Question

Abbie paints twice as fast as Beth and three times as fast as Cathie. If it takes them $60\text{min}$to paint a living room with all three working together, how long would it take Abbie if she worked alone?

Short answer

The time taken by Abbie if she worked alone is $110\text{min}$.

Step-by-step solution

Step 1. Write the given information.

The total time taken to complete the work by A,B and C working together is $60\text{min}$.

The time taken by A is twice of time take by B.

The time taken by A is thrice of time taken by C.

Step 2. Form the system of equations.

Let the time take by A painting alone by x, time taken by B painting alone be y and time taken by C painting alone be z.

So, the speed of A, B and C are,

$\begin{array}{l}Spee{d}_A=\frac{1}{x}\\ Spee{d}_B=\frac{1}{y}\\ Spee{d}_C=\frac{1}{z}\end{array}$

Now it is given that,

$\begin{array}{c}Spee{d}_A=2Spee{d}_B\\ \frac{1}{x}=2\left(\frac{1}{y}\right)\\ \frac{1}{2x}=\frac{1}{y}\end{array}$

Also,

$\begin{array}{c}Spee{d}_A=3Spee{d}_C\\ \frac{1}{x}=3\left(\frac{1}{z}\right)\\ \frac{1}{3x}=\frac{1}{z}\end{array}$

Step 3. Form the equation and solve for x.

The speed of all the three working together is,

$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{60}$

Substitute the values in the above expression.

$\begin{array}{c}\frac{1}{x}+\frac{1}{2x}+\frac{1}{3x}=\frac{1}{60}\\ \frac{6+3+2}{6x}=\frac{1}{60}\\ 11=\frac{x}{10}\\ x=110\end{array}$

So, the time taken by A is$110$ min.

Thus, the time taken by Abbie if she worked alone is $110\text{min}$.