A sum of 2,500 is distributed among X, Y and Z in the ratio 1/2, 3/4, 5/6. What is the difference between the maximum share and the minimum share?
UPSC-2020
To find the difference between the maximum and minimum shares, we first need to express the given ratio in whole numbers. Then, we can calculate the individual shares of X, Y, and Z from the total sum and find the difference.
Convert the ratio to whole numbers: The given ratio is \(1/2 : 3/4 : 5/6\). To convert this to whole numbers, we find the Least Common Multiple (LCM) of the denominators (2, 4, 6).
The LCM of 2, 4, and 6 is 12.
Now, multiply each fraction by the LCM (12):
\[ \text{X's part} = \frac{1}{2} \times 12 = 6 \] \[ \text{Y's part} = \frac{3}{4} \times 12 = 9 \] \[ \text{Z's part} = \frac{5}{6} \times 12 = 10 \]So, the ratio of shares among X, Y, and Z is \(6 : 9 : 10\).
Calculate the sum of the ratio parts: Sum of ratio parts \( = 6 + 9 + 10 = 25\).
Calculate each person's share: The total sum to be distributed is ₹2,500.
\[ \text{Share of X} = \frac{6}{25} \times 2500 = 6 \times 100 = \text{₹}600 \] \[ \text{Share of Y} = \frac{9}{25} \times 2500 = 9 \times 100 = \text{₹}900 \] \[ \text{Share of Z} = \frac{10}{25} \times 2500 = 10 \times 100 = \text{₹}1000 \]Identify the maximum and minimum shares: Maximum share \( = \text{₹}1000\) (Z's share) Minimum share \( = \text{₹}600\) (X's share)
Calculate the difference between the maximum and minimum shares: Difference \( = \text{Maximum share} - \text{Minimum share} = 1000 - 600 = \text{₹}400\).
Correct Option: C) ₹ 400
If 40% of a number is equal to two-third of another number, what is the ratio of first number to the second number ?
The problem states a relationship between 40% of a first number and two-thirds of a second number. We need to find the ratio of the first number to the second number. We will set up an equation based on the given information and then solve for the ratio.
Let the first number be \(A\) and the second number be \(B\).
According to the problem, 40% of the first number is equal to two-thirds of the second number. We can write this as an equation:
\[ 40\% \text{ of } A = \frac{2}{3} \text{ of } B \]Convert the percentage to a fraction:
\[ \frac{40}{100} \times A = \frac{2}{3} \times B \]Simplify the fraction \(\frac{40}{100}\):
\[ \frac{2}{5} \times A = \frac{2}{3} \times B \]Now, we need to find the ratio of the first number to the second number, which is \(\frac{A}{B}\). To do this, we can rearrange the equation:
\[ \frac{A}{B} = \frac{2}{3} \div \frac{2}{5} \]Perform the division by multiplying by the reciprocal:
\[ \frac{A}{B} = \frac{2}{3} \times \frac{5}{2} \]Cancel out the common factor (2):
\[ \frac{A}{B} = \frac{5}{3} \]Therefore, the ratio of the first number to the second number is \(5:3\).
B) 5:3 is the correct ratio derived from the given conditions.
The ratio compounded of (2 : 3), (9 : 4), (5 : 6) and (8 : 10) is :
To find the compounded ratio of several given ratios, we multiply the antecedents (first terms) of all the ratios to get the new antecedent, and multiply the consequents (second terms) of all the ratios to get the new consequent. Then, we simplify the resulting ratio to its lowest terms.
Further simplification:
\[\text{Compounded Ratio} = \frac{1}{\cancel{3}^{\,1}} \times \frac{\cancel{3}^{\,1}}{\cancel{2}^{\,1}} \times \frac{\cancel{5}^{\,1}}{2} \times \frac{\cancel{4}^{\,2}}{\cancel{5}^{\,1}}\]Let's re-do the simplification more systematically:
\[\text{Compounded Ratio} = \frac{2 \times 9 \times 5 \times 8}{3 \times 4 \times 6 \times 10}\]Numerator: \(2 \times 9 \times 5 \times 8 = 18 \times 40 = 720\)
Denominator: \(3 \times 4 \times 6 \times 10 = 12 \times 60 = 720\)
So, the compounded ratio is:
\[\text{Compounded Ratio} = \frac{720}{720}\]This can be written as \(1:1\).
B) 1:1 — The calculation shows that the compounded ratio is \(1:1\).
