21. Meera walked in the park. She walked half of the total time at a speed of $2 \mathrm{~km} / \mathrm{h}$. She then walked half of the total distance at a speed of $3 \mathrm{~km} / \mathrm{h}$. Finally, she completed the remainder of the walk at a speed of $3 \mathrm{~km} / \mathrm{h}$. For what fraction of the total time did she walk at a speed of $4 \mathrm{~km} / \mathrm{h}$ ?
A $\frac{1}{14}$
B $\frac{1}{12}$
C $\frac{1}{7}$
D $\frac{1}{5}$
$\mathrm{E} \frac{1}{4}$
SOLUTION
A

Let's denote the half of the time for Meera's walk with $t$ and the time she walks with a speed of $4 \mathrm{~km} / \mathrm{h}$ with $x$. Then, the time she walks with a speed of $3 \mathrm{~km} / \mathrm{h}$ is $t-x$. We have the following:
Speed: $2 \mathrm{~km} / \mathrm{h}$, Time: $t$, Distance : $2 t$
Speed: $3 \mathrm{~km} / \mathrm{h}$, Time: $t-x$, Distance : $3(t-x)$
Speed: $4 \mathrm{~km} / \mathrm{h}$, Time: $x$, Distance : $4 x$
Because Meera walks half the distance at $3 \mathrm{~km} / \mathrm{h}$, we have $3(t-x)=2 t+4 x$. Therefore $t=7 x$. Hence $x=\frac{1}{7} t$. The total time for the walk was $2 t$, so $x=\frac{1}{14} \times 2 t$. That is, $\frac{1}{14}$ of the time was spent walking with a speed of $4 \mathrm{~km} / \mathrm{h}$.