Math Benchmark

A store offers a 20% discount on all items, then an additional 10% off the discounted price for members. What is the total percentage discount a member receives off the original pr…

A car rental company charges $40 per day plus $0.25 per mile. If someone rents a car for 3 days and drives 450 miles, what is the total cost? If they split it among 4 people, how m…

A rectangular swimming pool is 25 meters long, 10 meters wide, and has an average depth of 2 meters. Calculate how many liters of water it holds (1 cubic meter = 1000 liters). Then…

A recipe requires a ratio of 3:2:1 for flour, sugar, and butter by weight. If you use 450 grams of flour, how many grams of sugar and butter do you need? What is the total weight o…

An investment of $10,000 earns 6% compound interest annually. How much is it worth after 5 years? Show the formula and calculation. Then explain how the result changes if interest …

You roll two fair six-sided dice. What is the probability that the sum is 7 or 11? What is the probability that the sum is even? Show your counting method.

A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. If you draw 2 marbles without replacement, what is the probability both are the same color? Calculate exactly.

A factory produces light bulbs, and 3% are defective. A quality control test correctly identifies defective bulbs 95% of the time, but incorrectly flags good bulbs as defective 2% …

Given a normal distribution with mean = 100 and standard deviation = 15, what percentage of values fall between 85 and 130? Show your z-score calculations and explain the 68-95-99.…

Solve the system of equations: 2x + 3y = 17 and 5x - y = 8. Show each step of your solution method and verify your answer by plugging back into both equations.

A projectile's height in meters after t seconds is given by h(t) = -4.9t^2 + 50t + 2. Find the maximum height reached. At what time does the projectile hit the ground? Show your wo…

Find all real roots of x^3 - 6x^2 + 11x - 6 = 0. Factor the polynomial and explain the Rational Root Theorem you used. Verify each root.

The sum of three consecutive integers is 87. Find the three integers. Then generalize: for any sum S, what condition must S satisfy for three consecutive integers to exist that sum…

A right triangle has one leg of length 8 cm and hypotenuse of length 17 cm. Find the length of the other leg, the area of the triangle, and the radius of the largest circle that ca…

A cylindrical water tank has a radius of 3 meters and a height of 8 meters. If water is filled at a rate of 2 cubic meters per minute, how long does it take to fill? If the tank is…

How many different ways can you arrange the letters in the word 'MISSISSIPPI'? Show the formula for permutations with repeated elements and compute the exact number.

A committee of 3 people must be chosen from a group of 5 women and 4 men. How many different committees are possible if: (a) there are no restrictions, (b) the committee must have …

Find the GCD of 123456 and 789012 using the Euclidean algorithm. Show each step. Then find their LCM using the formula GCD(a,b) * LCM(a,b) = a * b.

Prove that for any integer n, n^3 - n is divisible by 6. Break the proof into cases or use factorization and modular arithmetic.

Find the derivative of f(x) = x^2 * sin(x) * e^x using the product rule. Show each step. Then find the value of x in [0, π] where the derivative equals zero and classify the critic…

Evaluate the definite integral: ∫(from 0 to π/2) sin^2(x) dx. Show the trigonometric identity used and verify your result by relating it to the area under the curve.

A mortgage of $300,000 has a 30-year term at 6.5% annual interest, compounded monthly. Calculate the monthly payment using the annuity formula. Then determine the total interest pa…

You want to have $500,000 saved for retirement in 25 years. If your investment account earns 7% annually, how much do you need to deposit each month? Show the future-value-of-annui…

Is √2 + √3 a rational number? Prove your answer. (Hint: assume it is rational, then derive a contradiction.)

A banquet hall has tables that seat 8 people each. 220 guests have RSVP'd yes, but there is historically a 5% no-show rate. How many tables should be set up to accommodate everyone…