A man divides his property so that his son’s share to his wife’s share and the wife’s share to his daughter’s share are both in the ratio 3 : 1. If the daughter gets Rs 10,000 less than the son, find the total worth of the property.
We are given ratios of shares for son to wife and wife to daughter. We need to find a combined ratio for son, wife, and daughter. Then, using the given difference in shares between the son and daughter, we can find the value of one unit of the ratio and subsequently the total worth of the property.
Let the son's share be \(S\), the wife's share be \(W\), and the daughter's share be \(D\).
From the problem statement, we have two ratios:
To combine these ratios, we need to make the common term (\(W\)) equal in both ratios. The current ratios are:
\[ S : W = 3 : 1 \]\[ W : D = 3 : 1 \]Multiply the first ratio by 3 to make the wife's share 3 units:
\[ S : W = (3 \times 3) : (1 \times 3) = 9 : 3 \]Now, the combined ratio \(S : W : D\) is:
\[ S : W : D = 9 : 3 : 1 \]Let the shares be \(9x\), \(3x\), and \(x\) for the son, wife, and daughter, respectively, where \(x\) is a common multiplier.
We are given that the daughter gets Rs 10,000 less than the son. So, the difference between the son's share and the daughter's share is Rs 10,000.
\[ S - D = 10,000 \]\[ 9x - x = 10,000 \]\[ 8x = 10,000 \]Solve for \(x\):
\[ x = \frac{10,000}{8} \]\[ x = 1,250 \]Now, calculate the individual shares:
The total worth of the property is the sum of all shares:
\[ \text{Total Property} = S + W + D \]\[ \text{Total Property} = 11,250 + 3,750 + 1,250 \]\[ \text{Total Property} = 16,250 \]Correct Option: B) Rs 16,250
Nirmal and Kapil started a business investing Rs 9,000 and Rs 12,000 respectively. After 6 months, Kapil withdrew half of his investment. If after a year, the total profit was Rs 4,600,what was Kapil’s share in it ?
To find Kapil's share of the profit, we need to determine the ratio of the effective capital invested by Nirmal and Kapil over the entire year. The effective capital is calculated by summing the product of the investment amount and the duration for which it was invested.
Nirmal invested Rs 9,000 for the entire year (12 months).
\[ \text{Nirmal's effective investment} = 9000 \times 12 = \text{Rs } 108,000 \]Kapil initially invested Rs 12,000 for 6 months. After 6 months, he withdrew half, so his remaining investment was Rs 6,000 for the next 6 months.
\[ \text{Kapil's effective investment} = (12000 \times 6) + (6000 \times 6) \]\[ = 72000 + 36000 = \text{Rs } 108,000 \]The ratio of their effective investments is the ratio of their shares in the profit.
\[ \text{Ratio (Nirmal : Kapil)} = 108,000 : 108,000 = 1 : 1 \]The total profit is Rs 4,600. Since the ratio of their shares is 1:1, they will share the profit equally.
\[ \text{Kapil's share} = \frac{1}{1+1} \times 4600 \]\[ = \frac{1}{2} \times 4600 = \text{Rs } 2,300 \]C) Rs 2,300 is the correct answer. Kapil's effective investment over the year was Rs 108,000, which is the same as Nirmal's. Therefore, they share the total profit equally.
The average age of three boys is 25 years and their ages are in the proportion 3: 5 : 7. The age of the youngest boy is:
We are given the average age of three boys and the ratio of their ages. We need to find the age of the youngest boy. First, we will calculate the sum of their ages using the average. Then, we will use the given ratio to find the individual ages.
The average age of three boys is 25 years.
\[ \text{Sum of ages} = \text{Average age} \times \text{Number of boys} \] \[ \text{Sum of ages} = 25 \times 3 = 75 \text{ years} \]The ages of the three boys are in the proportion 3 : 5 : 7. Let their ages be \(3x\), \(5x\), and \(7x\) years, where \(x\) is a common factor.
The sum of their ages is \(3x + 5x + 7x\), which must be equal to 75 years.
\[ 3x + 5x + 7x = 75 \] \[ 15x = 75 \] \[ x = \frac{75}{15} \] \[ x = 5 \]The youngest boy's age corresponds to the smallest part of the ratio, which is \(3x\).
\[ \text{Age of youngest boy} = 3x = 3 \times 5 = 15 \text{ years} \]Correct Option: B) 15 years
