1 · Ratios as parts
A ratio of 3 : 5 means 3 parts of one thing for every 5 parts of another, so 8 parts in all. If a class of 32 has boys and girls in the ratio 3 : 5, each part is 32 ÷ 8 = 4 students: 12 boys and 20 girls.
The fraction of the class that is boys is 3/8, not 3/5. A ratio compares part to part; a fraction compares part to whole.
2 · The difference trick
If the ratio is 4 : 7 and there are 15 more of the second kind, the difference is 3 parts. 3 parts = 15, so 1 part = 5, and the total (11 parts) is 55.
3 · Proportions
A proportion says two ratios are equal. 5 notebooks cost $7.50; how much do 8 cost? 5/7.50 = 8/x, or simply: one notebook is $1.50, so 8 cost $12.
Check that the units line up: notebooks with notebooks, dollars with dollars.
4 · Unit rates and the better buy
A unit rate is the amount for one: 240 miles in 4 hours is 60 miles per hour. To compare prices, find the cost per ounce (or per item).
16 oz for $3.20 is $0.20 per ounce; 24 oz for $4.56 is $0.19 per ounce, so the bigger box is the better buy here, but only because its unit price is lower.
5 · Percent of a number and finding the whole
35% of 80 = 0.35 × 80 = 28. A quick way: 10% is 8, so 30% is 24 and 5% is 4.
Working backwards: 18 is 45% of what number? 45% is 18, so 5% is 2 and 100% is 40.
6 · Percent change
Percent change = change ÷ original. A price going from $40 to $50 is a 10 ÷ 40 = 25% increase. Going from $50 back to $40 is a 10 ÷ 50 = 20% decrease.
That is why a 20% rise followed by a 20% drop does not get you back where you started: $100 → $120 → $96.
7 · Discounts, tax, and tip
A $60 jacket at 25% off costs 75% of $60 = $45. Another 10% off the sale price is 90% of $45 = $40.50, not 35% off the original ($39).
Tax and tip are added: a $40 meal with 8% tax and a 20% tip on the $40 costs 40 + 3.20 + 8 = $51.20.
8 · Simple interest and commission
Simple interest = principal × rate × time. $500 at 4% for 3 years earns 500 × 0.04 × 3 = $60.
Commission is a percent of sales, often added to a base pay: $300 a week plus 5% of $2,000 in sales is 300 + 100 = $400.
9 · Distance, rate, and time
Distance = rate × time. 150 miles at 50 miles per hour takes 3 hours; 2.5 hours at 48 miles per hour covers 120 miles.
Average speed is total distance ÷ total time. Driving 2 hours at 30 mph and 1 hour at 60 mph is 120 miles in 3 hours: 40 mph, not the average of 30 and 60.
10 · Working together
If Ann paints a room in 3 hours and Ben in 6, in one hour Ann paints 1/3 of it and Ben 1/6. Together they paint 1/3 + 1/6 = 1/2 an hour, so the room takes 2 hours.
Working together is always faster than the faster worker alone. If your answer is longer, something went wrong.
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