Mathematical Reasoning Arithmetic and Number Sense Flashcards

A store sells a product at a 40% markup over its wholesale cost. During a clearance sale, the store discounts the marked-up price by 25%. What is the net percentage change from the original wholesale cost to the final sale price?

Answer: 5% profit

Starting with a wholesale cost of $100: a 40% markup gives $140. A 25% discount on $140 = 0.75 × $140 = $105. The net change from $100 to $105 is a 5% profit. Algebraically: 1.40 × 0.75 = 1.05, confirming a 5% net gain. A common error is averaging the percentages (40% − 25% = 15%), which ignores that the 25% discount applies to the already-inflated price.

Which of the following is equivalent to (2³ × 2⁻⁵) ÷ 2⁻⁴?

Answer: 4

Using exponent rules: 2³ × 2⁻⁵ = 2^(3+(−5)) = 2⁻². Then 2⁻² ÷ 2⁻⁴ = 2^(−2−(−4)) = 2^(−2+4) = 2² = 4. A common trap is mishandling the double negative when subtracting the exponent −4.

The ratio of red to blue marbles in a bag is 3:7. After 15 red marbles are added and 10 blue marbles are removed, the new ratio is 3:4. How many blue marbles were in the bag originally?

Answer: 70

Let red = 3x and blue = 7x. After changes: (3x + 15)/(7x − 10) = 3/4. Cross-multiplying: 4(3x + 15) = 3(7x − 10) → 12x + 60 = 21x − 30 → 90 = 9x → x = 10. Original blue count = 7x = 70.

A driver travels from City A to City B at 60 mph, then returns at 40 mph over the same route. What is the average speed for the entire round trip?

Answer: 48 mph

Average speed for equal distances uses the harmonic mean: 2(s₁ × s₂)/(s₁ + s₂) = 2(60 × 40)/(60 + 40) = 4800/100 = 48 mph. The arithmetic mean of 50 mph is wrong because more time is spent at the slower speed, pulling the true average below 50.

N is a two-digit integer where the tens digit is exactly twice the units digit, and the sum of the digits of N² equals 9. Which of the following is N?

Answer: 21

Two-digit numbers where tens = 2 × units: 21, 42, 63, 84. Checking squares: 21² = 441, digit sum = 4+4+1 = 9 ✓. 42² = 1764, digit sum = 1+7+6+4 = 18 ✗. 63² = 3969, digit sum = 3+9+6+9 = 27 ✗. 84² = 7056, digit sum = 7+0+5+6 = 18 ✗. Only 21 satisfies both conditions.

Pipe A fills a tank in 6 hours; Pipe B drains it in 12 hours. Both pipes are open simultaneously, and the tank starts at 1/3 full. How many hours until the tank is completely full?

Answer: 8

Pipe A fills at 1/6 per hour; Pipe B drains at 1/12 per hour. Net fill rate = 1/6 − 1/12 = 2/12 − 1/12 = 1/12 per hour. The tank is 1/3 full, so 2/3 remains. Time = (2/3) ÷ (1/12) = (2/3) × 12 = 8 hours.