A sum of 430 has been distributed among 45 people consisting of men, women and children. The total amounts given to men, women and children are in the ratio 12 :15 :16. But the amounts received by each man, woman and child are in the ratio 6 :5 4 . Find what each man, woman and child receives (in %).

Options

A. 12,10,8
B. 18,15,12
C. 120150160
D. 60,75,80

Answer

To solve the problem, we need to break it down step by step.

### Step 1: Understand the Ratios
We are given two sets of ratios:
1. The total amounts received by men, women, and children are in the ratio of 12:15:16.
2. The amounts received by each man, woman, and child are in the ratio of 6:5:4.

### Step 2: Calculate the Total Parts in Each Ratio
First, we calculate the total parts for the first ratio:
- Total parts for men, women, and children = 12 + 15 + 16 = 43 parts.

### Step 3: Calculate the Amount Distributed to Each Group
Now, we can find the total amount distributed to each group:
- Amount given to men = (12/43) * 430 = 120.
- Amount given to women = (15/43) * 430 = 150.
- Amount given to children = (16/43) * 430 = 160.

### Step 4: Calculate the Total Parts for the Second Ratio
Next, we calculate the total parts for the second ratio:
- Total parts for the amounts received by each person = 6 + 5 + 4 = 15 parts.

### Step 5: Find the Amount Received by Each Individual
Now we can find out how much each individual receives:
- Let the amount received by each man be 6x, each woman be 5x, and each child be 4x.
- The total amount distributed can also be expressed as:
- Total amount = (Number of men * Amount per man) + (Number of women * Amount per woman) + (Number of children * Amount per child)

### Step 6: Set Up the Equations
Let:
- Number of men = m
- Number of women = w
- Number of children = c

From the problem, we know:
- m + w + c = 45 (total number of people)
- Total amount = 6m + 5w + 4c = 430

### Step 7: Solve the Equations
We also have the ratios of the total amounts:
- From the first ratio, we can express:
- m = (12/43) * 430 / 6 = 20
- w = (15/43) * 430 / 5 = 15
- c = (16/43) * 430 / 4 = 10

### Step 8: Calculate the Percentage Received by Each
Now we can find the percentage of the total amount each group receives:
- Percentage received by each man = (6x / 430) * 100
- Percentage received by each woman = (5x / 430) * 100
- Percentage received by each child = (4x / 430) * 100

### Final Step: Calculate x
From the total amount received by men, women, and children, we can find x:
- 6m + 5w + 4c = 430
- Substitute the values of m, w, and c to find x.