ISAT Mathematical Problem-Solving 2 — Questions and Answers
Question 1: 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?
- 60 mph (Correct answer)
- 72 mph
- 80 mph
- 90 mph
Correct 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.
Question 2: If log₂(x) + log₂(x − 2) = 3, what is the value of x?
- 4 (Correct answer)
- 3
- 2
- 6
Correct 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.
Question 3: A rectangle's length is 3 more than twice its width. If its area is 44 square units, what is the width?
- 4 (Correct answer)
- 5.5
- 3
- 11
Correct answer: 4
Width w satisfies w(2w+3) = 44 → 2w² + 3w − 44 = 0 → (2w+11)(w−4) = 0; width = 4.
Question 4: What is the sum of all integers from 1 to 100 that are divisible by either 3 or 5?
- 2418 (Correct answer)
- 2310
- 2415
- 2500
Correct answer: 2418
Sum(div by 3) + Sum(div by 5) − Sum(div by 15) = 1683 + 1050 − 315 = 2418.
Question 5: If f(x) = x² − 4x + 7, what is the minimum value of f(x)?
- 3 (Correct answer)
- 4
- 7
- 2
Correct answer: 3
Completing the square: f(x) = (x−2)² + 3, so the minimum value is 3 at x = 2.
Question 6: A jar contains 5 red and 3 blue marbles. Two marbles are drawn without replacement. What is the probability both are red?
- 5/14 (Correct answer)
- 25/64
- 10/28
- 1/4
Correct answer: 5/14
P(both red) = (5/8) × (4/7) = 20/56 = 5/14.
Question 7: The first term of a geometric sequence is 3 and the common ratio is 2. What is the sum of the first 6 terms?
- 189 (Correct answer)
- 192
- 96
- 63
Correct answer: 189
S₆ = 3(2⁶ − 1)/(2 − 1) = 3 × 63 = 189.
ISAT Mathematical Problem-Solving 2
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?