Percent problems: interest, tax, markup, percent change

1. Change 8% to a decimal: 8\% = 0.08.
2. Tax: 0.08 \times 45 = \$3.60
3. Total: 45 + 3.60 = \$48.60

One-step version: 45 \times 1.08 = 48.60.

The total cost is $48.60.

Check: 10% of $45 is $4.50, so 8% should be a bit less — $3.60 fits, and the total is a little under $50 ✓

Tip: "Total cost" means price plus tax. Multiplying by 1 + \text{rate} (here 1.08) does both steps at once and avoids stopping at the tax.

1. Discount: 25\% \text{ of } \$80 = 0.25 \times 80 = \$20
2. Sale price: 80 - 20 = \$60

One-step version: paying 75% of the price, 80 \times 0.75 = 60.

The sale price is $60.

Check: \tfrac{1}{4} of $80 is $20, and \$60 + \$20 = \$80 ✓

Tip: "25% off" means you pay 100\% - 25\% = 75\%. Multiplying by 1 - \text{rate} gives the sale price in one step and keeps you from choosing the discount amount by mistake.

1. Find the discount: 35\% of $120. 0.35 \times 120 = \$42 By hand: 10\% of 120 is 12, so 30\% = 36; 5\% = 6; 36 + 6 = 42.
2. Subtract the discount from the regular price: \$120 - \$42 = \$78 Shortcut: 35% off means you pay 100\% - 35\% = 65\%: 0.65 \times 120 = \$78.

The sale price is $78.

Check: \$78 + \$42 = \$120 ✓ and \$42 \div \$120 = 0.35 = 35\% ✓

Tip: "Percent off" gives you the discount, not the price. Either subtract the discount or multiply by the percent you still pay (1 - \text{rate}). The sale price must be less than the original, and for any discount under 50% it must be more than half of the original.

1. Change 18% to a decimal: 18\% = 0.18.
2. Find the tip: 0.18 \times \$46 = \$8.28 By hand: 10\% of 46 is 4.60; 8\% is 8 \times 0.46 = 3.68; 4.60 + 3.68 = 8.28.
3. Add the tip to the bill: \$46.00 + \$8.28 = \$54.28 Shortcut: 46 \times 1.18 = 54.28.

Jordan will pay $54.28 in total.

Check: The tip is a bit less than 20% of $46, which is $9.20; $8.28 is a bit less ✓ so the total $54.28 is reasonable.

Tip: Read the last line of the question: "tip" and "total" are different answers, and both usually appear among the choices. Multiplying by 1 + \text{rate} (here 1.18) gives the total in one step.

1. Identify the values: P = 2{,}400, r = 3.5\% = 0.035.
2. Change months to years: t = \frac{18}{12} = 1.5\text{ years}
3. Apply the formula: I = Prt = 2{,}400 \times 0.035 \times 1.5 = 126

He will earn $126 in interest.

Check: One year of interest is 2{,}400 \times 0.035 = \$84; half a year is $42; 84 + 42 = 126 ✓

Tip: In I = Prt, the rate is per year, so time must be in years: divide months by 12. Also note the question asks for the interest, not the account balance ($2,526).

1. Marked price (40% markup on cost): 36 \times 1.40 = \$50.40
2. Sale price (25% off the marked price): 50.40 \times 0.75 = \$37.80

The sale price is $37.80.

Check: Discount = 0.25 \times 50.40 = \$12.60, and 50.40 - 12.60 = 37.80 ✓

Tip: Successive percents multiply: 1.40 \times 0.75 = 1.05, so the sale price is 5% above cost — not 15% (which you'd get by subtracting 40\% - 25\%). Each percent applies to the price at that moment.

1. After 15% off, the customer pays 100\% - 15\% = 85\% of the original price x: 0.85x = 51
2. Divide both sides by 0.85: x = \frac{51}{0.85} = 60

The original price was $60.

Check: 15% of $60 is $9, and 60 - 9 = 51 ✓

Tip: To undo a percent change, divide by the multiplier. Adding 15% back on (51 \times 1.15 = \$58.65) fails because the discount was 15% of the original price, which is bigger than $51.

1. Find the amount of change: 14{,}000 - 12{,}500 = 1{,}500 \text{ people}
2. Divide by the original (starting) value: \frac{1{,}500}{12{,}500} = 0.12
3. Change to a percent: 0.12 \times 100\% = 12\%

The population increased by 12%.

