Question

A can do a piece of work in 6 days while B alone can do it in 8 days. With the help of C, they can complete the work in 2 days. If they are paid Rupees 720 for the whole work, how much money A gets?

• A. Rupees 180
• B. Rupees 240
• C. Rupees 300
• D. Rupees 420

Hint:

In work and wages problems, always reduce the data to "1-day work" (or unit work). Then compare each person's contribution over the total period and divide wages in the same ratio.

Answer 1

The Correct Option is B

Step 1: Work rates of A and B 
A completes the work in 6 days. Hence, A's 1-day work = \( \frac{1}{6} \). 
B completes the work in 8 days. Hence, B's 1-day work = \( \frac{1}{8} \). 
Step 2: Combined rate of A, B, and C 
Together, A, B, and C complete the work in 2 days. Hence, their combined 1-day work = \( \frac{1}{2} \). 
Step 3: Finding C's rate 
\[ \frac{1}{6} + \frac{1}{8} + \text{C's 1-day work} = \frac{1}{2} \] 
Take LCM of 6 and 8, which is 24: 
\[ \frac{4}{24} + \frac{3}{24} + \text{C's 1-day work} = \frac{1}{2} \] 
\[ \frac{7}{24} + \text{C's 1-day work} = \frac{1}{2} \] 
\[ \text{C's 1-day work} = \frac{1}{2} - \frac{7}{24} = \frac{12}{24} - \frac{7}{24} = \frac{5}{24} \] 
Step 4: Work done individually in 2 days 
- Work done by A in 2 days = \( 2 \times \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \). 
- Work done by B in 2 days = \( 2 \times \frac{1}{8} = \frac{2}{8} = \frac{1}{4} \). 
- Work done by C in 2 days = \( 2 \times \frac{5}{24} = \frac{10}{24} = \frac{5}{12} \). 
Check: \(\frac{1}{3} + \frac{1}{4} + \frac{5}{12} = \frac{4}{12} + \frac{3}{12} + \frac{5}{12} = 1 \). ✅ 
Step 5: Payment distribution 
The total payment is distributed in the ratio of work done. 
Work fractions = A : B : C = \( \frac{1}{3} : \frac{1}{4} : \frac{5}{12} \). 
Take LCM of 12: \( \frac{4}{12} : \frac{3}{12} : \frac{5}{12} = 4 : 3 : 5 \). 
Step 6: A's share 
Total ratio sum = \( 4 + 3 + 5 = 12 \). 
A's share = \( \frac{4}{12} \times 720 = \frac{1}{3} \times 720 = 240 \). ✅ 
\[ \boxed{240} \]

Answer 2

Using whole work-units (instead of fractions) makes it easy to check each option against the actual work each person did.

1. Rupees 180: Taking total work as 24 units (LCM of 6, 8, 2), A's rate is \(24/6=4\) units/day, B's is \(24/8=3\) units/day, and the three together complete \(24/2=12\) units/day, so C's rate is \(12-4-3=5\) units/day. Over 2 days, A does \(8\) units out of the total \(24\), so A's fair share of payment should be \(\frac{8}{24}\times720=240\), not \(180\). This option understates A's share.
2. Rupees 240: As computed, A does 8 of the 24 total units of work, earning \(\frac{8}{24}\times720=240\). This matches exactly.
3. Rupees 300: This would require A to have done \(\frac{300}{720}\times24=10\) units, but A only did 8 units in 2 days, so this overstates A's contribution.
4. Rupees 420: This would require A to have done \(\frac{420}{720}\times24=14\) units, far more than the 8 units A actually completed, so this is inconsistent with A's work rate.

Work done is the fairest basis for splitting the payment, and A's actual share of the 24 total units is 8, giving a payment of 240.

So the correct answer is Rupees 240.