home, her speed is slower, 3.5 km/h3.5\,\text{km/h}. Her total walking time to get to Nabil’s and back was 4040 minutes. - how long does it take Fatima to get to Nabil’s house?
  1. Convert to Hours: First, let's convert the total walking time from minutes to hours, since the speeds are given in km/h.\newlineTotal walking time in hours = 4040 minutes ÷\div 6060 minutes/hour\newlineTotal walking time in hours = 23\frac{2}{3} hours
  2. Distance and Time Formula: Let's denote the distance from Fatima's house to Nabil's place as D kmD \, \text{km}. We can use the formula time=distancespeed\text{time} = \frac{\text{distance}}{\text{speed}} to express the time it takes for Fatima to travel to Nabil's house and back.\newlineTime to Nabil's house = D4.5 km/h\frac{D}{4.5 \, \text{km/h}}\newlineTime to return home = D3.5 km/h\frac{D}{3.5 \, \text{km/h}}
  3. Total Walking Time Equation: The total time spent walking is the sum of the time to get to Nabil's house and the time to return home.\newlineTotal walking time = Time to Nabil's house + Time to return home\newline23\frac{2}{3} hours = D4.5\frac{D}{4.5} km/h + D3.5\frac{D}{3.5} km/h
  4. Combine Fractions: Now we need to solve for DD. To do this, we can find a common denominator for the two fractions and combine them into a single equation.23\frac{2}{3} hours = D×3.5+D×4.54.5×3.5\frac{D \times 3.5 + D \times 4.5}{4.5 \times 3.5}
  5. Clear Fractions: Multiply both sides of the equation by the common denominator (4.5×3.5)(4.5 \times 3.5) to clear the fractions.\newline(23)×(4.5×3.5)=D×3.5+D×4.5(\frac{2}{3}) \times (4.5 \times 3.5) = D \times 3.5 + D \times 4.5
  6. Multiply Left Side: Perform the multiplication on the left side of the equation.\newline(23)×(4.5×3.5)=2×(4.5×3.5)/3(\frac{2}{3}) \times (4.5 \times 3.5) = 2 \times (4.5 \times 3.5) / 3\newline(23)×(4.5×3.5)=2×15.75/3(\frac{2}{3}) \times (4.5 \times 3.5) = 2 \times 15.75 / 3\newline(23)×(4.5×3.5)=31.5/3(\frac{2}{3}) \times (4.5 \times 3.5) = 31.5 / 3\newline(23)×(4.5×3.5)=10.5(\frac{2}{3}) \times (4.5 \times 3.5) = 10.5
  7. Distribute DD: Now distribute DD on the right side of the equation.10.5=D×(3.5+4.5)10.5 = D \times (3.5 + 4.5)10.5=D×810.5 = D \times 8
  8. Solve for D: Divide both sides by 88 to solve for D.\newlineD=10.58D = \frac{10.5}{8}\newlineD=1.3125D = 1.3125 km
  9. Find Time to Nabil's House: Now that we have the distance DD, we can find the time it took Fatima to get to Nabil's house using her speed of 4.54.5 km/h.\newlineTime to Nabil's house = D/4.5D / 4.5 km/h\newlineTime to Nabil's house = 1.31251.3125 km / 4.54.5 km/h
  10. Divide to Find Time: Perform the division to find the time.\newlineTime to Nabil's house = 1.31254.5\frac{1.3125}{4.5}\newlineTime to Nabil's house = 0.291666…0.291666\ldots hours
  11. Convert to Minutes: Finally, convert the time back to minutes, since the question prompt asks for the time in minutes.\newlineTime to Nabil's house in minutes = 0.291666…0.291666\ldots hours ×60\times 60 minutes/hour\newlineTime to Nabil's house in minutes = 17.517.5 minutes

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