A person divides a certain amount among his three sons in the ratio of 3 ∶ 4 ∶ 5. If he had divided this amount in the ratio of 1/3,1/4,1/5, his son, who had got the lowest share earlier, would get Rs.1,188 more. Find the amount (in Rs).
6,768
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.
\[\frac{20}{47}A - \frac{1}{4}A = 1188\]To solve for \(A\), we find a common denominator for the fractions, which is \(47 \times 4 = 188\).
\[\left(\frac{20 \times 4}{47 \times 4}\right)A - \left(\frac{1 \times 47}{4 \times 47}\right)A = 1188\] \[\frac{80}{188}A - \frac{47}{188}A = 1188\] \[\left(\frac{80 - 47}{188}\right)A = 1188\] \[\frac{33}{188}A = 1188\]Now, we can solve for \(A\):
\[A = 1188 \times \frac{188}{33}\]We can simplify the fraction \(\frac{1188}{33}\). Dividing 1188 by 33:
\[1188 \div 33 = 36\]So,
\[A = 36 \times 188\]Calculating \(36 \times 188\):
\[36 \times 188 = 36 \times (100 + 80 + 8)\] \[= 36 \times 100 + 36 \times 80 + 36 \times 8\] \[= 3600 + 2880 + 288\] \[= 6480 + 288\] \[= 6768\]The total amount is Rs. 6768.
Final Answer
The amount is Rs. 6,768.
Ratio Problem Revision Table
| Concept | Description | Application in Problem |
|---|---|---|
| Ratio | A comparison of two or more quantities of the same kind. | Given ratios 3:4:5 and 1/3:1/4:1/5. |
| Sum of Ratio Parts | Adding the numbers in a ratio to find the total parts. | \(3+4+5=12\); \(20+15+12=47\). |
| Finding Shares | Share = (Individual Ratio Part / Total Ratio Parts) × Total Amount. | Calculated shares like \(\frac{3}{12}A\) and \(\frac{20}{47}A\). |
| Ratio Conversion | Converting fractional ratios to whole number ratios by multiplying by the LCM of denominators. | Converted 1/3:1/4:1/5 to 20:15:12. |
| Setting up Equation | Translating the problem statement into a mathematical equation. | \(\frac{20}{47}A - \frac{1}{4}A = 1188\). |
| Solving Linear Equation | Using algebraic methods to find the unknown variable (Total Amount). | Solved for A by isolating it. |
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.
Important Questions from Ratio and Proportion
The cost of a diamond is directly proportional to the square of its weight. The cost of a 14 gm diamond is Rs. 2560. This diamond got broken down into two pieces in the ratio of 5 ∶ 9. How much loss percent is incurred due to this breakage ? (Correct to two decimal places)
Atul purchased Bread costing Rs.20 and gave a 100 rupee note to the shopkeeper. The shopkeeper gave the balance money in coins of denomination Rs.2, Rs.5 and Rs.10. If these coins are in the ratio 5 ∶ 4 ∶ 1, then how many Rs.5 coins did the shopkeeper give?
In a school 3/8 of the number of students are girls and the rest are boys. One-third of the number of boys are below 10 years and 2/3 the number if girls are also below 10 years. If the number of students of age 10 or more years is 260. then the number of boys in the school is:
If a : b : c = \(\frac{1}{4} : \frac{1}{3} : \frac{1}{2}, \) then \( \ \frac{a}{b} : \frac{b}{c} : \frac{c}{a} = ?\)
A, B and C divide a certain sum of money among themselves. The average of the amount with them is Rs.4520. Share of A is \(10\frac{2}{3}%\) % more than share of B and \(33\frac{1}{3}%\) % less than share of C. What is the share of B (in Rs.)?
