Quantitative Reasoning Flashcards

Two individuals, Alex and Ben, are tasked with painting a fence. Alex can paint the entire fence alone in 6 hours, while Ben can paint the same fence alone in 4 hours. If they work together, how long will it take them to paint the fence?

Answer: 2 hours and 24 minutes

To solve this, we must consider their combined work rate. Alex's rate is 1/6 of the fence per hour, and Ben's rate is 1/4 of the fence per hour. Their combined rate is the sum of their individual rates: 1/6 + 1/4. To add these fractions, find a common denominator, which is 12. So, the combined rate is 2/12 + 3/12 = 5/12 of the fence per hour. To find the total time to complete the job (1 fence), we take the reciprocal of the rate: Time = 1 / (5/12) = 12/5 hours. 12/5 hours is equal to 2.4 hours. To convert the 0.4 hours into minutes, multiply by 60: 0.4 * 60 = 24 minutes. Therefore, it will take them 2 hours and 24 minutes to paint the fence together.

A circular dartboard has a radius of 10 inches. The bullseye in the center is also a circle and has a radius of 2 inches. What is the area of the dartboard that is outside of the bullseye?

Answer: 96π sq. in.

The area of a circle is calculated using the formula A = πr². First, calculate the total area of the dartboard: A_total = π(10)² = 100π square inches. Next, calculate the area of the bullseye: A_bullseye = π(2)² = 4π square inches. To find the area of the dartboard outside the bullseye, subtract the bullseye's area from the total area: Area = A_total - A_bullseye = 100π - 4π = 96π square inches.

A bag contains 5 red marbles, 4 blue marbles, and 3 green marbles. If two marbles are drawn from the bag at random without replacement, what is the probability that both marbles are blue?

Answer: 1/11

First, determine the total number of marbles, which is 5 + 4 + 3 = 12. The probability of the first marble drawn being blue is 4/12. Since the marble is not replaced, there are now 11 marbles left in the bag, and only 3 of them are blue. The probability of the second marble drawn also being blue is 3/11. To find the probability of both events occurring in sequence, multiply their individual probabilities: P(Both Blue) = (4/12) * (3/11) = 12/132. This fraction simplifies to 1/11.

A 20-foot ladder leans against a vertical wall, with its base positioned 12 feet away from the bottom of the wall. What is the cosine of the angle that the ladder makes with the ground?

Answer: 3/5

The ladder, the wall, and the ground form a right-angled triangle. The ladder itself is the hypotenuse (20 feet). The distance from the wall to the base of the ladder is the adjacent side relative to the angle with the ground (12 feet). The cosine of an angle in a right triangle is defined as the ratio of the adjacent side to the hypotenuse (SOH CAH TOA). Therefore, cos(θ) = Adjacent / Hypotenuse = 12 / 20. This fraction simplifies to 3/5.

Which of the following is the correct solution for x in the equation log₃(x + 5) = 2?

Answer: 4

The definition of a logarithm states that log_b(y) = a is equivalent to bᵃ = y. Applying this to the given equation, the base b is 3, a is 2, and y is (x + 5). Therefore, we can rewrite the equation in exponential form as 3² = x + 5. Calculating 3² gives 9, so the equation becomes 9 = x + 5. To solve for x, subtract 5 from both sides: x = 9 - 5 = 4.

A retail item originally priced at $80 is put on sale for $68. What is the percent decrease in the price of the item?

Answer: 15%

The formula for percent decrease is [(Original Price - New Price) / Original Price] × 100. First, find the amount of the decrease: $80 - $68 = $12. Next, divide this decrease by the original price: $12 / $80. This fraction simplifies to 3/20, which is 0.15 as a decimal. Finally, multiply by 100 to convert the decimal to a percentage: 0.15 × 100 = 15%.