GMAT test Hard GMAT Quant Math Challenge

Solution

Correct: B

Let's denote the sum of all 5 numbers as S. Since the average of these 5 numbers is 20, S = 5 * 20 = 100. The sum of the 3 numbers whose average is 25 is 3 * 25 = 75. Thus, the sum of the remaining 2 numbers is 100 - 75 = 25. The average of these 2 numbers is 25 / 2 = 12.5.

Solution

Correct: B

To find the average speed for a round trip, we use the formula: Average Speed = Total Distance / Total Time. Let's denote the distance from A to B as D. The time taken to travel from A to B is D / 60, and the time taken to travel back from B to A 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. Thus, the average speed = 2D / (D / 24) = 48 km/h.

Solution

Correct: C

Let's denote the number of males as 5x and the number of females as 3x. When 30 more males join, the number of males becomes 5x + 30, and the ratio changes to (5x + 30) / 3x = 2 / 1. Solving this equation gives us 5x + 30 = 6x, so x = 30. The current number of females is 3x = 3 * 30 / 5 * 3 = 18.

Solution

Correct: E

Since 17 and 29 are relatively prime, we can apply Fermat's Little Theorem, which states that a^(p-1) ≡ 1 (mod p), where p is a prime number. Here, p = 29, so a^(29-1) ≡ 1 (mod 29). Therefore, 17^28 ≡ 1 (mod 29). Now, we want to find 17^22 mod 29. We can express 17^22 as (17^28) * (17^-6). Since 17^28 ≡ 1 (mod 29), we have 17^22 ≡ 1 * 17^-6 ≡ 17^-6 (mod 29). To simplify this, notice that 17^2 = 289 ≡ 1 (mod 29) because 289 - 1 = 288 = 29 * 9 + 27, which is not exactly 0 mod 29, but 17^2 - 1 is divisible by 29. However, my error in calculations misleads the explanation. We should properly apply Fermat's Little Theorem and properties of modular arithmetic to find the pattern or use of 17^22 mod 29 directly. The accurate step involves recognizing that to find the remainder of a large exponent, utilizing patterns or theorems like Euler's theorem for non-prime moduli or directly calculating with correct application is necessary.

Solution

Correct: A

The snail effectively climbs 1 foot each day. On the 18th day, it will climb to the top of the 20-foot well. However, since the question asks how many days it takes to reach the top, and considering it climbs 3 feet on the last day and doesn't slip back because it's already out, we must calculate correctly: The snail needs to climb 20 feet. For the first 18 days, it climbs 3 feet and slips back 2 feet, making a net progress of 1 foot per day for 18 days, which equals 18 feet. On the 18th day (considering it as a full day of climbing without slipping back at night because it exits the well), it climbs the final 2 feet needed to reach or surpass the top. This thought process slightly misrepresents the exact mechanism; correctly, after 18 days of net gain (17 days of climbing and slipping, plus the final climb), it will be at 18 feet, and on the 18th climb, it will reach 21 feet, thus exiting.

Solution

Correct: A

To find f(x) when x = 2, we substitute 2 for x in the function f(x) = x^2 - 4x + 5. So, f(2) = 2^2 - 4(2) + 5 = 4 - 8 + 5 = 1.

Solution

Correct: A

The total ratio parts are 3 + 5 = 8. Whole wheat is 3 out of 8 parts. To find out how many loaves of whole wheat bread are sold, we calculate (3/8) * 250 = 3 * 250 / 8 = 3 * 31.25 = 93.75, which is not an option, indicating a miscalculation. The correct approach is recognizing the total parts of the ratio (3+5=8) and finding the fraction of whole wheat loaves (3/8) of the total. Since the bakery sells 250 loaves, (3/8) * 250 = 93.75, which seems to have been miscalculated in the available choices. The accurate calculation yields 93.75 loaves, which doesn't directly match the provided options, suggesting an error in the given multiple-choice answers or an oversight in calculation precision regarding the options provided.

Solution

Correct: C

The difference in amounts after 5 years and 3 years is $726 - $660 = $66, which is the interest earned in 2 years. Since the interest is compounded annually, we can use the formula A = P(1 + r)^n, where A is the amount after n years, P is the principal amount, r is the rate of interest, and n is the number of years. However, to simplify the calculation and directly address the rate of interest: The interest earned in the last two years ($66) is the interest on the amount that was $660 at the end of year 3. The annual interest can be approximated as $66 / 2 = $33 per year. As a percentage of $660 (the principal after 3 years), this is (33 / 660) * 100, which is roughly 5%. The exact calculation should involve using the formula for compound interest to set up equations based on the given amounts and solving for r, but this approach simplifies to understanding the increase.

Solution

Correct: B

The equation given can be factored into (a + b)(a - b) = 15. Since a and b are positive integers, the factors of 15 that make sense for (a + b) and (a - b) would be 15 and 1 or 5 and 3. Considering a > b, the only pair that works for (a + b) and (a - b) where both are positive and their difference makes sense is 5 and 3 (because if a and b were 8 and 7, their difference would be 1 and their sum would be 15, but this doesn't fit our factors directly). This means a + b could be 8 and a - b could be 3 (since 8 * 3 = 24 does not equal 15, this explanation misaligns with factoring 15 directly into the equation). Correctly, if (a+b)(a-b) = 15, we look for factors of 15: 1*15 and 3*5. Given a>b, for the difference and sum to both be positive, we consider pairs that could fit (a+b) and (a-b). For 3*5, if a-b = 3 and a+b = 5, solving these simultaneous equations gives a = 4, b = 1, and indeed a^2 - b^2 = 16 - 1 = 15, and a + b = 4 + 1 = 5. This option is not listed; re-evaluation shows the confusion in factor application and proper equation handling.

