Question

You and a friend volunteer to paint a small house as a community service project. Working alone, you can paint the house in 15 hours. Your friend can paint the house in 18 hours working alone. How long will it take both of you, working together, to paint the house?

Short answer

It will take approximately 8.18 hours for both you and your friend to paint the house together.

Step-by-step solution

Determine Individual Rate of Work

Rate of work is defined as the amount of work done per unit time. Therefore, the rate of work for you is \( \frac{1}{15} \) of the house per hour. Similarly, the rate of work for your friend is \( \frac{1}{18} \) of the house per hour.

Determine Combined Rate of Work

When two individuals work together, their rates of work are additive. Therefore, the combined rate of work is \( \frac{1}{15} + \frac{1}{18} = \frac{11}{90} \) of the house per hour.

Determine Time for Combined Work

Finally, we can calculate the time it would take for both to paint the house together. This can be done by dividing the total work (1 house in this case) by the combined rate of work. Therefore, time taken is \( \frac{1}{(\frac{11}{90})} = \frac{90}{11} \approx 8.18 \) hours.

Key concepts

Rate of Work

The rate of work is a concept that describes how much work is completed per unit of time. In our example, this is how much of the house is painted per hour. Imagine this rate as a tiny worker inside a clock; it moves a little bit closer to finishing the task every tick of the second hand.
For you, the rate of work is expressed as \( \frac{1}{15} \). This means you can paint one complete house in 15 hours if you are working all by yourself. Every hour, you manage to cover \( \frac{1}{15} \) of the house.
Similarly, your friend's rate of work is \( \frac{1}{18} \), implying they take longer, 18 hours, to finish painting one house alone. Every hour, they paint \( \frac{1}{18} \) of it.

• The smaller the denominator in the rate of work, the faster the worker is (e.g., \( \frac{1}{10} \) is faster than \( \frac{1}{15} \)).
• Keep in mind that these rates are fractions, which makes addition easier, as we'll see in combining efforts.

Combined Work Rate

When joining forces, two or more workers' individual rates can be added together. This is because each person's effort contributes to the completion of the whole task.
For you and your friend, your combined work rates add up to \( \frac{1}{15} + \frac{1}{18} \). By finding a common denominator and performing the arithmetic, you get \( \frac{11}{90} \). This result tells us, together, you paint \( \frac{11}{90} \) of the house in one hour.

• Notice how combining work rates is similar to combining speeds in a relay race; everyone is working toward finishing together.
• The sum of work rates should always reflect all contributions, making sure no work effort is missed out!

Algebraic Problem Solving

Solving work problems often involves breaking them into parts using algebra. With rates expressed as fractions, algebra helps us solve for unknowns, like the time it takes for a combined effort.
To find how long it takes when working together, you apply a simple formula. Divide the total work (1 full house) by the combined rate. This can be set up as \( \frac{1}{\text{combined rate}} = \text{total hours} \).
Substituting the combined rate, we get \( \frac{1}{\left( \frac{11}{90} \right)} \), which simplifies to \( \frac{90}{11} \approx 8.18 \). This computation gives the exact time needed for both of you to finish the task.

• When solving such equations, maintaining consistency—like converting fractions without error—is crucial.
• This process showcases how algebra can systematically tackle real-life situations by breaking them down methodically.