Mathematical Reasoning Flashcards

A rectangle's length is increased by 20% and its width is decreased by 15%. What is the net percentage change in the rectangle's area?

Answer: Increase of 2%

Multiply the two scale factors: 1.20 × 0.85 = 1.02, meaning the new area is 102% of the original — a 2% increase. The common mistake is subtracting the percentages (20% − 15% = 5%), which ignores that percentage changes to dimensions multiply rather than add when computing area.

A retailer marks up the wholesale cost of an item by 40%, then advertises a 25% sale on the marked price. Compared to the original wholesale cost, what is the customer's final price?

Answer: 5% above wholesale

Starting from wholesale price W, the marked price is 1.40W. After the 25% discount: 1.40W × 0.75 = 1.05W — which is 5% above wholesale. The trap answer is 15% above wholesale (from incorrectly computing 40% − 25%), but that ignores that the discount is taken on the already-inflated price, not the original wholesale cost.

Given f(x) = x² − 4x + 3, what is the value of f(5) − f(−1)?

Answer: 0

f(5) = 25 − 20 + 3 = 8. f(−1) = 1 + 4 + 3 = 8. Therefore f(5) − f(−1) = 8 − 8 = 0. This occurs because the parabola f(x) = (x−1)(x−3) is symmetric about x = 2, and the inputs 5 and −1 are both exactly 3 units away from that axis of symmetry, producing identical outputs.

A water tank is 3/4 full. After pumping in 15 more gallons, it is 9/10 full. What is the total capacity of the tank in gallons?

Answer: 100

Let C = total capacity. The equation is (3/4)C + 15 = (9/10)C. Subtracting (3/4)C from both sides: 15 = (9/10 − 3/4)C. Finding the common denominator: 9/10 − 3/4 = 18/20 − 15/20 = 3/20. So 15 = (3/20)C, giving C = 15 × (20/3) = 100 gallons.

Pipe A can fill an empty tank in 4 hours. Pipe B drains the same tank in 6 hours. If both pipes are open at the same time starting with an empty tank, how many hours will it take to fill the tank completely?

Answer: 12 hours

Pipe A's fill rate is 1/4 tank per hour. Pipe B's drain rate is 1/6 tank per hour. The net fill rate is 1/4 − 1/6 = 3/12 − 2/12 = 1/12 tank per hour. Time to fill the full tank = 1 ÷ (1/12) = 12 hours. A common error is averaging the two times (4 and 6) or simply adding them.

At an event, the ratio of adults to children is 5:3. After 10 adults leave and 10 children arrive, the ratio becomes 1:1. How many people were at the event originally?

Answer: 80

Let adults = 5k and children = 3k. After the changes: (5k − 10) = (3k + 10), so 2k = 20 and k = 10. Original adults = 50, original children = 30, giving a total of 80. Verify: after changes, 40 adults and 40 children — a 1:1 ratio. ✓