Test payment 1 math olympiad Math Challenge

Solution

Correct: B

To solve for x, we need to isolate the variable x. We can do this by subtracting 5 from both sides of the equation, which gives us 2x = 11 - 5, so 2x = 6. Then, we divide both sides by 2, which gives us x = 6/2, so x = 3.

Solution

Correct: A

To find the total amount of money the bakery makes in a day, we need to multiply the number of loaves sold by the cost of each loaf. So, 250 loaves * $2 = $500.

Solution

Correct: A

To solve for y, we need to multiply both sides of the equation by 4, which gives us y = 9 * 4, so y = 36.

Solution

Correct: A

To find the average speed for the entire trip, we can use the formula: average speed = total distance / total time. Since the distance from A to B is the same as from B to A, we can let the distance be d. The time it takes to travel from A to B is d/60 and the time it takes to travel from B to A is d/40. So, the total time is d/60 + d/40. The total distance is 2d. Therefore, the average speed is 2d / (d/60 + d/40) = 2 / (1/60 + 1/40) = 2 / (4/240 + 6/240) = 2 / (10/240) = 2 * 240 / 10 = 48 km/h.

Solution

Correct: A

If 1/4 of the tank is already filled, then 1/4 * 1000 = 250 liters are already in the tank. To find the amount of water that can still be added, we subtract the amount already in the tank from the total capacity: 1000 - 250 = 750 liters.

Solution

Correct: A

Using the Pythagorean theorem, a^2 + b^2 = c^2, where c is the length of the hypotenuse and a and b are the lengths of the legs. We are given c = 10 cm and one of the legs, let's say a, is 6 cm. We need to find b. So, 6^2 + b^2 = 10^2. This means 36 + b^2 = 100. Subtracting 36 from both sides gives us b^2 = 64. Taking the square root of both sides, we get b = 8 cm.

Solution

Correct: A

The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius. Given C = 36π, we can set up the equation 36π = 2πr. Dividing both sides by 2π gives us r = 36π / 2π = 18 cm.

Solution

Correct: A

To find the area of the path, we first find the area of the larger rectangle including the path and then subtract the area of the garden. The dimensions of the larger rectangle are (10+2*1) by (5+2*1), which is 12 meters by 7 meters. The area of this larger rectangle is 12 * 7 = 84 square meters. The area of the garden itself is 10 * 5 = 50 square meters. So, the area of the path is 84 - 50 = 34 square meters. However, this calculation seems slightly off based on the provided options; correctly, it should be recalculated as the difference between the outer and inner areas. The outer area is indeed 12*7 = 84 square meters, and the inner area is 10*5 = 50 square meters. The correct difference, and thus the area of the path, is 84 - 50 = 34 square meters, but considering the given options and reevaluating, if we calculate the area of the path by adding the lengths of the sides of the garden and the path and then subtracting the area of the garden, we might find the closest answer. However, based on the calculation provided, none of the given options match exactly, suggesting a recalibration towards the exact question might yield a different approach or indicate an error in calculation or options provided.

Solution

Correct: C

To solve the quadratic equation x^2 + 4x + 4 = 0, we notice it is a perfect square trinomial that can be factored as (x + 2)^2 = 0. This means x + 2 = 0. Solving for x gives x = -2.

Solution

Correct: A

The sum of the angles in a triangle is always 180 degrees. Given the first angle is 60 degrees and the second angle is 80 degrees, we can find the third angle by subtracting the sum of these two angles from 180 degrees. So, 180 - (60 + 80) = 180 - 140 = 40 degrees.

Solution

Correct: A

To find the equation of the line, we first need to find the slope (m) of the line. The formula for slope is m = (y2 - y1) / (x2 - x1), where (x1, y1) = (2, 3) and (x2, y2) = (4, 5). Substituting these values gives m = (5 - 3) / (4 - 2) = 2 / 2 = 1. Now that we have the slope, we can use the point-slope form of the line equation, y - y1 = m(x - x1), and substitute m = 1 and (x1, y1) = (2, 3) to get y - 3 = 1(x - 2), which simplifies to y - 3 = x - 2. Adding 3 to both sides gives y = x + 1.

Solution

Correct: A

To solve for x, we first subtract 5 from both sides of the equation, which gives us 3x = 20 - 5, so 3x = 15. Then, we divide both sides by 3, which gives us x = 15 / 3, so x = 5.

Solution

Correct: A

To find the discount amount, we calculate 15% of the original price. 15% of $2 is 0.15 * $2 = $0.30. Subtracting the discount from the original price gives us $2 - $0.30 = $1.70.

Solution

Correct: C

To solve for x, we first add 5 to both sides of the equation, which gives us 2x = 11 + 5, so 2x = 16. Then, we divide both sides by 2, which gives us x = 16 / 2, so x = 8.

Solution

Correct: C

To find out how many pieces of candy each friend will get, we need to divide the total number of pieces of candy by the number of friends. So, 48 pieces / 8 friends = 6 pieces per friend.

Solution

Correct: B

The rate at which the machines produce widgets is constant. Since 5 machines can make 5 widgets in 5 minutes, this means that each machine can make 1 widget in 5 minutes. If we have 100 machines, they can make 100 widgets in the same amount of time it takes 5 machines to make 5 widgets because the production rate per machine does not change. Thus, it would still take 5 minutes for 100 machines to make 100 widgets.

Solution

Correct: B

To find the average speed, we divide the total distance traveled by the total time taken. So, average speed = total distance / total time = 250 miles / 5 hours = 50 mph.

Solution

Correct: B

Using the Pythagorean theorem, a^2 + b^2 = c^2, where c is the length of the hypotenuse (5 inches) and one of the legs (let's say a) is 3 inches. We need to find b. So, 3^2 + b^2 = 5^2, which gives us 9 + b^2 = 25. Subtracting 9 from both sides gives us b^2 = 16. Taking the square root of both sides, we get b = 4 inches.

Solution

Correct: C

Let's denote the speed of the plane in still air as P and the speed of the wind as W. When flying with the wind, the effective speed is P + W, and against the wind, it's P - W. The time to travel 300 miles with the wind is 2 hours, so the speed with the wind is 300 / 2 = 150 mph. The time against the wind is 6 hours, so the speed against the wind is 300 / 6 = 50 mph. Now, we have two equations: P + W = 150 and P - W = 50. Adding these two equations eliminates W and gives us 2P = 200, so P = 100 mph. Substituting P = 100 into P + W = 150 gives us 100 + W = 150, which means W = 50 mph.

Solution

Correct: A

To find the equation of the line, first, find the slope (m) using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) = (1, 2) and (x2, y2) = (3, 4). So, m = (4 - 2) / (3 - 1) = 2 / 2 = 1. The slope is 1. Now, use the point-slope form y - y1 = m(x - x1) with (x1, y1) = (1, 2) to get y - 2 = 1(x - 1), which simplifies to y = x + 1.

Solution

Correct: D

To solve for x, first subtract 5 from both sides of the equation to get x/2 = 11 - 5, which simplifies to x/2 = 6. Then, multiply both sides by 2 to get x = 6 * 2, so x = 12.

Solution

Correct: B

If $80 is 80% of the original price (since 20% is off), we can find the original price by dividing $80 by 0.8 (which represents 80% as a decimal). So, the original price = $80 / 0.8 = $100.