Understanding the Ratio Division Problem

This problem involves dividing a total amount of money among three sons according to two different ratios. We are given the difference in the share of the son who received the lowest amount in the first division when the amount is divided using the second ratio. Our goal is to find the total amount of money.

Analyzing the First Ratio Division

In the first division, the amount is divided among the three sons in the ratio \(3 \ratio 4 \ratio 5\).

• The total number of ratio parts is \(3 + 4 + 5 = 12\).

Let the total amount be \(A\) rupees. The shares of the three sons in the first division are:

• Son 1's share: \(\frac{3}{12} \times A = \frac{1}{4}A\)
• Son 2's share: \(\frac{4}{12} \times A = \frac{1}{3}A\)
• Son 3's share: \(\frac{5}{12} \times A = \frac{5}{12}A\)

To identify the lowest share among \(\frac{1}{4}, \frac{1}{3}, \frac{5}{12}\), we can find a common denominator, which is 12.

• \(\frac{1}{4} = \frac{3}{12}\)
• \(\frac{1}{3} = \frac{4}{12}\)
• \(\frac{5}{12} = \frac{5}{12}\)

Comparing \(\frac{3}{12}, \frac{4}{12}, \frac{5}{12}\), the lowest fraction is \(\frac{3}{12}\). Thus, the lowest share in the first division is Son 1's share, which is \(\frac{1}{4}A\).

Converting and Calculating the Second Ratio Shares

In the second division, the amount is divided in the ratio \(1/3 \ratio 1/4 \ratio 1/5\).

To work with whole numbers, we find the Least Common Multiple (LCM) of the denominators 3, 4, and 5. The LCM of 3, 4, and 5 is \(3 \times 4 \times 5 = 60\).

Multiplying each part of the ratio by 60:

• \(\left(\frac{1}{3} \times 60\right) \ratio \left(\frac{1}{4} \times 60\right) \ratio \left(\frac{1}{5} \times 60\right)\)
• \(20 \ratio 15 \ratio 12\)

The total number of ratio parts in the second division is \(20 + 15 + 12 = 47\).

Assuming the shares correspond to the same sons in the order of the ratio, the shares of the three sons in the second division are:

• Son 1's share: \(\frac{20}{47} \times A = \frac{20}{47}A\)
• Son 2's share: \(\frac{15}{47} \times A = \frac{15}{47}A\)
• Son 3's share: \(\frac{12}{47} \times A = \frac{12}{47}A\)

The son who got the lowest share earlier is Son 1, whose share was \(\frac{1}{4}A\). In the second division, Son 1 gets \(\frac{20}{47}A\). The problem states that this son gets Rs. 1188 more in the second division.

Setting up and Solving for the Total Amount

The difference between Son 1's share in the second division and his share in the first division is Rs. 1188.

To solve for \(A\), we find a common denominator for the fractions, which is \(47 \times 4 = 188\).

Now, we can solve for \(A\):

We can simplify the fraction \(\frac{1188}{33}\). Dividing 1188 by 33:

So,

Calculating \(36 \times 188\):

The total amount is Rs. 6768.

Final Answer

The amount is Rs. 6,768.

Ratio Problem Revision Table

Additional Information on Ratio and Proportion

Ratio and Proportion is a fundamental concept in mathematics that deals with comparing quantities. A ratio \(a \ratio b\) is a comparison of two quantities \(a\) and \(b\). Proportion is an equality between two ratios.

• Types of Ratios:
  • Simple Ratio: A ratio of two quantities (e.g., 2:3).
  • Compound Ratio: The ratio of the product of antecedents to the product of consequents of two or more simple ratios (e.g., if ratios are a:b and c:d, the compound ratio is ac:bd).
• Direct Proportion: Two quantities are in direct proportion if an increase in one quantity leads to a proportional increase in the other, and vice versa. If \(x\) is directly proportional to \(y\), then \(\frac{x}{y} = k\) (a constant).
• Inverse Proportion: Two quantities are in inverse proportion if an increase in one quantity leads to a proportional decrease in the other, and vice versa. If \(x\) is inversely proportional to \(y\), then \(xy = k\) (a constant).
• Applications: Ratios and proportions are widely used in various fields, including business (profit sharing, mixing ingredients), physics (speed, distance, time), chemistry (mixing solutions), and everyday life (scaling recipes, maps). Understanding how to divide amounts according to given ratios is a common application tested in quantitative aptitude exams.