Math 8 · Percents, ratios and rates
Parts, wholes, and how much for one
A percent compares a number with 100. A ratio compares two amounts of the same kind, and a rate compares two different kinds, like dollars and grams. All three work by multiplying rather than adding: double one amount and the other doubles too. That one idea handles a price over 100%, a paint mix for any size of job, and the better buy at the grocery store.
- 1. The words, first
- 2. Percents, including over 100% and under 1%
- 3. Ratios and proportions
- 4. Rates and unit rates
- 5. What costs marks
The words, first
The idea: Part-to-part and part-to-whole are the pair to keep straight. The same bag of marbles gives both.
| Word | What it means |
|---|---|
| Percent (%) | Per hundred. 45% means 45/100, or 0.45. 100% is the whole amount. |
| Percent increase, decrease | A change written as a percent of the original amount, the amount before the change. |
| GST | The 5% federal Goods and Services Tax. Alberta has no provincial sales tax, so GST is the only sales tax added at the till. |
| Ratio | A comparison of two amounts of the same kind, written 3 : 2 and read “3 to 2”. Part-to-part compares two parts; part-to-whole compares a part with the whole. |
| Equivalent ratios, proportion | Ratios that make the same comparison, like 3 : 2 and 12 : 8. A proportion says two ratios are equal: 3 : 5 = 24 : 40. |
| Rate, unit rate | A comparison of two different kinds of quantity, like 270 km in 3 hours. A unit rate is for one: 90 km/h. |
| Scale | The ratio of a length on a map or drawing to the real length it stands for. |
Percents, including over 100% and under 1%
The idea: A percent is a number of hundredths. 100% is the whole amount, so 250% is two and a half times it, and ½% is half of one hundredth.
150% = 1.5 · 250% of 40 = 2.5 × 40 = 100 · 0.5% = 0.005 · 0.5% of $3000 = $15
Over 100%. A town grows from 800 people to 1000. It is now 1000 ÷ 800 = 1.25 times as big, or 125% of what it was: the whole 100%, plus 25% more.
Under 1%. ½% is 0.5%, which is 0.005. An account paying ½% a year on $400 pays 0.005 × 400 = $2. Writing 0.5% as 0.5 would give $200.
Percent increase and decrease. Divide the change by the original amount. A ticket goes from $40 to $50: 10 ÷ 40 = 0.25, a 25% increase. If it falls back from $50 to $40, the change is the same $10, but now the original is $50: 10 ÷ 50 = 0.2, a 20% decrease.
Worked example: a discount, then GST. A jacket is $80, marked 25% off, and 5% GST is added at the till. What do you pay?
- Discount: 0.25 × 80 = 20, so the sale price is 80 − 20 = $60.
- GST on the sale price: 0.05 × 60 = 3, so the tax is $3.
- Total: 60 + 3 = $63.
In one line, 75% of the price and then 105% of that: 80 × 0.75 × 1.05 = 63. Taking 25% − 5% = 20% off the $80 instead gives $64, because the two percents are of different amounts.
Ratios and proportions
The idea: A ratio compares amounts by multiplying. Multiply or divide both terms by the same number to get an equivalent ratio, and use that to fill the gap in a proportion.
Part to part, and part to whole. A bag holds 12 red and 8 green marbles. Red to green is 12 : 8, which simplifies to 3 : 2 when both terms are divided by 4. That is part to part. Red to all the marbles is 12 : 20, or 3 : 5. That is part to whole, and it can also be written 3/5 = 60%.
The trap: 3 : 2 does not mean 3/2 of the marbles are red. Add the parts to find the whole first.
Solving a proportion. In 3 : 5 = x : 40, the 5 was multiplied by 8 to make 40. Do the same to the 3: x = 3 × 8 = 24. As fractions, 3/5 = 24/40 ✓.
Worked example. A green paint is mixed 3 parts blue to 2 parts yellow. How much of each is in 15 L of it?
- Parts in the whole: 3 + 2 = 5 parts.
- One part: 15 ÷ 5 = 3, so one part is 3 L.
- Each colour: blue is 3 × 3 = 9 L and yellow is 2 × 3 = 6 L. Check: 9 + 6 = 15, and 9 : 6 = 3 : 2 ✓.
The ratio says how to share, not how much: 3 L of blue and 2 L of yellow would make only 5 L.
Rates and unit rates
The idea: A rate compares two different kinds of quantity. Divide to find the amount for one — the unit rate — and any two rates can be compared directly.
Worked example: the better buy. A 750 g box of cereal costs $5.25. A 1.2 kg box costs $7.80. Which is cheaper for the amount?
- Same units first: 1.2 kg is 1200 g.
- Price per 100 g. The small box is 7.5 lots of 100 g, and 5.25 ÷ 7.5 = 0.70, so $0.70 per 100 g. The large box is 12 lots, and 7.80 ÷ 12 = 0.65, so $0.65 per 100 g.
- Compare: the large box is 5 cents cheaper per 100 g — the better buy, if it gets eaten before it goes stale.
Speed is a rate. A car travels 270 km in 3 hours: 270 ÷ 3 = 90 km/h. At that rate, 5 hours covers 5 × 90 = 450 km, and 360 km takes 360 ÷ 90 = 4 hours.
Scale is a ratio used like a rate. On a map with a scale of 1 : 50 000, 1 cm stands for 50 000 cm, which is 500 m. A trail 6.4 cm long on the map is 6.4 × 500 = 3200 m, or 3.2 km.
What costs marks
The idea: Most of these are dividing by the wrong amount, or comparing things that are not the same kind.
- Writing 0.5% as 0.5. It is 0.005, half of one hundredth.
- Dividing by the new amount for a percent change. Always divide by the original.
- Adding and subtracting percents of different amounts. 25% off and then 5% GST is not 20% off.
- Treating a part-to-part ratio as a fraction of the whole. Red to green of 3 : 2 means 3/5 are red, not 3/2.
- Mixing units in a scale. 1 : 50 000 means 1 cm to 50 000 cm, not 50 000 m.