2. Percents and interest
Identify the percent BASE before computing. Successive percent changes multiply. Simple vs compound interest.
Core ideas
- Percent OF WHAT? Always name the base first; "25% more than x" means 1.25x, and the base of a change is the starting value, not the ending one.
- Successive percent changes multiply: +20% then -20% is 1.2 * 0.8 = 0.96, a net 4% loss, never zero.
- Simple interest grows linearly: A = P(1 + rt). Compound interest multiplies each period: A = P(1 + r)^t.
- To undo a p% increase you need a smaller percent decrease: dividing by 1.25 is the same as multiplying by 0.8.
Worked example 1
A jacket's price is raised 25%, then the new price is discounted 20%. The final price is what percent of the original?
Show solution
Use multipliers on the original base. The raise gives multiplier 1.25 and the discount gives 0.80, so the final price is 1.25 * 0.80 = 1.00 times the original, exactly 100%. The trap is subtracting percents (25 - 20 = 5% up); percents of different bases never add. Here the 20% discount acts on the larger raised price, so it removes more dollars than the 25% added.
Worked example 2
$1,000 is invested for 2 years at 10% annual interest. How much more does it earn with annual compounding than with simple interest?
Show solution
Simple interest: 1000 * 0.10 * 2 = 200, so the account earns $200. Compound: 1000 * 1.1^2 = 1000 * 1.21 = 1210, earning $210. The difference is 210 - 200 = $10. Shortcut: the extra is interest on the first year's interest, 10% of $100 = $10.
Practice set
Question 1
15 percent of 60 is what percent of 45 ?
Compute the first quantity, then be careful that 45 is the base for the second percent.
15 percent of 60 is 0.15 * 60 = 9. Then 9 as a percent of 45 is 9/45 = 1/5 = 20 percent. The answer is C. Choice A gives the number 9 itself rather than the percent.
Question 2
Alan's salary is 25 percent greater than Beth's salary. Beth's salary is what percent less than Alan's salary?
The two percents use different bases, so pick a concrete number for Beth's salary and compute.
Let Beth's salary be 100. Then Alan's salary is 125. Beth earns 25 less than Alan, and as a percent of Alan's salary that is 25/125 = 20 percent. The answer is B. Choice C is the trap of reusing the same 25 percent with the wrong base.
Question 3
Pat invests $1,000 at 10 percent annual interest compounded annually, and Quinn invests $1,000 at 10 percent simple annual interest. After 2 years, how much more money does Pat have than Quinn?
The only extra money from compounding is the interest earned on the first year's interest.
Pat has 1000 * 1.1 * 1.1 = $1,210. Quinn has 1000 + 2 * 100 = $1,200. The difference is $10, choice B. That $10 is exactly 10 percent interest on the $100 of first-year interest, the only place compounding differs from simple interest in year two.
Question 4
The price of a jacket was increased by 20 percent and the new price was then decreased by 20 percent. The final price is what percent of the original price?
Successive percent changes multiply; they do not cancel just because the percents match.
A 20 percent increase multiplies the price by 1.2 and a 20 percent decrease multiplies it by 0.8. Together: 1.2 * 0.8 = 0.96, so the final price is 96 percent of the original. The answer is A. Choice B is the trap of assuming the changes cancel.