Quantitative (Problem Solving) Flashcards

Jed has more than twice as many strawberries as Dianna, but only half as many as Lisa. If Dianna has ‘a' number of strawberries, how many strawberries do Jed, Dianna, and Lisa have in terms of ‘a'?

Answer: 7a+6

Let Dianna have 'a' strawberries. If Jed has `2a+2` strawberries, this satisfies the condition of having "more than twice as many" as Dianna (since `2a+2 > 2a`). If Jed has `2a+2` strawberries, then Lisa has twice that amount, which is `2 * (2a+2) = 4a+4` strawberries. The total number of strawberries for Jed, Dianna, and Lisa combined is `a + (2a+2) + (4a+4)`, which simplifies to `7a+6`.

Milk should be thinned to a 3:2 ratio of milk to water. The milkman accidentally added water, resulting in 8 liters of milk that is half water and half milk. What must he add to correct the proportions of the mixture?

Answer: 2 liters milk

The desired milk to water ratio is 3:2. The current mixture is 8 liters, half milk and half water, meaning 4 liters milk and 4 liters water. To achieve a 3:2 ratio with 4 liters of water, we need (3/2) * 4 = 6 liters of milk. Since there are currently 4 liters of milk, the milkman must add 6 - 4 = 2 liters of milk to correct the proportions.

A rectangle's width is 2/3 times its length. What is the perimeter of this rectangle if the length is calculated to be 9?

Answer: 30

The length of the rectangle is given as 9. The width is 2/3 times its length, so the width is (2/3) * 9 = 6. The perimeter of a rectangle is calculated as 2 * (length + width). Therefore, the perimeter is 2 * (9 + 6) = 2 * 15 = 30.

A line l passes through the point and is parallel to the y-axis (2,3). What are the gradient (m) and x-intercept?

Answer: m= ∞ , x= (2,0)

A line parallel to the y-axis is a vertical line. Vertical lines have an undefined or infinite gradient (m = ∞). Since the line passes through the point (2,3) and is vertical, every point on the line will have an x-coordinate of 2. Therefore, it intersects the x-axis at the point where y=0, which is (2,0).

What is the equation of the new parabola formed by shifting y = x2 three units to the positive y-axis?

Answer: y = x2 + 3

To shift a parabola vertically on the y-axis, you add or subtract a constant from the entire function. Shifting 'k' units to the positive y-axis means adding 'k' to the original equation. Therefore, shifting y = x² three units up results in the new equation y = x² + 3.

A sphere with a diameter of 1 unit is enclosed in a cube with sides of 1 unit each. Find the remaining unoccupied volume inside the cube.

Answer: 1-π/6

First, calculate the volume of the cube: side length is 1 unit, so volume is 1³ = 1 cubic unit. Next, calculate the volume of the sphere: with a diameter of 1 unit, the radius is 0.5 units. The volume of a sphere is (4/3)πr³, which becomes (4/3)π(0.5)³ = (4/3)π(1/8) = π/6 cubic units. The remaining unoccupied volume is the cube's volume minus the sphere's volume, which is 1 - π/6.

If a triangle's greatest side is A, and the other two sides are B and C. What is the connection between them?

Answer: A>|B-C|

This question refers to the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Additionally, the absolute difference between the lengths of any two sides must be less than the length of the third side. Given A is the greatest side, the condition A > |B-C| ensures that the difference between the other two sides is always less than the longest side, a necessary condition for a valid triangle.

The City Opera House is growing. The city block containing the opera house is currently rectangular in shape, with a total volume of 9600 feet. What is the new volume if the expanded Opera House is 2.5 times as long, wide, and deep as the original structure?

Answer: 150,000

The original volume of the opera house is 9600 cubic feet. If the expanded opera house is 2.5 times as long, wide, and deep, each dimension is multiplied by 2.5. Therefore, the new volume will be the original volume multiplied by 2.5 for each dimension: 9600 * 2.5 * 2.5 * 2.5. This calculates to 9600 * (2.5)³ = 9600 * 15.625, which equals 150,000 cubic feet.

In a university of 200 people, the number of Psychology majors is 50 less than four times that of Computer Science majors. How many club members are Computer Science majors if one-fifth of the club members are neither Psychology nor Computer Science majors, and no club member is majoring in both Psychology and Computer Science?

Answer: 42

Out of 200 people, one-fifth (40 people) are neither majors, leaving 160 people majoring in Psychology (P) or Computer Science (CS). Let CS majors be 'x'. Psychology majors are '4x - 50'. Since no one majors in both, the total P and CS majors is x + (4x - 50) = 160. Solving this equation gives 5x - 50 = 160, so 5x = 210, and x = 42. Thus, there are 42 Computer Science majors.

If the total cost of 20 pairs of shoes is equal to the total revenue generated by the sale of 25 pairs of shoes, what is the percentage of profit or loss made on each pair of shoes sold, assuming each pair of shoes costs the same dollar amount and is sold for the same dollar amount?

Answer: 20% loss

Let C be the cost per pair of shoes and S be the selling price per pair. The total cost of 20 pairs is 20C, and the total revenue from 25 pairs is 25S. Given that 20C = 25S, we can simplify this to 4C = 5S. This means S = (4/5)C = 0.8C. Since the selling price (0.8C) is less than the cost price (C), there is a loss. The loss per pair is C - 0.8C = 0.2C. The percentage loss is (Loss / Cost) * 100% = (0.2C / C) * 100% = 20% loss.