Check: 12\% of 12,500 is 0.12 \times 12{,}500 = 1{,}500, and 12{,}500 + 1{,}500 = 14{,}000 ✓

Tip: Always divide the change by the value you started from (the "old" value). Dividing by the new value (1{,}500 \div 14{,}000 \approx 10.7\%) is the most common mistake, and it often appears as a wrong answer choice.

1. Read the table: February (the start) = \$105; April (the end) = \$70.
2. Find the change: 70 - 105 = -35 The bill went down by $35.
3. Divide by the starting value (February): \frac{-35}{105} = -\frac{1}{3} \approx -0.333
4. Change to a percent: -0.333 \times 100\% \approx -33.3\%

The bill decreased by about 33.3%.

Check: 33.3\% of $105 is about \tfrac{1}{3} \times 105 = \$35, and 105 - 35 = \$70 ✓

Tip: With a table, circle the two values the question names (and ignore the others). The month that comes first in time is the original value, even if it is not the first row of the table.

1. Find the markup amount: 60\% \text{ of } \$45 = 0.60 \times 45 = \$27
2. Add the markup to the store's cost: \$45 + \$27 = \$72 Shortcut: a 60% markup means the price is 100\% + 60\% = 160\% of the cost: 1.6 \times 45 = \$72.

The selling price is $72.

Check: \$72 - \$45 = \$27, and \$27 \div \$45 = 0.6 = 60\% ✓

Tip: A markup percent is always a percent of the cost (what the store paid). Multiply the cost by 1 + \text{markup rate} to get the selling price in one step.

1. Store A: the discount is 0.25 \times 160 = \$40, so the price is \$160 - \$40 = \$120
2. Store B: the price is \$160 - \$35 = \$125
3. Compare: \$125 - \$120 = \$5.

Store A is cheaper, by $5.

Check: 25% is one quarter, and one quarter of $160 is $40, which is a bigger discount than $35 ✓ So Store A must be cheaper, by 40 - 35 = \$5.

Tip: To compare a percent discount with a dollar discount, turn the percent into dollars using the actual price. The answer can flip for a different price: on a $100 item, 25% off is only $25, and $35 off would win.

1. First discount: 20% off $250. You pay 80%: 0.80 \times \$250 = \$200
2. Second discount: 10% off the sale price of $200. You pay 90%: 0.90 \times \$200 = \$180
3. In one step: 250 \times 0.80 \times 0.90 = 250 \times 0.72 = \$180

The final price is $180.

Check: The total discount is \$250 - \$180 = \$70, which is 70 \div 250 = 28\%, not 30%, because the second 10% was taken from a smaller amount ✓

Tip: Successive percents multiply; they don't add. "20% off, then 10% off" is a 28% discount, not 30%. Use the multipliers (1 - r_1)(1 - r_2).

1. Start with $100 (any starting value gives the same percent).
2. After a 20% rise: 100 \times 1.20 = \$120
3. After a 20% fall: 120 \times 0.80 = \$96
4. Overall change: \frac{96 - 100}{100} = -0.04 = -4\%

The stock had an overall 4% decrease.

Check: Using multipliers: 1.20 \times 0.80 = 0.96, and 0.96 = 1 - 0.04 → a 4% decrease ✓

Tip: Successive percent changes multiply; they don't cancel or add. The 20% drop is taken from a larger number ($120), so it removes more dollars ($24) than the rise added ($20).

1. Job B starts $800 behind: \$2{,}000 - \$1{,}200 = \$800
2. The 5% commission must make up that $800. Let s be the monthly sales: 0.05s = 800
3. Divide by 0.05: s = \frac{800}{0.05} = \$16{,}000 (Equation version: 1{,}200 + 0.05s = 2{,}000 gives the same result.)

The jobs pay the same when monthly sales are $16,000.

Check: Job B pay = 1{,}200 + 0.05 \times 16{,}000 = 1{,}200 + 800 = \$2{,}000, equal to Job A ✓ With more sales than $16,000, Job B pays more.

Tip: For a "break-even" question, set the two pay expressions equal and solve. Dividing by 0.05 is the same as multiplying by 20, a quick mental check: 800 \times 20 = 16{,}000.

1. Let p be the price before tax. The total is the price plus 6.5% of the price: p + 0.065p = 1.065p
2. Set it equal to the total: 1.065p = 85.20
3. Divide both sides by 1.065: p = \frac{85.20}{1.065} = \$80.00

The price before tax was $80.00.

