Interest, Discount, and Markup: Practice Questions & Explanations

Simple interest formula: I = P × r × t, where P is principal, r is rate (as decimal), t is time (in years). I = 1000 × 0.05 × 3 = $150. The simple interest formula is straightforward: just multiply principal × rate × time. Watch units: rate must be in the same time unit as time (annual rate with years, monthly rate with months). For ASVAB problems, rates are almost always annual. Compound interest is different and more complex: I = P(1 + r/n)^(nt) - P, where n is compounding periods per year. Most ASVAB interest problems use simple interest. The total amount after interest = principal + interest = $1,000 + $150 = $1,150. The same formula handles: bank interest, loan interest, bond interest. For half-year periods, just adjust t (e.g., 18 months = 1.5 years). Quick mental math: 5% per year for 3 years is approximately 15% total in simple interest; 15% of $1,000 = $150.

Markup is added to cost to determine selling price. Markup = 40% of $30 = 0.40 × $30 = $12. Selling price = cost + markup = $30 + $12 = $42. Alternative: selling price = cost × (1 + markup percentage as decimal) = $30 × 1.40 = $42. The second method is faster — multiplying by 1.40 is one step. Distinguish markup from margin: markup is calculated based on cost; margin is calculated based on selling price. A 40% markup on $30 cost gives selling price $42; the margin (profit ÷ selling price) is 12/42 ≈ 28.6%. Real-world business context: retailers buy at wholesale, mark up to retail. Common markups vary by industry. Discount problems work similarly but subtract: 20% discount on $50 = $50 × (1 - 0.20) = $50 × 0.80 = $40. The 'multiply by 1.X' or 'multiply by 0.X' shortcuts save time on the ASVAB.

Let P = original principal. Simple interest formula: A = P(1 + rt), where A is the final amount, r is the rate, t is the time. $1,080 = P(1 + 0.04 × 2) = P(1.08). Solve: P = $1,080/1.08 = $1,000. Verification: interest = $1,000 × 0.04 × 2 = $80; total = $1,000 + $80 = $1,080 ✓. This 'reverse' interest problem asks for the principal given the ending amount, rate, and time. Strategy: compute the total growth factor (1 + rt) = (1 + 0.08) = 1.08; divide the final amount by this factor to find the original. Same approach works for tax problems ('what was the pre-tax price if final price was $X?'), discount problems ('what was the original price if discount was X%?'), and compound interest problems (more complex factor). Always identify what you're solving for and set up the equation accordingly.

25% of $80 = $20 discount. Sale price = $80 - $20 = $60. Two-step discount problems: find the discount amount first (25% of 80 = 20), then subtract it. The distractor $20 is the discount itself, not the price; $65 subtracts only $15 (which would be 18.75%); $55 subtracts $25 — a common slip of treating '25% off' as '$25 off'.

Tax = 6% × $1,200 = $72. Total = $1,200 + $72 = $1,272. A shortcut: total with tax = price × 1.06 = 1,200 × 1.06 = 1,272. The distractor $1,206 adds only $6 (treating 6% as $6), $1,260 adds 5%, and $1,268 is a near-miss. Sales tax is always applied to the full pre-tax price, then added on top of it.

SALE PRICE = original − discount. The discount is 25% of $40 = 0.25 × 40 = $10. Sale price = $40 − $10 = $30. ASVAB Arithmetic Reasoning tests percentage problems.

Shortcut: 25% off means you pay 75% (100% − 25%): 0.75 × $40 = $30. Either method gives $30. Strategy: convert the percentage to a decimal (25% = 0.25), find the discount, then subtract — or directly find the percentage you pay. Both approaches are reliable.

A 25% discount means you subtract 25% of the price. First find the discount: 25% of $80 = 0.25 × 80 = $20. Then subtract from the original: $80 − $20 = $60. A faster method is to recognize that after a 25% discount you pay 75% of the original: 0.75 × $80 = $60. Discount problems test whether you can convert a percent to a decimal and decide whether to subtract (discount) or add (markup/tax). The sale price here is $60; the $20 answer is the discount amount, not the final price, so read the question carefully.

Simple interest uses the formula I = P × r × t, where P is the principal, r is the annual rate as a decimal, and t is the time in years. Here I = $1,500 × 0.04 × 3 = $60 × 3 = $180. The interest earned is $180. Note the question asks only for the interest, not the total balance — the total would be $1,500 + $180 = $1,680, which is offered as a distractor. Memorize I = Prt for simple interest problems, convert the percent to a decimal (4% = 0.04), and multiply through carefully.

A markup is added to the cost. First find the markup: 30% of $40 = 0.30 × 40 = $12. Then add it to the cost: $40 + $12 = $52. Equivalently, a 30% markup means selling at 130% of cost: 1.30 × $40 = $52. Markup problems are the mirror image of discount problems — you add the percentage instead of subtracting it. The $12 distractor is just the markup amount, not the selling price. The item sells for $52.