Introduction / Context:
This is an algebraic time and work problem in which the number of men and the time taken are expressed in terms of a variable x. Two different conditions about completing the full work and three-fourths of the work are given. You must determine x first and then use it to find how long (x + 10) men would take to complete the whole work.
Given Data / Assumptions:
- (x - 2) men complete the full work in x days.
- (x + 7) men complete 75% (that is, 3/4) of the work in (x - 10) days.
- All men are assumed to have equal and constant efficiency.
- We need the time for (x + 10) men to complete the full work.
Concept / Approach:
Convert both conditions into algebraic equations for the total amount of work W. Equate the expressions and solve for x. Then compute the total work using the first condition. Finally, use the work W and the rate of (x + 10) men to find the required time. This is a classic use of simultaneous equations in a work context.
Step-by-Step Solution:
Let total work = W units.
From the first condition: (x - 2) men * x days = W.
So W = x(x - 2).
From the second condition: (x + 7) men * (x - 10) days = 3/4 of W.
So (x + 7)(x - 10) = (3/4) * W = (3/4) * x(x - 2).
Write equation: (x + 7)(x - 10) = (3/4)x(x - 2).
Expand left side: x^2 - 10x + 7x - 70 = x^2 - 3x - 70.
Right side: (3/4)x(x - 2) = (3/4)(x^2 - 2x).
Multiply both sides by 4 to remove denominator:
4(x^2 - 3x - 70) = 3(x^2 - 2x).
4x^2 - 12x - 280 = 3x^2 - 6x.
Bring all terms to one side: 4x^2 - 12x - 280 - 3x^2 + 6x = 0.
Simplify: x^2 - 6x - 280 = 0.
Solve quadratic: x^2 - 6x - 280 = 0.
Discriminant = 6^2 + 4 * 280 = 36 + 1120 = 1156.
Square root of 1156 = 34.
So x = (6 ± 34) / 2, giving x = 20 or x = -14.
x must be positive and larger than 10 (because of x - 10), so x = 20.
Now total work W = x(x - 2) = 20 * 18 = 360 units.
(x + 10) men = 20 + 10 = 30 men.
Rate of 30 men = 30 units per day if one man does 1 unit per day.
Time required = W / 30 = 360 / 30 = 12 days.
Verification / Alternative check:
Check second condition with x = 20: (x + 7)(x - 10) = 27 * 10 = 270 units of work.
3/4 of W = 3/4 * 360 = 270, which matches, so x = 20 is consistent.
Why Other Options Are Wrong:
27, 25 or 18 days do not correspond to the correct x value and total work.
Any other x would violate at least one of the two conditions about workers and days.
Common Pitfalls:
Students often forget to discard the negative root of the quadratic even though a negative number of days or men is impossible.
Another error is mishandling the 3/4 factor for 75% of the work.
Always double check algebraic expansion and fraction multiplication.
Final Answer:
(x + 10) men will finish the work in 12 days.
Discussion & Comments