If a and b are positive integers and lcm(a,b)=60 and gcd(a,b)=6, what is the value of ab?
Answer: 360
This is a Multiple Choice, Select One GRE Quantitative Reasoning problem. The core quantitative principle involved is the relationship between least common multiple (LCM), greatest common divisor (GCD), and the product of two numbers, where LCM(a, b) GCD(a, b) = a b. Given LCM(a, b) = 60 and GCD(a, b) = 6, the product ab = 60 * 6 = 360. The correct answer of 360 satisfies the relationship because it directly follows from multiplying the given LCM and GCD. A representative incorrect option like 120 might result from dividing instead of multiplying the values. Another error, such as choosing 180, could stem from misapplying prime factorizations or assuming incorrect factors.
In a certain class, the ratio of juniors to seniors is 5:3. If there are 48 students total in the class, how many seniors are in the class?
Answer: 18
This is a Multiple Choice, Select One GRE Quantitative Reasoning problem. The core quantitative principle involved is using ratios to divide a total into parts. The ratio of juniors to seniors is 5:3, so total parts are 5 + 3 = 8, and with 48 students, each part is 48 / 8 = 6. Seniors are then 3 parts, or 3 * 6 = 18. The correct answer of 18 satisfies the ratio because 30 juniors and 18 seniors give 5:3 and total 48. A representative incorrect option like 20 might result from misdividing the total or assuming equal parts. Another error, such as choosing 24, could stem from inverting the ratio accidentally.
3 bagels and 4 muffins cost $15; 4 bagels and 3 muffins cost $13. What is the cost of one muffin?
Answer: $3
Comparing the two purchases, the first has one extra muffin and one fewer bagel and costs $2 more, so a muffin costs $2 more than a bagel. If a bagel is b, the muffin is b + 2; then 3b + 4(b + 2) = 15 gives b = 1, so the muffin costs $3. Don't stop at $1; that is the bagel's cost, not the muffin's.
What is the value of 50−18?
Answer: 22
This is a Multiple Choice, Select One GRE Quant problem requiring simplification of square roots. The key principle is to factor out perfect squares from under the radical sign. We have √50 = √(25×2) = 5√2 and √18 = √(9×2) = 3√2. Therefore, √50 - √18 = 5√2 - 3√2 = 2√2. The Borrect answer B (2√2) properly applies the simplification rules. A common error would be to incorrectly simplify the radicals, such as writing √50 as 25√2 or attempting to subtract the numbers under the radicals before simplifying.
A bag contains 6 red marbles and 4 blue marbles. One marble is selected at random from the bag. What is the probability of selecting a blue marble?
Answer: 52
This is a Multiple Choice, Select One GRE Quant problem involving basic probability. The core principle is that probability equals the number of favorable outcomes divided by the total number of possible outcomes. The bag contains 6 red marbles and 4 blue marbles, giving a total of 10 marbles. The probability of selecting a blue marble is 4/10, which simplifies to 2/5. The Borrect answer B (2/5) satisfies this calculation perfectly. A common error would be to calculate the probability of red marbles (6/10 = 3/5) instead of blue marbles, or to incorrectly simplify the fraction.
A line in the coordinate plane passes through the points (2,5) and (6,13). What is the slope of the line?
Answer: 2
This is a Multiple Choice, Select One GRE Quantitative Reasoning problem. The core quantitative principle involved is calculating the slope of a line using the formula (y2 - y1)/(x2 - x1). For points (2,5) and (6,13), slope = (13 - 5)/(6 - 2) = 8/4 = 2. This measures the rate of change. The correct answer of 2 satisfies the formula because the rise over run is 8 over 4. A representative incorrect option like 1/2 might result from inverting the fraction. Another error, such as choosing 19/20, could stem from using wrong coordinates.
A right circular cylinder has radius 3 cm and height 10 cm. What is the volume of the cylinder, in cubic centimeters?
Answer: 90π
This is a Multiple Choice, Select One GRE Quant problem. It involves the formula for the volume of a right circular cylinder, V = πr2 h. Given radius 3 cm and height 10 cm, substitute to get V = π (3)2 (10) = π 9 10 = 90π cubic centimeters. This computation directly applies the formula without additional complexities. The value 90π satisfies the requirements because it correctly multiplies the base area π*9 by the height 10. A representative incorrect option like 30π (choice A) might result from an assumption error, such as using V = πr h instead of πr2 h, yielding π*3*10 = 30π. Another error leading to 60π (choice B) could be using 2πr h for surface area instead of volume.
