Work and Rate Problems with Step-by-Step Solutions | Math Practice

This page presents a collection of work and rate problems in mathematics, covering scenarios such as lawn mowing, tank filling, swimming pools, and pumps. Each problem is accompanied by step-by-step solutions to help you understand the methods used to calculate time, rates, and combined work. These examples are ideal for students, teachers, and anyone looking to practice work and time exercises or improve problem-solving skills in work rate mathematics.

Problem 1 - Lawn Mowing Time

It takes 1.5 hours for Tim to mow the lawn. Linda can mow the same lawn in 2 hours. How long will it take John and Linda, working together, to mow the lawn?

Solution:

We first calculate the rate of work of John and Linda:
John: \( \frac{1}{1.5} \), Linda: \( \frac{1}{2} \)

Let \(t\) be the time for John and Linda to mow the lawn. The work done by John alone is given by:

\( t \times \frac{1}{1.5} \)

The work done by Linda alone is given by:

\( t \times \frac{1}{2} \)

When the two work together, their work will be added. Hence:

\( t \times \frac{1}{1.5} + t \times \frac{1}{2} = 1 \)

Multiply all terms by 6 and simplify:

\( 6 \left(t \times \frac{1}{1.5} + t \times \frac{1}{2} \right) = 6 \)
\( 4t + 3t = 6 \)

Solve for \(t\):

\( t = \frac{6}{7} \text{ hours } \approx 51.5 \text{ minutes} \)

Problem 2 - Pump Rates

It takes 6 hours for pump A, used alone, to fill a tank of water. Pump B used alone takes 8 hours to fill the same tank. We want to use three pumps: A, B, and another pump C to fill the tank in 2 hours. What should be the rate of pump C? How long would it take pump C, used alone, to fill the tank?

Solution:

The rates of pumps A and B can be calculated as follows:
A: \( \frac{1}{6} \), B: \( \frac{1}{8} \)

Let \(R\) be the rate of pump C. When working together for 2 hours, we have:

\( 2 \left( \frac{1}{6} + \frac{1}{8} + R \right) = 1 \)

Solve for \(R\):

\( R = \frac{1}{4.8} \)

Let \(t\) be the time it takes pump C, used alone, to fill the tank. Hence:

\( t \times \frac{1}{4.8} = 1 \Rightarrow t = 4.8 \text{ hours} \)

Problem 3 - Tank Filling with Drainage

A tank can be filled by pipe A in 5 hours and by pipe B in 8 hours, each pump working on its own. When the tank is full and a drainage hole is open, the water is drained in 20 hours. If initially the tank was empty and someone started the two pumps together but left the drainage hole open, how long does it take for the tank to be filled?

Solution:

Let's first find the rates of the pumps and the drainage hole:
Pump A: \( \frac{1}{5} \), Pump B: \( \frac{1}{8} \), Drainage: \( \frac{1}{20} \)

Let \(t\) be the time for the pumps to fill the tank. The pumps add water into the tank while the drainage hole removes water, hence:

\( t \left( \frac{1}{5} + \frac{1}{8} - \frac{1}{20} \right) = 1 \)

Solve for \(t\):

\( t = 3.6 \text{ hours} \)

Problem 4 - Swimming Pool Filling Time

A swimming pool can be filled by pipe A in 3 hours and by pipe B in 6 hours, each pump working on its own. At 9 am pump A is started. At what time will the swimming pool be filled if pump B is started at 10 am?

Solution:

The rates of the two pumps are:
Pump A: \( \frac{1}{3} \), Pump B: \( \frac{1}{6} \)

Working together, if pump A works for \(t\) hours then pump B works \(t - 1\) hours since it started 1 hour late. Hence:

\( t \times \frac{1}{3} + (t - 1) \times \frac{1}{6} = 1 \)

Solve for \(t\):

\( t = \frac{7}{3} \text{ hours} \approx 2 \text{ hours } 20 \text{ minutes} \)

The swimming pool will be filled at:

9:00 + 2:20 = 11:20