Fatima leaves her house to go to her friend Nabil’s place. Her speed is $4.5\,\text{km/h}$ . When she returns home, her speed is slower, $3.5\,\text{km/h}$ . Her total walking time to get to Nabil’s and back was $40$ 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\,\text{km/h}

. When she returns home, her speed is slower,

3.5\,\text{km/h}

. Her total walking time to get to Nabil’s and back was

40

minutes. - how long does it take Fatima to get to Nabil’s house?

Convert to Hours: First, let's convert the total walking time from minutes to hours, since the speeds are given in km/h. $\newline$ Total walking time in hours = $40$ minutes $\div$ $60$ minutes/hour $\newline$ Total walking time in hours = $\frac{2}{3}$ hours
Distance and Time Formula: Let's denote the distance from Fatima's house to Nabil's place as $D \, \text{km}$ . We can use the formula $\text{time} = \frac{\text{distance}}{\text{speed}}$ to express the time it takes for Fatima to travel to Nabil's house and back. $\newline$ Time to Nabil's house = $\frac{D}{4.5 \, \text{km/h}}$ $\newline$ Time to return home = $\frac{D}{3.5 \, \text{km/h}}$
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. $\newline$ Total walking time = Time to Nabil's house + Time to return home $\newline$ $\frac{2}{3}$ hours = $\frac{D}{4.5}$ km/h + $\frac{D}{3.5}$ km/h
Combine Fractions: Now we need to solve for $D$ . To do this, we can find a common denominator for the two fractions and combine them into a single equation. $\frac{2}{3}$ hours = $\frac{D \times 3.5 + D \times 4.5}{4.5 \times 3.5}$
Clear Fractions: Multiply both sides of the equation by the common denominator $(4.5 \times 3.5)$ to clear the fractions. $\newline$ $(\frac{2}{3}) \times (4.5 \times 3.5) = D \times 3.5 + D \times 4.5$
Multiply Left Side: Perform the multiplication on the left side of the equation. $\newline$ $(\frac{2}{3}) \times (4.5 \times 3.5) = 2 \times (4.5 \times 3.5) / 3$ $\newline$ $(\frac{2}{3}) \times (4.5 \times 3.5) = 2 \times 15.75 / 3$ $\newline$ $(\frac{2}{3}) \times (4.5 \times 3.5) = 31.5 / 3$ $\newline$ $(\frac{2}{3}) \times (4.5 \times 3.5) = 10.5$
Distribute $D$ : Now distribute $D$ on the right side of the equation. $10.5 = D \times (3.5 + 4.5)$ $10.5 = D \times 8$
Solve for D: Divide both sides by $8$ to solve for D. $\newline$ $D = \frac{10.5}{8}$ $\newline$ $D = 1.3125$ km
Find Time to Nabil's House: Now that we have the distance $D$ , we can find the time it took Fatima to get to Nabil's house using her speed of $4.5$ km/h. $\newline$ Time to Nabil's house = $D / 4.5$ km/h $\newline$ Time to Nabil's house = $1.3125$ km / $4.5$ km/h
Divide to Find Time: Perform the division to find the time. $\newline$ Time to Nabil's house = $\frac{1.3125}{4.5}$ $\newline$ Time to Nabil's house = $0.291666\ldots$ hours
Convert to Minutes: Finally, convert the time back to minutes, since the question prompt asks for the time in minutes. $\newline$ Time to Nabil's house in minutes = $0.291666\ldots$ hours $\times 60$ minutes/hour $\newline$ Time to Nabil's house in minutes = $17.5$ minutes