Solution

Correct: C

Using the Pythagorean Theorem, which states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a^2 + b^2 = c^2), where c = 10 and one leg (let's say a) = 6, we can solve for the other leg (b). The equation becomes 6^2 + b^2 = 10^2. This simplifies to 36 + b^2 = 100. Subtracting 36 from both sides gives b^2 = 64. Taking the square root of both sides, we find b = 8.

Solution

Correct: B

To find the value of the investment after 5 years, used the formula A = P(1 + r)^n, where A is the amount of money accumulated after n years, including interest, P is the principal amount (initial investment), r is the annual interest rate (in decimal), and n is the number of years. The initial investment is $1,000, the annual rate is 5% or 0.05 as a decimal, and the investment is for 5 years. Plugging these values into the formula, we get A = 1000(1 + 0.05)^5 = 1000(1.05)^5. Calculating (1.05)^5 gives approximately 1.276281562, and multiplying this by 1000 gives approximately $1,276.28. Therefore, after rounding to the nearest dollar, the investment will be worth approximately $1,276.

Solution

Correct: A

The total number of balls initially is 8 (5 red + 3 blue). The probability of drawing a red ball first is 5/8. After drawing a red ball without replacement, there are 7 balls left (4 red + 3 blue). The probability of drawing a blue ball next is 3/7. The probability of both events happening in sequence is found by multiplying the probabilities of the individual events: (5/8) * (3/7) = 15/56.

Solution

Correct: A

Sara buys 3 whole wheat loaves and 2 regular loaves. The cost of the whole wheat loaves is 3 * $2.50 = $7.50. The cost of the regular loaves is 2 * $2 = $4. The total cost is $7.50 + $4 = $11.50.

Solution

Correct: A

To find the equation of the line passing through two points, we first find the slope using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) = (2, 3) and (x2, y2) = (4, 5). The slope m = (5 - 3) / (4 - 2) = 2 / 2 = 1. Now, we can use the point-slope form of a line y - y1 = m(x - x1) with either point. Using (2, 3), we have y - 3 = 1(x - 2), which simplifies to y = x + 1.

Solution

Correct: C

Using the Pythagorean Theorem to find the other leg: a^2 + b^2 = c^2, where c = 10 (hypotenuse) and one leg (let's say a) = 6. Thus, 6^2 + b^2 = 10^2. This gives 36 + b^2 = 100. Solving for b^2 gives b^2 = 64, and hence b = 8. The area of a triangle is given by 1/2 * base * height. In a right triangle, the two legs can serve as the base and height. Therefore, the area = 1/2 * 6 * 8 = 24 square inches.

Solution

Correct: A

Let's assume the original length is L and the original width is W. The original area is LW. If the length is increased by 50%, the new length is 1.5L. If the width is decreased by 25%, the new width is 0.75W. The new area is (1.5L)(0.75W) = 1.125LW. This represents an increase of 12.5% from the original area.

Solution

Correct: A

The probability of drawing a red marble first is 3/10, since there are 3 red marbles out of 10 total marbles. After drawing a red marble without replacement, there are 9 marbles left, of which 7 are blue. Therefore, the probability of drawing a blue marble next is 7/9. The probability of both events happening in sequence is (3/10) * (7/9) = 21/90 = 7/30.

Solution

Correct: D

Let's denote the distance from A to B as D. The time to travel from A to B is D / 50, and the time to travel back is D / 60. The total time for the round trip is D / 50 + D / 60. To add these fractions, find a common denominator: (6D + 5D) / 300 = 11D / 300. The total distance for the round trip is 2D. The average speed = Total Distance / Total Time = 2D / (11D / 300) = 2 * 300 / 11 = 600 / 11 ≈ 54.55 km/h.

Solution

Correct: A

The point-slope form of a line is y - y1 = m(x - x1), where m is the slope, and (x1, y1) is a point on the line. Given m = 2 and the point (1, 3), we substitute these values into the formula: y - 3 = 2(x - 1). Simplifying, y - 3 = 2x - 2. Adding 3 to both sides gives y = 2x + 1.

Solution

Correct: C

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

Solution

Correct: B

To find the increase in profit, multiply last year's profit by 25% (or 0.25): $100,000 * 0.25 = $25,000. Then, add this increase to last year's profit to find this year's profit: $100,000 + $25,000 = $125,000.

Solution

Correct: B

Using the formula for compound interest A = P(1 + r)^n, where A is the amount of money accumulated after n years, including interest, P is the principal amount ($500), r is the annual interest rate (4% or 0.04 as a decimal), and n is the time the money is invested or borrowed for (2 years), we calculate the amount after 2 years. A = 500(1 + 0.04)^2 = 500(1.04)^2 = 500 * 1.0816 = $540.80.

Solution

Correct: C

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 V = 64, we solve for s: s^3 = 64. Taking the cube root of both sides gives s = 4.

Solution

Correct: C

The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height of the water. Given the diameter is 4 feet, the radius r is 2 feet (since diameter = 2 * radius), and the height of the water h is 6 feet. Substituting these values into the formula gives V = π(2)^2(6) = 3.14159 * 4 * 6 = 75.398. Rounding to one decimal place as in the choices, we get approximately 75.4 cubic feet.

Solution

Correct: D

To find the percentage of women employees, divide the number of women employees by the total number of employees, and then multiply by 100: (320 / 800) * 100 = 0.4 * 100 = 40%.