A cube has volume 125 cubic units. What is the surface area of the cube, in square units?
Answer: 150
This is a Multiple Choice, Select One GRE Quantitative Reasoning problem. The core quantitative principle involved is relating volume to side length and then to surface area for a cube. Volume 125 = s³, so s = 5; surface area = 6s² = 6*25 = 150. This uses the cube root and squaring. The correct answer of 150 satisfies the calculations because 5³ = 125 and 6*25 = 150. A representative incorrect option like 100 might result from using 4 sides instead of 6. Another error, such as choosing 75, could stem from halving the area mistakenly.
If f(x)=2x2−5x+1, what is the value of f(3)?
Answer: 4
This is a Multiple Choice, Select One GRE Quant problem. It involves evaluating a quadratic function by substituting a given value. For f(x) = 2x2 - 5x + 1, substitute x = 3: f(3) = 2*(3)2 - 5*3 + 1 = 2*9 - 15 + 1 = 18 - 15 + 1 = 4. This direct computation follows the order of operations. The value 4 satisfies the requirements because it accurately results from plugging in x = 3 into the function. A representative incorrect option like 7 (choice C) might result from an arithmetic slip, such as computing 2*9 - 5*3 + 1 as 18 - 15 + 4 or misadding. This fails to correctly perform the subtraction and addition steps.
A car travels 120 miles at an average speed of 48 miles per hour and then travels another 60 miles at an average speed of 30 miles per hour. What is the car's average speed, in miles per hour, for the entire 180-mile trip?
Answer: 40
This is a Multiple Choice, Select One GRE Quant problem. It involves calculating average speed as total distance divided by total time, not the arithmetic mean of speeds. The time for the first 120 miles at 48 mph is 120/48 = 2.5 hours, and for the next 60 miles at 30 mph is 60/30 = 2 hours, for a total time of 4.5 hours. The average speed is then total distance 180 miles divided by 4.5 hours, equaling 40 mph. The value 40 satisfies the requirements because it correctly accounts for the different times spent at each speed over the entire 180-mile trip. A representative incorrect option like 39, though not listed, or approximating to 38 (choice B) might result from an arithmetic slip, such as averaging the speeds directly (48 + 30)/2 = 39 without weighting by distance. This fails because it does not consider the longer time spent at the slower speed.
A rectangle has perimeter 28 and area 48. If its length is greater than its width, what is the length?
Answer: 8
From perimeter, length + width = 14. From area, the two positive numbers with sum 14 and product 48 are 6 and 8. Since length is greater than width, the length is 8. The tempting 6 is only the width, not the answer.
If 2x−3=5x+12, what is the value of x?
Answer: −5
This is a Multiple Choice, Select One GRE Quant problem requiring solving a linear equation. The fundamental principle involves isolating the variable by combining like terms and using inverse operations. Starting with 2x - 3 = 5x + 12, we subtract 2x from both sides to get -3 = 3x + 12, then subtract 12 from both sides to get -15 = 3x. Dividing both sides by 3 yields x = -5. The Dorrect answer D (x = -5) satisfies the original equation: 2(-5) - 3 = -10 - 3 = -13, and 5(-5) + 12 = -25 + 12 = -13. A common mistake would be to incorrectly combine terms, such as getting x = 3 by making a sign error when moving terms across the equals sign.
A and B together can finish a job in 6 days; A alone can finish it in 10 days. How many days would B alone need?
Answer: 15
Together A and B complete 1/6 of the job per day. A alone completes 1/10 per day, so B completes 1/6 - 1/10 = 1/15 of the job per day. Thus B alone needs 15 days. The tempting wrong answer is 4 days, but that just subtracts 10 - 6; rates add, not completion times.
If x5=72, what is the value of x?
