Ratios, rates, speed and percentage change
What this lesson covers
- Ratios can be scaled up or down or simplified.
- A rate proportionally compares two quantities that have different units of measure; when working with rates, ‘per’ means ‘for every’ in day-to-day contexts.
A ratio and a rate are not the same thing
- A ratio compares two amounts of the same kind. No units.
- A rate compares two different kinds. Kilometres and hours.
- That difference decides what you are allowed to do with it.
Simplifying a ratio
- Twelve to eighteen. Both divide by six.
- Two to three. Same relationship, smaller numbers.
- Whatever you do to one side, do to the other.
| Ratio | Working | Simplified |
|---|---|---|
| 12 : 18 | both divide by 6 | 2 : 3 |
| 24 : 36 | both divide by 12 | 2 : 3 |
| 50 : 30 : 20 | all divide by 10 | 5 : 3 : 2 |
| The relationship | has not changed | only the numbers |
Scaling one up
- A recipe for four uses two to three.
- For eight, double both sides: four to six.
- It is the same ratio. Only the serving changed.
| Ratio | Scaled | Serves |
|---|---|---|
| 2 : 3 | for four people | |
| Double it | 4 : 6 | for eight |
| Triple it | 6 : 9 | for twelve |
| All the same ratio | written differently |
Which mix is stronger?
- Two to three against three to five. You cannot tell by looking.
- Turn each into a fraction of the whole: two fifths, three eighths.
- Nought point four beats nought point three seven five.
| Mix | Of the whole | Value |
|---|---|---|
| Mix A, 2 : 3 | 2 parts in 5 | 0.4 |
| Mix B, 3 : 5 | 3 parts in 8 | 0.375 |
| So A is stronger | by a little | |
| Compare fractions | not the raw numbers |
Rates, and the word "per"
- Kilometres per hour. Dollars per kilogram. Words per minute.
- Two different units, and the units carry the meaning.
- Read "per" as "for every" and rates get much easier.
| Rate | Read it as | Measuring |
|---|---|---|
| 80 km/h | kilometres per hour | speed |
| $18.00/kg | dollars per kilogram | price |
| 45 words/min | words per minute | typing |
| "per" | means "for every" |
For every
- Eighty kilometres per hour means eighty for every hour.
- So two hours is two lots of eighty.
- Said that way, you do not need a formula to work it out.
Which pack is cheaper?
- Three for four fifty, or five for seven dollars.
- Different sizes, so put both on the same basis: price each.
- One fifty against one forty. The bigger pack wins here.
| Offer | Working | Each |
|---|---|---|
| 3 for $4.50 | 4.50 ÷ 3 | $1.50 each |
| 5 for $7.00 | 7.00 ÷ 5 | $1.40 each |
| The five-pack wins | by ten cents each | |
| Same basis | is the whole trick |
Speed, distance and time
- A hundred and twenty kilometres in an hour and a half.
- Speed is the distance for every hour: eighty kilometres an hour.
- Going back the other way, multiply the speed by the time.
| Given | Working | Answer |
|---|---|---|
| 120 km in 1.5 h | 120 ÷ 1.5 | 80 km/h |
| 80 km/h for 2.5 h | 80 × 2.5 | 200 km |
| 300 km at 80 km/h | 300 ÷ 80 | 3.75 h |
| 3.75 hours | is 3 h 45 min | not 3 h 75 |
Three point seven five hours
- That is three hours and three quarters of an hour.
- Three quarters of sixty minutes is forty-five.
- So three hours forty-five, and never three hours seventy-five.
Applying a change to a number
- Up fifteen percent means a hundred and fifteen percent of it.
- So multiply by one point one five. Two hundred and forty becomes 276.
- Down eight percent leaves ninety-two percent. Multiply by 0.92.
| Change | Working | Result |
|---|---|---|
| 240 up 15% | 240 × 1.15 | 276 |
| 340 up 12% | 340 × 1.12 | 380.8 |
| 500 down 8% | 500 × 0.92 | 460 |
| One multiply | not an add-on |
Percentage change between two numbers
- Fifty goes up to seventy-five. The change is twenty-five.
- Now divide by where you started, which is fifty.
- Nought point five, so it is a fifty percent increase.
| From, to | Change and base | Answer |
|---|---|---|
| 50 to 75 | change 25, base 50 | +50% |
| 80 to 60 | change 20, base 80 | −25% |
| 250 to 200 | change 50, base 250 | −20% |
| Divide by | where you started |
Why the starting point decides
- Fifty up to seventy-five is a fifty percent increase.
- Seventy-five back down to fifty is a thirty-three percent decrease.
- Same two numbers, same gap. Different base, different percent.
| Direction | Change on base | Percent |
|---|---|---|
| 50 up to 75 | 25 on a base of 50 | +50% |
| 75 back to 50 | 25 on a base of 75 | −33% |
| Same two numbers | same gap | |
| Different starting point | different percentage |
One question to ask yourself
- Where did this start? That is what you divide by.
- Not the bigger number, and not the one you finished on.
- Get the base right and the rest is a division.
What to take away
- Seven lines, and the last one is where the marks are.
| A ratio has no units | so it can be scaled |
| Simplify | divide both sides by the same |
| To compare ratios | turn them into fractions |
| A rate has two units | "per" means "for every" |
| Speed | is distance divided by time |
| To apply a change | multiply once |
| Percentage change | divide by where you started |
Your turn: four questions
- Pause here and work all four.
- Question 3 is one multiplication, not two steps.
- Question 4 is asking you which number is the base.
| Question | |
|---|---|
| 1 | Simplify the ratio 24 : 36. |
| 2 | A car does 210 km in 2.5 hours. Speed? |
| 3 | Increase 340 by 12%. |
| 4 | A price falls from 250 to 200. By what %? |
Show answer
1. 2 : 3. Twelve divides into both, and dividing both sides by the same number leaves the relationship untouched. 2. 84 km/h. Speed is distance over time, so 210 divided by 2.5. 3. 380.8. A 12% increase means 112% of it, so multiply by 1.12 in one step. 4. 20%. The change is 50, and the base is where it started, which is 250. 50 over 250 is 0.2. Dividing by 200 would give 25%, and that is the answer to a different question: it is the rise from 200 back up to 250.
The answers
- Twenty-four to thirty-six is two to three.
- Two hundred and ten over two point five is eighty-four km an hour.
- 340 times 1.12 is 380 point 8.
- Fifty over two hundred and fifty is twenty percent.
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