1 · Percents as multipliers
Turn every percent change into one multiplication. 15% off means you pay 85%, so multiply by 0.85. An 8% tax means multiply by 1.08.
A $60 jacket at 15% off costs 60 × 0.85 = $51, in one step.
2 · Percent change
Percent change = change ÷ original. From 40 to 50 is a change of 10, and 10 ÷ 40 = 25% increase. From 50 back to 40 is 10 ÷ 50 = 20% decrease.
The same change of 10 gives different percents because the starting amount is different. Always divide by where you started.
3 · Back to the original
After a 20% discount a game costs $64. The $64 is 80% of the original, so the original is 64 ÷ 0.8 = $80.
Adding 20% to $64 gives $76.80, which is wrong: the 20% was taken from the original price, not from $64.
4 · Changes in a row
Multiply the multipliers. A 20% rise and then a 20% drop: 1.2 × 0.8 = 0.96, a 4% decrease, not "no change."
Two 10% raises: 1.1 × 1.1 = 1.21, a 21% raise. The second raise is taken on the bigger amount.
5 · Simple and compound interest
Simple interest = principal × rate × time: $1,000 at 5% for 3 years earns 1,000 × 0.05 × 3 = $150.
Compound interest multiplies each year: $1,000 at 5% for 2 years becomes 1,000 × 1.05 × 1.05 = $1,102.50. The second year earns interest on the first year's interest.
6 · Sharing in a ratio
Add the parts to find what one part is worth. Split $120 in the ratio 2 : 3 : 5: there are 10 parts, so one part is $12, and the shares are $24, $36, and $60.
7 · Ratios that change
Write each amount with the same multiplier k. Boys to girls is 3 : 4, so 3k boys and 4k girls. If 6 boys join and then the numbers are equal, 3k + 6 = 4k, so k = 6: there were 18 boys and 24 girls.
8 · Mixtures
Follow the part that does not change. 20 liters of a mix that is 30% juice has 6 liters of juice. To make it 20% juice, those 6 liters must be 20% of the total, so the total is 30 liters: add 10 liters of water.
9 · Average speed
Average speed = total distance ÷ total time. Drive 60 miles at 30 mph (2 hours) and 60 miles back at 60 mph (1 hour): 120 miles in 3 hours is 40 mph, not 45.
You spend more time at the slower speed, so the average leans toward it.
10 · Working together and converting rates
Add the work done in one hour. A fills a tank in 3 hours (1/3 per hour) and B in 6 hours (1/6 per hour). Together they do 1/3 + 1/6 = 1/2 per hour, so the tank takes 2 hours.
To convert a rate, multiply by fractions equal to 1: 20 meters per second × 3,600 seconds per hour ÷ 1,000 meters per kilometer = 72 km per hour.
Ready to practice?
The quiz gives you new questions each time and explains every answer. You'll also find more quizzes and full practice tests.