How to Solve Work Problem Part 2

This is the second part of the Solving Work Problems Series. In the previous post, we have discussed in detail the concept behind how to solve work problems.  In this post, we are going to learn more examples and solve more complicated problems.

Problem 2

A hose can fill a pool in 3 hours, while a smaller hose can fill it in 5 hours. If the hoses are opened together the same time, how many hours will they be able to fill the pool?

Solution

House A can fill the pool in 3 hours, so it can fill 1/3 of the pool in 1 hour.

House A can fill the pool in 5 hours, so it can fill 1/5 of the pool in 1 hour.

Together, they can fill 1/3 + 1/5 of the pool in 1 hour.

Let x be the number of hours to fill the pool. As we have done in the previous post, we set up the following equation (read the previous post for details).  That is, $\displaystyle \frac{1}{3}x + \frac{1}{5}x = 1$

Multiplying both sides by 15, the least common denominator of 1/5 and 1/3, we have $15(\displaystyle \frac{1}{3}x) + 15( \frac{1}{5}x) = 15(1)$ $\frac{15}{3}x + \frac{15}{5}x = 15$ $5x + 3x = 15$ $8x = 15$ $x = \frac{15}{8}$

This means that the two hoses will fill the pool in 15/8 or 1 and 7/8 hours.

Problem 3

Chloe and Diane are gown designers in a prestigious company. Chloe and Diane can embellish a gown in 4 hours. Chloe alone can do the same task in 6 hours. How long will Diane be able to do the same task if she were to work alone?

Solution

Chloe and Diane can finish the task in 4 hours, so they can finish 1/4 of the task in 1 hour.

Chloe alone can finish the task in 6 hours, so she can finish 1/6 of the task in 1 hour.

Diane can finish the task in x hours, so she can finish 1/x of the task in 1 hour.

Note that if we combine the work of Chloe (1/6) and Diane (1/x), their rate is 1/4 of the task. That is $\displaystyle \frac{1}{6} + \frac{1}{x} = \frac{1}{4}$

We multiply the equation by 12x, the least common denominator of 6, 4, and x, we have $\displaystyle 12x (\frac{1}{6}) + 12x (\frac{1}{6}) =12x( \frac{1}{4})$ $\displaystyle \frac{12x}{6} + \frac{12x}{x} = \frac{12x}{4}$ $2x + 12 = 3x$ $x = 4$

This means that Diane can finish the task alone in 4 hours.

In the next post, we will have more examples. • Aang

The same solution for problem 1 and 2 can also be used for problem 3, I think…

1/6 (4) + 1/x (4) = 1

Simplify:
4/6 + 4/x = 1

Multiplied by 6x to get rid of fraction:
24x/6 + 24x/x = 6x

4x + 24 = 6x

24 = 2x

12 = x

Using the same formula makes it easier for me to remember 🙂

• Aki_

Isn’t the lcd for the problem #3 is 12?

Either way 12 or 24 yields the same answer.