Problem 50 - Introductory Algebra for College Students

Question

An experienced carpenter can panel a room 3 times faster than an apprentice can. Working together, they can panel the room in 6 hours. How long would it take each person working alone to do the job?

Step-by-Step Solution

Answer

When working alone, the apprentice would take 24 hours and the carpenter would take 8 hours to panel the room.

Step 1: Define the Variables

Let's call the amount of time it takes the apprentice to panel the room by himself 'A' hours, and the time it takes the carpenter 'C' hours. Since we know that the carpenter works three times as fast as the apprentice, we can say C = A/3.

Step 2: Use the Combined Work Formula

The formula we'll use is 1/A + 1/C = 1/T, where T is the total time it takes them to do the job working together (in this case T=6). Substituting C = A/3 into the equation, we get 1/A + 3/A = 1/6.

Step 3: Solve for A

Adding the two fractions on the left side we get 4/A = 1/6. Solving for 'A' gives us A = 24. Which means the apprentice would take 24 hours to panel the room by himself.

Step 4: Solve for C

Using the relation defined in step 1, we substitute A = 24 into the equation, getting C = 24/3. Which means the carpenter would take 8 hours to panel the room by himself.