Question

A group of N people worked on a project.They finished \(35\%\) of the project by working 7 hours a day for 10 days.Thereafter,10 people left the group and the remaining people finished the rest of the project in 14 days by working 10 hours a day.Then the value of N is

• A. 23
• B. 140
• C. 36
• D. 150

Answer 1

The Correct Option is B

A group finishes 35% of a project in 10 days working 7 hours per day. Then, 10 people leave. The remaining group completes the remaining 65% of the project in 14 days working 10 hours per day. Find the original number of people in the group.

Step 1: Let Total Work = \( W \)

First Phase:

• Work done = \( 0.35W \)
• Let initial number of people = \( N \)
• Total man-hours = \( N \times 7 \times 10 \)
• So, \( 0.35W = N \times 7 \times 10 \)

Second Phase:

• Work done = \( 0.65W \)
• Remaining people = \( N - 10 \)
• Total man-hours = \( (N - 10) \times 10 \times 14 \)
• So, \( 0.65W = (N - 10) \times 10 \times 14 \)

Step 2: Eliminate \( W \) by Equating Both Expressions

From the first phase: \[ W = \frac{N \times 7 \times 10}{0.35} \] From the second phase: \[ W = \frac{(N - 10) \times 10 \times 14}{0.65} \]

Equating both: \[ \frac{N \times 7 \times 10}{0.35} = \frac{(N - 10) \times 10 \times 14}{0.65} \]

Step 3: Simplify

Multiply both sides: \[ N \times 7 \times 10 \times 0.65 = (N - 10) \times 10 \times 14 \times 0.35 \] \[ 70N \times 0.65 = 140(N - 10) \times 0.35 \] \[ 45.5N = 49(N - 10) \]

Step 4: Solve for \( N \)

\[ 45.5N = 49N - 490 \] \[ 49N - 45.5N = 490 \] \[ 3.5N = 490 \Rightarrow N = \frac{490}{3.5} = 140 \]

Final Answer:

\[ \boxed{140} \] So, the initial number of people in the group is 140.

Correct Option: (B)

Answer 2

A group of \( N \) people works 7 hours per day for 10 days and completes 35% of a project. Then, 10 people leave. The remaining people complete the rest of the work by working 10 hours a day for 14 days. Find the initial number of people, \( N \).

Step 1: Work Done in the First Phase

Work done by \( N \) people in 10 days at 7 hours per day: \[ \text{Work} = N \times 7 \times 10 = 70N \text{ units} \] According to the question, this represents 35% of the total work: \[ 0.35 \times \text{Total Work} = 70N \Rightarrow \text{Total Work} = \frac{70N}{0.35} = 200N \]

Step 2: Work Remaining

Remaining work: \[ \text{Remaining Work} = 200N - 70N = 130N \text{ units} \]

Step 3: Second Phase Work Done by Remaining People

After 10 people leave, number of people = \( N - 10 \)
They work 14 days, 10 hours per day: \[ \text{Work} = (N - 10) \times 14 \times 10 = 140(N - 10) \] This equals the remaining work: \[ 140(N - 10) = 130N \]

Step 4: Solve the Equation

Expand: \[ 140N - 1400 = 130N \Rightarrow 10N = 1400 \Rightarrow N = \frac{1400}{10} = 140 \]

Final Answer:

\[ \boxed{140} \] So, the initial number of people in the group is 140.

Correct Option: (B)