IAS Prelims test Math for IAS Prelims Math Challenge

Solution

Correct: B

To find the value of x, we need to isolate x on one side of the equation. We can do this by subtracting 5 from both sides of the equation: 2x + 5 - 5 = 11 - 5, which simplifies to 2x = 6. Then, we divide both sides by 2 to get x = 6/2 = 3.

Solution

Correct: B

First, calculate the total amount of money made from selling 250 loaves at $2 each: 250 * $2 = $500. Then, calculate the total amount of money made from selling 150 loaves at $3 each: 150 * $3 = $450. Finally, add these two amounts together to get the total: $500 + $450 = $950. However, the closest answer is $900, but since $950 is not an option, it seems there might be an error in the given options as the correct calculation yields $950.

Solution

Correct: A

To find the average speed for the entire trip, we use the formula: Average Speed = Total Distance / Total Time. Since the distance from A to B is the same as from B to A, let's denote this distance as D. The time taken to travel from A to B is D/60 and the time taken to travel back is D/40. The total time is D/60 + D/40. Finding a common denominator, we get (2D + 3D)/120 = 5D/120 = D/24. The total distance is 2D. So, the average speed = 2D / (D/24) = 2 * 24 = 48 km/h.

Solution

Correct: A

First, find the slope (m) of the line using the formula m = (y2 - y1) / (x2 - x1), which gives m = (5 - 3) / (4 - 2) = 2 / 2 = 1. Then, use the point-slope form of a line equation, y - y1 = m(x - x1), with one of the points, for example, (2,3): y - 3 = 1(x - 2). Simplify this to get y - 3 = x - 2, and then y = x + 1.

Solution

Correct: @

To determine the nature of the line, we look at its slope. The line goes from point A (50,100) to (200,150). The slope m = (y2 - y1) / (x2 - x1) = (150 - 100) / (200 - 50) = 50 / 150 = 1/3. Since the slope is not infinite (which would indicate a vertical line) or 0 (which would indicate a horizontal line), and it's not 1, the correct answer is that the slope of the line is neither vertical, horizontal, nor 1.

Solution

Correct: A

The rate at which the machines work is the key. If 5 machines can make 5 widgets in 5 minutes, this means that each machine can make 1 widget in 5 minutes. Therefore, 100 machines can make 100 widgets in the same amount of time it takes for 1 machine to make 1 widget, which is 5 minutes.

Solution

Correct: A

Let's denote the cost of the ball as B. Since the bat costs $1.00 more than the ball, the bat costs B + $1.00. The total cost of both is $1.10, so B + (B + $1.00) = $1.10. Simplifying, 2B + $1.00 = $1.10. Subtract $1.00 from both sides to get 2B = $0.10. Divide both sides by 2 to find B = $0.05.

Solution

Correct: @

First, multiply every term by 4 to clear the fraction: x + 12 = 8x - 20. Then, subtract x from both sides to get 12 = 7x - 20. Add 20 to both sides to get 32 = 7x. Finally, divide both sides by 7 to find x = 32/7. However, it seems there was a mistake in solving the equation as none of the provided options match the correct solution. Let's correct that: x + 12 = 8x - 20, subtract x from both sides to get 12 = 7x - 20, add 20 to both sides to get 32 = 7x, and then divide by 7 to get x = 32/7, which does not match any given option. The error seems to be in the provided options as the correct calculation yields x = 32/7.

Solution

Correct: B

Use the Pythagorean theorem: a^2 + b^2 = c^2, where c is the length of the hypotenuse (10 cm), and one of the sides (let's say a) is 6 cm. So, 6^2 + b^2 = 10^2. This simplifies to 36 + b^2 = 100. Subtract 36 from both sides to get b^2 = 64. Taking the square root of both sides gives b = 8.

Solution

Correct: B

The formula for the volume of a cube is V = s^3, where V is the volume and s is the length of a side. Given that V = 64, we have 64 = s^3. Taking the cube root of both sides gives s = 4.

Solution

Correct: B

The sum of the interior angles of any polygon can be found using the formula (n-2)*180 degrees, where n is the number of sides. For a hexagon, n = 6. So, the sum of the interior angles is (6-2)*180 = 4*180 = 720 degrees.

Solution

Correct: B

First, find the speed of the car: Speed = Distance / Time = 250 km / 5 hours = 50 km/h. Then, use the speed to find the distance traveled in 8 hours: Distance = Speed * Time = 50 km/h * 8 hours = 400 km.

Solution

Correct: A

To find 3/4 of the tank's capacity, multiply the tank's capacity by 3/4: 1000 * (3/4) = 750 liters.

Solution

Correct: B

Since 32 = 2^5, we can rewrite the equation as 2^x = 2^5. Therefore, x = 5.

Solution

Correct: B

First, calculate the discount amount: 20% of $2.50 = 0.20 * $2.50 = $0.50. Then, subtract the discount from the original price: $2.50 - $0.50 = $2.00.

Solution

Correct: B

The sum of the lengths of the two given sides is 5 + 7 = 12 cm, and their difference is 7 - 5 = 2 cm. So, the third side must be greater than 2 cm and less than 12 cm.

Solution

Correct: D

The perimeter P of a rectangle is given by the formula P = 2(length + width). Substituting the given values: P = 2(8 + 5) = 2*13 = 26 cm.

Solution

Correct: C

First, determine the car's speed: Speed = Distance / Time = 40 km / 1 hour = 40 km/h. Then, use the speed to find the distance traveled in 7 hours: Distance = Speed * Time = 40 km/h * 7 hours = 280 km.

Solution

Correct: B

To find the percentage of the bottle that is filled, divide the amount of water in the bottle by the bottle's capacity, then multiply by 100: (0.5 / 2) * 100 = 25%.

Solution

Correct: B

To solve for x, add 3 to both sides of the equation: x - 3 + 3 = 7 + 3, which simplifies to x = 10.

Solution

Correct: C

Let the original price be P. The sale price is 20% off, so the bicycle is sold at 80% of its original price. This can be expressed as 0.80*P = $80. To find P, divide both sides by 0.80: P = $80 / 0.80 = $100.

Solution

Correct: C

First, determine the rate at which the people paint: 3 people can paint the fence in 4 hours, meaning the total work hours required for 1 person to paint the fence is 3*4 = 12 hours. Since we want the fence painted in 2 hours, we use the formula: Total work hours = Number of people * Hours to complete. So, 12 = Number of people * 2. Solving for the number of people gives Number of people = 12 / 2 = 6.

Solution

Correct: C

To find out how many pieces of candy each friend will get, divide the total number of candies by the number of friends: 48 / 8 = 6.

Solution

Correct: A

First, find the slope of the line using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) = (1, 2) and (x2, y2) = (3, 4): m = (4 - 2) / (3 - 1) = 2 / 2 = 1. Then, use the point-slope form of the line equation y - y1 = m(x - x1), with one of the points, for example, (1, 2): y - 2 = 1(x - 1). Simplify this to get y = x + 1.