Mathematical Problem-Solving Flashcards

A train travels 360 miles at a constant speed. If the speed were increased by 20 mph, the trip would take 1 hour less. What is the original speed?

Answer: 60 mph

Setting up 360/v − 360/(v+20) = 1 gives v² + 20v − 7200 = 0, which factors to (v−60)(v+120) = 0, so v = 60 mph.

If log₂(x) + log₂(x − 2) = 3, what is the value of x?

Answer: 4

log₂(x(x−2)) = 3 means x(x−2) = 8, so x² − 2x − 8 = 0 → (x−4)(x+2) = 0; since x > 2, x = 4.

A rectangle's length is 3 more than twice its width. If its area is 44 square units, what is the width?

Answer: 4

Width w satisfies w(2w+3) = 44 → 2w² + 3w − 44 = 0 → (2w+11)(w−4) = 0; width = 4.

What is the sum of all integers from 1 to 100 that are divisible by either 3 or 5?

Answer: 2418

Sum(div by 3) + Sum(div by 5) − Sum(div by 15) = 1683 + 1050 − 315 = 2418.

If f(x) = x² − 4x + 7, what is the minimum value of f(x)?

Answer: 3

Completing the square: f(x) = (x−2)² + 3, so the minimum value is 3 at x = 2.

A jar contains 5 red and 3 blue marbles. Two marbles are drawn without replacement. What is the probability both are red?

Answer: 5/14

P(both red) = (5/8) × (4/7) = 20/56 = 5/14.

The first term of a geometric sequence is 3 and the common ratio is 2. What is the sum of the first 6 terms?

Answer: 189

S₆ = 3(2⁶ − 1)/(2 − 1) = 3 × 63 = 189.