Check: Tax on $80 is 0.065 \times 80 = \$5.20, and 80 + 5.20 = \$85.20 ✓

Tip: To undo a percent increase, divide by the multiplier (1 + \text{rate}). Do not subtract the percent of the final amount; the tax was 6.5% of the smaller, pre-tax price.

1. First tier: 2% of the first $50,000: 0.02 \times 50{,}000 = \$1{,}000
2. Second tier: sales above $50,000 are 82{,}000 - 50{,}000 = \$32{,}000. At 3.5%: 0.035 \times 32{,}000 = \$1{,}120
3. Add the two tiers: \$1{,}000 + \$1{,}120 = \$2{,}120

Her commission was $2,120.

Check: The answer should be between 2% of $82,000 ($1,640) and 3.5% of $82,000 ($2,870), because part of her sales earn each rate ✓ $2,120 is in that range.

Tip: In tiered ("bracket") problems, each rate applies only to its own slice of the total. Write the slices first and confirm they add up to the total (50{,}000 + 32{,}000 = 82{,}000).

1. Convert the time to years: t = \frac{9}{12} = 0.75 \text{ year}
2. Find the interest with I = Prt (formula sheet): I = 950 \times 0.084 \times 0.75 = 79.80 \times 0.75 = \$59.85
3. Add the interest to the amount borrowed: \$950 + \$59.85 = \$1{,}009.85

Tomás pays $1,009.85 in total.

Check: One full year of interest is 0.084 \times 950 = \$79.80; three-quarters of that is \$59.85 ✓ and 950 + 59.85 = 1{,}009.85 ✓

Tip: An annual rate needs time in years: months \div 12. Using 9 as the time would charge 9 years of interest; a 9-month loan should cost less than one year's interest.

1. Time in years: 6 months = \tfrac{6}{12} = 0.5 year, so t = 2.5.
2. Substitute into I = Prt: 360 = 3{,}200 \times r \times 2.5
3. Simplify the right side: 360 = 8{,}000r
4. Solve: r = \frac{360}{8{,}000} = 0.045 = 4.5\%

The annual interest rate is 4.5%.

Check: 3{,}200 \times 0.045 = \$144 per year, and 144 \times 2.5 = \$360 ✓

Tip: Rearranged, the formula is r = \dfrac{I}{Pt}. Convert the time carefully: 2 years 6 months is 2.5 years, not 2.6 (there are 12 months in a year, not 10).

1. Use I = Prt (formula sheet) with I = 810, r = 0.045, t = 4: 810 = P \times 0.045 \times 4
2. Multiply the known factors: 0.045 \times 4 = 0.18, so 810 = 0.18P
3. Divide both sides by 0.18: P = \frac{810}{0.18} = \$4{,}500

Grace must deposit $4,500.

Check: 4{,}500 \times 0.045 \times 4 = 202.50 \times 4 = \$810 ✓

Tip: When solving for P, r, or t, multiply the two known factors on the right first, then divide the interest by that product. Watch the decimal: 4.5% is 0.045, not 0.45.

1. Let c be the store's cost. A 40% markup means the price is the cost plus 40% of the cost: c + 0.40c = 1.40c
2. Set it equal to the selling price: 1.40c = 91
3. Divide by 1.40: c = \frac{91}{1.40} = \$65

The store paid $65 for the jacket.

Check: 40% of $65 is 0.4 \times 65 = \$26, and 65 + 26 = \$91 ✓

Tip: The markup percent is based on the cost, which is unknown here. So you can't take 40% of $91. Undo a markup by dividing by 1 + \text{markup rate}.

The store did two things in order: (1) took 40% off the original price, then (2) subtracted $12. Undo them in reverse order.

1. Undo the coupon: the sale price before the coupon was \$51 + \$12 = \$63
2. Undo the 40% discount. After 40% off, the customer pays 60%, so 0.60 \times \text{original} = 63 \text{original} = \frac{63}{0.60} = \$105

The original price was $105.

Check: 40% off $105: 0.6 \times 105 = \$63. Coupon: 63 - 12 = \$51 ✓

Tip: To work backwards through several steps, undo the last step first (like taking off shoes before socks). And undo a percent discount by dividing by the percent you paid (1 - \text{rate}), never by the discount rate.