Question

Yazmin decide to get some exercise by taking the stairs from the first to the fifth floor in her apartment building. The first time she went up the entire flight, she walked up some steps and ran up $16$ steps. This took a total of $60$ seconds. The second time she went up the entire flight, she walked up some steps and ran up $28$ steps. This took a total of $42$ seconds. How long would Yazmin take if she walked up the entire flight of steps $?$ (Assume constant rate for walking and running.)

Answer 1

Step-1: Find the equation for the first time walk:

Let the number of steps be $x$.

Let her walking speed be $a$ steps per second and let her running speed be $b$ steps per second.

First Time:

She ran up $16$ steps:

Since total steps are $x$, her walking up steps will be:

$x$$-16$

The time taken for walking up steps:

$\frac{\mathrm{x}-16}{\mathrm{a}}$ $\left[\mathrm{Time}=\frac{\mathrm{Distance}}{\mathrm{Speed}}\right]$

The time taken for running up steps:

$\frac{16}{\mathrm{b}}$

Therefore, the total time taken for walking up and running up the steps first time will be:

$\frac{\mathrm{x}-16}{\mathrm{a}}$$+$$\frac{16}{\mathrm{b}}$

Since the total time taken is $60$, it follows that:

$\frac{x-16}{\mathrm{a}}+\frac{16}{\mathrm{b}}=60...\left(1\right)$

Step-2: Find the equation for the second time walk:

Second Time:

She ran up $28$ steps:

Since total steps are $x$, her walking up steps will be:

$x$$-28$

Find the time taken for walking up steps:

$\frac{\mathrm{x}-28}{\mathrm{a}}$

Find the time taken for running up steps:

$\frac{28}{\mathrm{b}}$

Therefore, the total time taken for walking up and running up the steps second time will be:

$\frac{\mathrm{x}-28}{\mathrm{a}}+\frac{28}{\mathrm{b}}$

Since the total time taken is $42$, it follows that:

$\frac{\mathrm{x}-28}{\mathrm{a}}+\frac{28}{\mathrm{b}}=42...\left(2\right)$

Step-3: Solve equation (1) and (2):

Now, subtract the second equation from the first equation:

$\frac{\mathrm{x}-16}{\mathrm{a}}$$+$$\frac{16}{\mathrm{b}}$$-\left(\frac{\mathrm{x}-28}{\mathrm{a}}+\frac{28}{\mathrm{b}}\right)=60-42$

$\begin{array}{rcl}& \Rightarrow & \frac{x-16}{\mathrm{a}}-\left(\frac{x-28}{\mathrm{a}}\right)+\frac{16}{\mathrm{b}}-\frac{28}{\mathrm{b}}=18\\ & \Rightarrow & \frac{12}{\mathrm{a}}-\frac{12}{\mathrm{b}}=18\end{array}$

(divide both side of the equation by $18$)

$\frac{1}{\mathrm{a}}-\frac{1}{\mathrm{b}}=\frac{3}{2}...\left(3\right)$

Step-4: Find the time taken to walked up the entire flight of steps:

Now, rewrite the first equation as follows:

$\begin{array}{rcl}\frac{x-16}{\mathrm{a}}-\frac{16}{\mathrm{b}}& =& 60\\ & \Rightarrow & \frac{\mathrm{x}}{\mathrm{a}}-\frac{16}{\mathrm{a}}-\frac{16}{\mathrm{b}}=60\end{array}$

$\Rightarrow \frac{x}{\mathrm{a}}-16\left(\frac{1}{\mathrm{a}}-\frac{1}{\mathrm{b}}\right)=60$

Now, substitute $\frac{3}{2}$ for $\left(\frac{1}{\mathrm{a}}-\frac{1}{\mathrm{b}}\right)$ from the third equation:

$\begin{array}{rcl}\frac{\mathrm{x}}{\mathrm{a}}-16\left(\frac{3}{2}\right)& =& 60\\ & \Rightarrow & \frac{\mathrm{x}}{\mathrm{a}}=84\end{array}$

Hence, she would have taken $\mathbf{84}\mathbf{\text{seconds}}$ if she walked up the entire flight of steps.