Here's the problem: It takes Stan 20 days to build a house by himself. When his wife, Brenda, joins him, it takes only 14 days. How long does it take Brenda to build a house on her own?
Is there some sort of formula to solve this? I have no idea where to begin. Thanks for the help!
Consider the following:
<< If it takes me 2 hours to paint a room and you 3 hours, ow long will it take to paint it together? >>
Method 1:
1--A can paint the house in 5 hours.
2--B can paint the house in 3 hours.
3--A's rate of painting is 1 house per A hours (5 hours) or 1/A (1/5) houses/hour.
4--B's rate of painting is 1 house per B hours (3 hours) or 1/B (1/3) houses/hour.
5--Their combined rate of painting is 1/A + 1/B (1/5 + 1/3) = (A+B)/AB (8/15) houses /hour.
6--Therefore, the time required for both of them to paint the 1 house is 1 house/(A+B)/AB houses/hour = AB/(A+B) = 5(3)/(5+3) = 15/8 hours = 1 hour-52.5 minutes.
Note - T = AB/(A + B), where AB/(A + B) is one half the harmonic mean of the individual times, A and B.
Method 2:
Consider the following diagram -
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A
1--Let c represent the area of the house to be painted.
2--Let A = the number of hours it takes A to paint the house.
3--Let B = the number of hours it takes B to paint the house.
4--A and B start painting at the same point but proceed in opposite directions around the house.
5--Eventually they meet in x hours, each having painted an area proportional to their individual painting rates.
6--A will have painted y square feet and B will have painted (c-y) square feet.
7--From the figure, A/c = x/y or Ay = cx.
8--Similarly, B/c = x/(c-y) or by = bc - cx.
9--From 7 & 8, y = cx/a = (bc - cx)/b from which x = AB/(A+B), one half of the harmonic mean of A and B.
I think this should give you enough of a clue as to how to solve your particular problem.
Thank you!
To solve this problem, we can use the concept of "work rate" or "efficiency." The idea is that the work they do is directly proportional to the time it takes.
Let's denote Stan's work rate as S (house/day) and Brenda's work rate as B (house/day).
We know that when Stan works alone for 20 days, he can complete one house. So, his work rate is:
S = 1 house / 20 days,
S = 1/20 house/day.
Now, when both Stan and Brenda work together for 14 days, they can complete one house. So, the combined work rate is:
S + B = 1 house / 14 days,
1/20 + B = 1/14.
To find Brenda's work rate, we subtract Stan's work rate from the combined work rate:
B = 1/14 - 1/20.
To simplify this equation, we need to find a common denominator. The least common multiple of 14 and 20 is 140. So, multiplying each fraction by the appropriate factor, we get:
B = 10/140 - 7/140,
B = 3/140.
This means Brenda's work rate is 3/140 house per day. To determine how long it takes her to build a house alone, we divide the work rate into 1 house:
Time taken by Brenda = 1 house / (3/140 house/day).
To divide by a fraction, we can multiply by its reciprocal:
Time taken by Brenda = 1 house * (140/3 house/day).
The house units cancel out, leaving us with:
Time taken by Brenda = 140/3 days.
Therefore, it takes Brenda approximately 46.67 (rounded to two decimal places) days to build a house alone.
So, the answer is that it takes Brenda approximately 46.67 days to build a house on her own.
Note: The exact answer would involve repeating decimals, but it's customary to round to a reasonable number of decimal places based on practicality.