Question

A group of 15 carpenters are hired to build a house, and they estimate that they can complete the job in 210 days. After working for 10 days, to speed up the process, the owner decides to hire 15 more carpenters, but these new carpenters are twice as efficient as the original team. How many days (approx.) in total will it take to complete the house?

• A. 72 days
• B. 85 days
• C. 77 days
• D. 70 days

Answer 1

Answer: (C)

77 days

Answer 2

Let the efficiency of one original carpenter be $E_o$ units of work per day.
The efficiency of one new carpenter is $E_n = 2 E_o$ units of work per day.

The total work required to build the house is estimated to be done by 15 original carpenters in 210 days.
Total work = Number of carpenters $\times$ Time $\times$ Efficiency per carpenter
Total work = $15 \times 210 \times E_o$ units.
Let's assume $E_o = 1$ unit/day for simplicity.
Total work = $15 \times 210 = 3150$ units.

In the first 10 days, 15 original carpenters worked.
Work done in the first 10 days = $15 \times 10 \times E_o = 15 \times 10 = 150$ units.

Remaining work = Total work - Work done in the first 10 days
Remaining work = $3150 - 150 = 3000$ units.

After 10 days, the team consists of 15 original carpenters and 15 new carpenters.
The daily work rate of the original 15 carpenters = $15 \times E_o = 15 \times 1 = 15$ units/day.
The daily work rate of the new 15 carpenters = $15 \times E_n = 15 \times (2 E_o) = 15 \times 2 = 30$ units/day.

The combined daily work rate of the new team = Work rate of original carpenters + Work rate of new carpenters
Combined daily work rate = $15 + 30 = 45$ units/day.

Let $d$ be the number of days taken to complete the remaining work.
Remaining work = Combined daily work rate $\times$ Time
$3000 = 45 \times d$
$d = \frac{3000}{45}$

To simplify the fraction $\frac{3000}{45}$:
Divide both numerator and denominator by 5: $\frac{600}{9}$
Divide both numerator and denominator by 3: $\frac{200}{3}$
So, $d = \frac{200}{3}$ days.

The total time taken to complete the house is the sum of the time spent in the first phase and the time spent in the second phase.
Total time = Time in first 10 days + Time for remaining work
Total time = $10 + \frac{200}{3}$ days
Total time = $\frac{10 \times 3}{3} + \frac{200}{3} = \frac{30}{3} + \frac{200}{3} = \frac{230}{3}$ days.

Now, we need to approximate the total time.
$\frac{230}{3} = 76$ with a remainder of 2.
So, $\frac{230}{3} = 76 \frac{2}{3}$ days.
As a decimal, $\frac{2}{3} \approx 0.666...$
Total time $\approx 76.67$ days.

We need to find the closest option to 76.67 days. A 72 days (Difference = 4.67) B 85 days (Difference = 8.33) C 77 days (Difference = 0.33) D 70 days (Difference = 6.67)

The closest option is 77 days.