Question

The work done by a man, a woman and a child is in the ratio $3:2:1$. There are $20$ men, $30$ women and $36$ children. Their weekly wages (₹ $780$) are divided in the ratio of work done by men, women and children. What will be the wages of $15$ men, $21$ women and $30$ children for $2$ weeks? 

• A. ₹$585$
• B. ₹$292.5$
• C. ₹$1170$
• D. ₹$900$

Hint:

In mixed-work wage problems, first find each group's share via (count × rate) ratios, then convert to per-head and scale by time.

Answer 1

The Correct Option is C

Group work per week is proportional to (number) × (per-head rate):

Men: $20 \times 3 = 60$, Women: $30 \times 2 = 60$, Children: $36 \times 1 = 36$.  
Thus weekly group-share ratio $= 60 : 60 : 36 = 5 : 5 : 3$ (sum = 13).  

One part $= \dfrac{780}{13} = ₹60$. Hence weekly group wages:  
Men ₹300, Women ₹300, Children ₹180.  

Per-head per week: Men ₹300/20 = 15, Women ₹300/30 = 10, Children ₹180/36 = 5.  

For 2 weeks, per head: Men ₹30, Women ₹20, Children ₹10.  

Therefore total for 15 men, 21 women, 30 children:  
\[15 \times 30 + 21 \times 20 + 30 \times 10 = 450 + 420 + 300 = \boxed{₹1170}.\] 

Answer 2

Instead of first reducing the total-work ratio and then dividing it among heads, we can assign a per-head weekly wage rate directly using the given \(3:2:1\) ratio and solve for the common constant, then test each option.

1. Option A (₹585): Let the per-head weekly wage of a man, woman and child be \(3k,2k,k\) respectively. Since there are \(20\) men, \(30\) women and \(36\) children, the total weekly wage is \[ 20(3k)+30(2k)+36(k)=60k+60k+36k=156k=780\;\Rightarrow\;k=5. \] So per-head weekly wages are man \(=₹15\), woman \(=₹10\), child \(=₹5\). For \(15\) men, \(21\) women, \(30\) children over \(2\) weeks: \[ 2\big(15\times15+21\times10+30\times5\big)=2(225+210+150)=2(585)=₹1170. \] This does not match ₹585, so option A is incorrect.
2. Option B (₹292.5): This is exactly a quarter of the computed ₹1170 and does not arise from the calculation, so it is incorrect.
3. Option C (₹1170): As computed above, the total wage for the given group over \(2\) weeks is exactly ₹1170.
4. Option D (₹900): This does not match the computed value ₹1170, so it is incorrect.

The per-head rate method confirms the total wages come to exactly ₹1170.

Hence, the correct answer is option C: ₹1170.