John takes 3 hours longer than Andrew to peel 400 pounds of apples. Together they take 8 hours. How long will it take John to peel 400 pounds by himself
This should give you the information you need.
If it takes one person 5 hours to paint a room and another person 3 hours, how long will it take to paint the room working together?
Method 1:
1--A can paint a room in 5 hours.
2--B can paint a room in 3 hours.
3--A's rate of painting is 1 room per A hours (5 hours) or 1/A (1/5) room/hour.
4--B's rate of painting is 1 room per B hours (3 hours) or 1/B (1/3) room/hour.
5--Their combined rate of painting is therefore 1/A + 1/B = (A+B)/AB = (1/5 + 1/3) = (8/15) rooms /hour.
6--Therefore, the time required for both of them to paint the 1 room working together is 1 room/(A+B)/AB rooms/hour = AB/(A+B) = 5(3)/(5+3) = 15/8 hours = 1 hour-52.5 minutes.
Note - Generally speaking (if the derivation is not specifically required), if it takes one person A units of time and another person B units of time to complete a specific task working alone, the time it takes them both to complete the task working together is T = AB/(A + B), where AB/(A + B) is one half the harmonic mean of the individual times, A and B.
You might like to derive the equivalant expression involving 3 people working alone and together which results in T = ABC/(AB + AC + BC).
To find out how long it will take John to peel 400 pounds of apples by himself, we can use the information given. Let's break down the problem step by step:
1. Let's assume that Andrew takes x hours to peel 400 pounds of apples. This means that John will take x + 3 hours to peel the same amount of apples.
2. Together, Andrew and John can peel 400 pounds of apples in 8 hours. This allows us to create the following equation:
1/x + 1/(x + 3) = 1/8.
3. To solve this equation, we need to find a common denominator and combine the fractions on the left side:
(x + 3 + x)/(x(x + 3)) = 1/8
(2x + 3)/(x^2 + 3x) = 1/8.
4. Cross-multiply to eliminate the fractions:
8(2x + 3) = x^2 + 3x.
5. Simplify the equation:
16x + 24 = x^2 + 3x.
6. Rearrange the equation to form a quadratic equation:
x^2 - 13x - 24 = 0.
7. Factorize the quadratic equation or use the quadratic formula to find the values of x.
8. Once you find the value of x, which represents the time it takes Andrew to peel 400 pounds of apples, add 3 hours to this value to find the time it would take John to peel the same amount of apples by himself.
Remember, step 7 involves finding the values of x using either factoring or the quadratic formula, and there may be multiple solutions.