UPSC CISF-AC-EXE

In a game between X and Y, X has to give ₹ 10 each time he loses to Y. If he wins, then he gets ₹ 50 from Y. If they play 15 times and X earns ₹ 450, how many times does X win ?

[amp_mcq option1=”8″ option2=”9″ option3=”10″ option4=”12″ correct=”option3″]

Let W be the number of times X wins and L be the number of times X loses. The total number of games played is W + L = 15. When X wins, he gains ₹ 50, so the total gain from wins is 50W. When X loses, he gives ₹ 10, so the total loss from losses is 10L. X’s net earning is the total gain minus the total loss, which is 50W – 10L. We are given that X earns ₹ 450, so 50W – 10L = 450. We now have a system of two linear equations: 1) W + L = 15 and 2) 50W – 10L = 450. From equation 1, L = 15 – W. Substituting this into equation 2: 50W – 10(15 – W) = 450. 50W – 150 + 10W = 450. 60W – 150 = 450. 60W = 600. W = 600 / 60 = 10. X wins 10 times.

– Define variables for the number of wins and losses.
– Set up one equation based on the total number of games played.
– Set up a second equation based on the total net earning (total gain from wins minus total loss from losses).
– Solve the system of linear equations for the number of wins.

This is an algebra word problem that requires setting up and solving a system of equations. It’s important to correctly represent the gain and loss per game and relate the total gain/loss to the net earning. Checking the answer (10 wins, 5 losses: 10*50 – 5*10 = 500 – 50 = 450) confirms the solution.

A man takes a loan at 5% simple interest for a period of 2 years. He immediately gives this money on loan at 5% compound interest for 2 years. What is the amount of loan he has taken if he makes a profit of ₹ 2,100 ?

[amp_mcq option1=”₹ 8,00,000″ option2=”₹ 4,20,000″ option3=”₹ 1,00,000″ option4=”₹ 8,40,000″ correct=”option4″]

Let the principal amount be P. The simple interest (SI) paid by the man for 2 years at 5% is SI = (P * 5 * 2) / 100 = 0.10P. The compound interest (CI) received by the man for 2 years at 5% is CI = P(1 + 5/100)^2 – P = P(1.05)^2 – P = P(1.1025) – P = 0.1025P. The man’s profit is the difference between the CI received and the SI paid: Profit = CI – SI = 0.1025P – 0.10P = 0.0025P. Given the profit is ₹ 2,100, we have 0.0025P = 2100. Solving for P: P = 2100 / 0.0025 = 2100 / (1/400) = 2100 * 400 = ₹ 8,40,000.

– Simple Interest formula: SI = PRT/100.
– Compound Interest formula: CI = Amount – Principal = P(1 + R/100)^T – P.
– The profit in this scenario is the difference between the interest earned (CI) and the interest paid (SI).
– Set up an equation with the profit difference equal to the given profit amount and solve for the principal P.

This question tests the understanding of simple and compound interest calculations and how to find the profit earned by an intermediary who borrows at one rate/type and lends at another rate/type. The difference in interest earned versus interest paid constitutes the profit.

A businessman lost 20% of his goods due to rain. The remaining items were sold at a profit of 50%. The businessman earned an overall profit of:

[amp_mcq option1=”10%” option2=”30%” option3=”20%” option4=”40%” correct=”option3″]

Assume the total cost price of the goods was ₹ 100. Let the total quantity be 100 units, so the cost per unit is ₹ 1. 20% of goods were lost, meaning 20 units were lost. Remaining goods are 100 – 20 = 80 units. These 80 units are sold at a profit of 50% on their cost. The cost of the 80 units is ₹ 80 (since cost per unit was ₹ 1). Selling price of these 80 units = Cost * (1 + Profit%) = ₹ 80 * (1 + 0.50) = ₹ 80 * 1.50 = ₹ 120. The businessman invested ₹ 100 initially (the cost of the total goods) and received ₹ 120 from the sale. The overall profit is ₹ 120 – ₹ 100 = ₹ 20. Overall profit percentage = (Overall Profit / Total Cost) * 100 = (₹ 20 / ₹ 100) * 100 = 20%.

– The profit/loss is calculated on the original total cost of the goods, even if a portion is lost.
– The selling price is calculated based on the goods that *are* sold, applying the given profit margin to their cost.
– Overall profit = Total Revenue – Total Cost.
– Overall profit percentage = (Overall Profit / Total Cost) * 100.

Setting the initial cost price to a convenient value like ₹ 100 simplifies calculations for percentage problems. The loss of goods means the cost invested in those goods is not recovered, and this must be accounted for in the overall profit calculation.

‘A’ and ‘B’ have pocket money in the ratio of 3 : 4. After the day’s work, ‘A’ earned ₹ 600 while ‘B’ earned ₹ 500. However, ‘A’ spent ₹ 150 and ‘B’ spent ₹ 100 during the day. If they have equal amount of money at the end of the day, then the pocket money ‘A’ had in the morning is:

[amp_mcq option1=”₹ 150″ option2=”₹ 200″ option3=”₹ 250″ option4=”₹ 100″ correct=”option1″]

Let A’s initial pocket money be 3x and B’s initial pocket money be 4x. At the end of the day, A’s total money is 3x + ₹ 600 – ₹ 150 = 3x + ₹ 450. B’s total money is 4x + ₹ 500 – ₹ 100 = 4x + ₹ 400. Since they have equal amounts at the end, 3x + 450 = 4x + 400. Solving for x, we get x = 50. A’s initial pocket money was 3x, which is 3 * 50 = ₹ 150.

– Represent the initial amounts using the given ratio with a variable (e.g., 3x and 4x).
– Formulate expressions for the final amounts for A and B by adding earnings and subtracting spending.
– Set the final amounts equal to each other based on the problem statement.
– Solve the resulting linear equation for the variable x.
– Calculate the initial pocket money for A using the value of x.

This is a typical word problem involving ratios and linear equations. Careful tracking of money earned and spent for each person is crucial. The phrase “equal amount of money at the end of the day” provides the equation needed to solve for the unknown variable.