Rose spent the entire day sightseeing in the Great Britain. Rose boarded a bus from her hotel and traveled 30 miles through the countryside at an average speed of 15 miles per hour. The bus then stopped in London for lunch before continuing on a three-hour tour of the city's sights at a speed of ten miles per hour. Finally, the bus exited the city and drove the 40 miles back to the hotel. Rose arrived at her hotel exactly two hours after she left London. What was the average rate of the bus for the entire journey?

Answer: 14

To find the average rate, we need the total distance and total time. Rose traveled 30 miles at 15 mph (Time = 30/15 = 2 hours), then for 3 hours at 10 mph (Distance = 10*3 = 30 miles), and finally 40 miles in 2 hours. The total distance is 30 + 30 + 40 = 100 miles. The total time is 2 + 3 + 2 = 7 hours. The average rate is Total Distance / Total Time = 100 miles / 7 hours, which is approximately 14.28 mph. The closest whole number option is 14 mph.

Joy spent $16 on each of the necklaces she purchased at the jewelry store. Princess also purchased a number of necklaces for $20 each. What is the average cost of the necklaces Joy and Princess bought if the ratio of the number of necklaces Joy bought to the number of necklaces Princess bought is 3 to 2?

Answer: 17.6

This is a weighted average problem. Let the number of necklaces Joy bought be 3n and Princess bought 2n, based on the 3 to 2 ratio. Joy's total cost is 3n * $16 = $48n. Princess's total cost is 2n * $20 = $40n. The total cost for all necklaces is $48n + $40n = $88n. The total number of necklaces is 3n + 2n = 5n. The average cost is Total Cost / Total Number = $88n / 5n = $17.6.

Josh, Daniel, and Clark all purchased flowers. Josh purchased a single digit number of flowers. Only one of the flowers purchased by Josh, Daniel, and Clark was divisible by 3. The number of flowers purchased by one of them was an even number. Which of the following could represent the number of flowers purchased by each person?

Answer: 9, 10, 13

We need to find an option where Josh's flowers are a single digit, only one number is divisible by 3, and only one number is even. Let's check option A: 9, 10, 13. Josh's flowers (9) is a single digit. Only 9 is divisible by 3. Only 10 is an even number. All conditions are met, making this the correct choice.

Given two positive integers x and y, what percentage of three more than y is twice the value of x?

Answer: 200x/(y + 3)

To express 'what percentage of A is B', the formula is (B/A) * 100%. In this case, 'A' is 'three more than y', which is (y + 3). 'B' is 'twice the value of x', which is (2x). So, the expression is (2x / (y + 3)) * 100, which simplifies to 200x / (y + 3).

Which of the following is greater than 1/2?

Answer: 4/7

To determine which fraction is greater than 1/2, we can compare each fraction to 1/2. A simple way is to double the numerator and see if it's greater than the denominator. For 4/7, doubling the numerator gives 8, which is greater than the denominator 7 (8 > 7), so 4/7 is greater than 1/2. For the other options, 2*6=12 (not > 13), 2*4=8 (not > 9), and 2*2=4 (not > 5), so they are not greater than 1/2.

What is the average (arithmetic mean) of all tens from 10 to 190 inclusive?

Answer: 100

The numbers are an arithmetic progression: 10, 20, ..., 190. The average of an arithmetic progression is simply the sum of the first and last term divided by 2. Therefore, the average is (10 + 190) / 2 = 200 / 2 = 100.

A metal cubical block weighs 6 pounds. How much heavier will a cube of the same metal be if its sides are twice as long?

Answer: 48

The weight of an object made of a uniform material is directly proportional to its volume. If the sides of a cube are twice as long, its new side length is 2s. The new volume will be (2s)³ = 8s³. Since the original volume was s³, the new cube's volume is 8 times greater. Therefore, its weight will also be 8 times greater: 8 * 6 pounds = 48 pounds.

A fence is to be built with 6 inch wide posts separated by 5 feet lengths of chain. Which of the following cannot be the length of a fence in feet if it begins and ends with a post? (1 foot = 12 inches)

Answer: 35

Let 'N' be the number of posts. Since the fence begins and ends with a post, there will be 'N-1' chain sections. Each post is 6 inches (0.5 feet) wide, and each chain section is 5 feet long. The total length (L) of the fence is L = N * (0.5 ft) + (N-1) * (5 ft) = 0.5N + 5N - 5 = 5.5N - 5. For a valid fence length, N must be an integer. Testing the options: for L=35, 35 = 5.5N - 5 => 40 = 5.5N => N = 40/5.5 = 80/11, which is not an integer. Therefore, 35 feet cannot be the length of such a fence.

A die is rolled, followed by a coin toss. What is the probability that the die will show an even number AND the coin will show a tail?

Answer: 1/4

The probability of rolling an even number on a standard six-sided die is 3 favorable outcomes (2, 4, 6) out of 6 total outcomes, which is 3/6 = 1/2. The probability of a coin showing a tail is 1 favorable outcome out of 2 total outcomes, which is 1/2. Since these events are independent, the probability of both occurring is the product of their individual probabilities: (1/2) * (1/2) = 1/4.

Company XYZ manufactures toy trucks for a shopping mall at a cost of $7.00 per truck for the first 500 trucks and $5.00 per truck after that. What was Company XYZ's gross profit if 600 trucks were produced and sold for $15.00 each?

Answer: $5000

To calculate the gross profit, first determine the total revenue and total cost. Total revenue from selling 600 trucks at $15 each is 600 * $15 = $9000. For the cost, the first 500 trucks cost $7 each (500 * $7 = $3500). The remaining 100 trucks (600 - 500) cost $5 each (100 * $5 = $500). The total cost is $3500 + $500 = $4000. Gross profit is Total Revenue - Total Cost = $9000 - $4000 = $5000.