Answer: 235
This is a Multiple Choice, Select One GRE Quantitative Reasoning problem. The core quantitative principle involved is solving a proportion by cross-multiplying or inverting. Given 5/x = 2/7, cross-multiplying gives 2x = 35, so x = 35/2. Alternatively, x = 5 (7/2) = 35/2. The correct answer of 35/2 satisfies the equation because 5/(35/2) = 5 2/35 = 10/35 = 2/7. A representative incorrect option like 14/5 might result from inverting incorrectly. Another error, such as choosing 7/10, could stem from swapping numerator and denominator.
If n is a positive integer and n2 is divisible by 6, what is the greatest integer that must divide n?
Answer: 6
Since n2 is divisible by 6, it contains at least one factor 2 and one factor 3. In a square, prime factors appear in pairs, so n itself must contain at least one 2 and one 3. Therefore 6 must divide n. The tempting wrong answer is 12: n = 6 makes n2 divisible by 6, but 12 does not divide 6, so 12 is not guaranteed.
A jar contains only water and sugar. The mixture is 20% sugar by volume. If the jar contains 300 milliliters of mixture, how many milliliters of sugar does it contain?
Answer: 60
This is a Multiple Choice, Select One GRE Quant problem involving percentage calculations. The fundamental principle is that to find a percentage of a quantity, multiply the quantity by the percentage expressed as a decimal. The mixture is 20% sugar by volume, and the total volume is 300 milliliters. Therefore, the volume of sugar is 0.20 × 300 = 60 milliliters. The Correct answer C (60) satisfies this calculation. A common mistake would be to calculate 80% of 300 (giving 240) by confusing the sugar percentage with the water percentage, or to divide 300 by 20 instead of multiplying by 0.20.
What is the value of 43+65?
Answer: 1219
This is a Multiple Choice, Select One GRE Quantitative problem that tests fraction addition with different denominators. The core principle requires finding a common denominator before adding fractions. To add 3/4 + 5/6, we first find the least common denominator, which is 12. Converting: 3/4 = 9/12 and 5/6 = 10/12. Therefore, 3/4 + 5/6 = 9/12 + 10/12 = 19/12. The correct answer is 19/12 because it represents the sum in its simplest form as an improper fraction. Answer choice A (19/24) incorrectly assumes 24 as the common denominator without properly converting the fractions, yielding 18/24 + 20/24 = 38/24, which doesn't equal 19/24.
A data set consists of the numbers 4, 7, 7, 10, and x. If the mean of the data set is 8, what is the value of x?
Answer: 12
This is a Multiple Choice, Select One GRE Quantitative Reasoning problem. The core quantitative principle involved is finding the mean of a data set and solving for an unknown value. The mean is 8 for five numbers, so the sum is 5 * 8 = 40; the known numbers sum to 4 + 7 + 7 + 10 = 28, so x = 40 - 28 = 12. The correct answer of 12 satisfies the mean because the total sum becomes 40, averaging 8. A representative incorrect option like 10 might result from miscounting the data points. Another error, such as choosing 14, could stem from an arithmetic slip in summation.
If f(x)=x2−4x+1, what is the value of f(5)?
Answer: 6
This is a Multiple Choice, Select One GRE Quantitative problem that tests function evaluation. The core principle involves substituting a given value into a function and performing the arithmetic operations. For f(x) = x² - 4x + 1, we substitute x = 5: f(5) = 5² - 4(5) + 1 = 25 - 20 + 1 = 6. The correct answer is 6 because it results from correctly applying the order of operations to evaluate the function at x = 5. Answer choice C (10) might result from an error such as calculating 5² - 4(5) + 5 or misapplying the function formula.
A store sells a jacket for 25% off its original price. If the sale price of the jacket is $72, what was the original price of the jacket?
Answer: $96
This is a Multiple Choice, Select One GRE Quant problem involving percent discount calculations. The core principle is that if an item is 25% off, the customer pays 75% of the original price. Since the sale price is $72 and this represents 75% of the original price, we can set up the equation: 0.75 × (original price) = $72. Solving for the original price: original price = $72 ÷ 0.75 = $96. The Borrect answer B ($96) satisfies the requirement because 0.75 × $96 = $72. A common error would be to add 25% of $72 to $72 (giving $90), which incorrectly treats the sale price as the base for the percentage calculation rather than recognizing that the discount is based on the original price.
