Question

Rs. 4536 is divided among 4 men, 5 women and 2 boys. The ratio of share of a man, a woman and a boy is \(7: 4: 3\). What is the share of a woman? (a) Rs. 336 (b) Rs. 498 (c) Rs. 166 (d) Rs. 256

Short answer

Answer: The share of each woman is Rs. 336.

Step-by-step solution

Determine the total number of parts

We have the ratio for each category of people (man, woman, and boy): \(7: 4: 3\). Since there are 4 men, 5 women, and 2 boys, we multiply the ratio for each category by the respective number of people in the category: - Number of man parts: \(7 \times 4 = 28\) - Number of woman parts: \(4 \times 5 = 20\) - Number of boy parts: \(3 \times 2 = 6\) The total number of parts can be determined by adding the number of parts in all three categories: Total parts = \(28 + 20 + 6 = 54\)

Find the value of each part

To find the value of a single part, we divide the total amount of money (Rs. 4536) by the total number of parts (54): Value of each part = \(\frac{4536}{54} = 84\)

Calculate the share of a woman

Now that we know the value of each part (Rs. 84), we can calculate the share of each woman by multiplying the value of each part by the number of parts assigned to a woman (4): Share of a woman = \(84 \times 4 = 336\) The share of each woman is Rs. 336, which corresponds to option (a).

Key concepts

Distribution of Wealth

Distribution of wealth often involves dividing resources or money among different individuals or groups according to a predetermined ratio or formula. This process ensures equity, taking into account various factors like contribution or need. In the context of the problem, wealth is distributed among men, women, and boys according to the ratio \(7:4:3\). The ratio signifies the relative share each group should receive. To achieve this:

• Calculate the total number of parts based on the given ratio.
• Allocate portions of the total wealth to each category based on these parts.

Breaking down the total money, Rs. 4536, into shares of a predetermined formula can highlight fairness in distribution while considering demographic proportions. Clear understanding of ratios and parts can simplify such operations for effective wealth allocation.

Shares and Dividends

Shares can be thought of as how the total sum is divided among different groups or individuals. In financial terms, a share might represent ownership. Here, shares symbolize the portion of Rs. 4536 allotted to men, women, and boys. Dividends, in this mathematical context, are similar to the earnings each person or group receives from the total amount.
To calculate these:

• Determine how many shares each participant should receive based on the ratio.
• Calculate the monetary value of each part or share.
• Multiply the number of shares by the value of each share to find out the amount each group or individual obtains.

This exercise underlines the significance of understanding shares, which helps make informed decisions regarding equitable resource allocation and financial planning.

Mathematical Problem Solving

Mathematical problem solving involves using logical and structured steps to arrive at a solution. This approach is essential when handling ratios and proportions. It provides clarity and ensures accuracy.

• First, identify the problem and what is required — here, the share of a woman.
• Second, gather relevant data useful for solving the problem — for instance, number of each group and their respective ratio.
• Third, transform verbal descriptions into mathematical expressions or equations.
• Fourth, perform calculations systematically — like finding the total parts, evaluating the value of each part, and finally determining the share based on parts allocated to each category.

Solving mathematical problems through well-defined steps leads not only to the correct results but also to a deeper comprehension of the underlying concepts involved in ratios and proportions.