Ratios you can scale, and rates that carry units
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.
Two ideas that look alike and are not
- Ratios and rates are both written as one thing against another.
- One of them carries units. The other does not.
- Knowing which is which is most of this lesson.
A ratio is a recipe, not a pair of amounts
- 3:2 does not mean three of this and two of that.
- It means three for every two. At any size at all.
- Six and four is the same recipe. So is thirty and twenty.
One relationship, four ways to write it
- Divide both sides by the same number and you simplify.
- Multiply both sides by the same number and you scale up.
- It is one move. The direction is the only difference.
| Ratio | What was done | Note |
|---|---|---|
| 30:20 | divide both by 10 | |
| 6:4 | divide both by 2 | |
| 3:2 | simplest form | no common factor left |
| 12:8 | multiply both by 4 | still the same recipe |
Simplest form: nothing left to divide by
- Find what both numbers share and take it out.
- 12 and 18 both divide by 6, giving 2:3.
- Simplest form means there is no common factor left.
| Ratio | What divides both | Simplest |
|---|---|---|
| 12:18 | both divide by 6 | 2:3 |
| 25:15 | both divide by 5 | 5:3 |
| 30:20 | both divide by 10 | 3:2 |
Now the question ratios exist for
- Share sixty dollars between two people in the ratio 3:2.
- Do not start by dividing sixty by anything.
- Start by counting how many parts there are.
Count the parts, then size one part
- Three parts and two parts makes five parts altogether.
- Sixty dollars over five parts, so one part is twelve dollars.
- Three parts is thirty-six, two parts is twenty-four.
| Step | Working | Result |
|---|---|---|
| Share $60 as 3:2 | first, count the parts | 3 + 2 = 5 parts |
| One part | $60 ÷ 5 | $12 |
| Three parts | 3 × $12 | $36 |
| Two parts | 2 × $12 | $24 |
| Check | 36 + 24 | $60, correct |
The same thing, seen from last lesson
- Five parts, and the first share is three of them.
- So it is three fifths of the total. Part to whole.
- Three fifths of sixty is thirty-six. It agrees.
| Step | Working | |
|---|---|---|
| 3:2 means 5 parts | so the first share is 3/5 | |
| 3/5 of $60 | 0.6 × 60 | $36 |
| 2/5 of $60 | 0.4 × 60 | $24 |
| Same answer | part-to-whole from last lesson |
The method does not care how many shares
- Concrete mixed 1:2:3, and you have twelve buckets.
- Six parts in total, so one part is two buckets.
- Two, four and six. Same method, three shares.
| Step | Working | Result |
|---|---|---|
| Concrete 1:2:3, 12 buckets | 1 + 2 + 3 | 6 parts |
| One part | 12 ÷ 6 | 2 buckets |
| The three shares | 2, 4 and 6 | adding to 12 |
Add your shares back up
- Thirty-six and twenty-four make sixty. Two, four and six make twelve.
- If they do not add to what you started with, it is wrong.
- One line, and it catches nearly every slip in this topic.
So what makes a rate different?
- A ratio compares like with like. Boys and girls. The units cancel.
- A rate compares unlike things. Kilometres and hours.
- Nothing cancels, so both units survive into the answer.
| Written | Comparing | What happens to units |
|---|---|---|
| 3:2 | boys to girls | same units, they cancel |
| 80 km/h | distance to time | different units, both stay |
| $1.30 each | money to apples | different units, both stay |
Per means for every
- 240 km in 3 hours. Divide, and you get 80 km per hour.
- Which reads: for every one hour, eighty kilometres.
- Every rate is a sentence like that.
| Situation | Working | The rate |
|---|---|---|
| 240 km in 3 hours | 240 ÷ 3 | 80 km per hour |
| $6.50 for 5 apples | 6.50 ÷ 5 | $1.30 per apple |
| 450 g for $9 | 450 ÷ 9 | 50 g per dollar |
And rates scale the same way ratios do
- Eighty kilometres for every one hour.
- So five hours is five times eighty. Four hundred kilometres.
- Half an hour is half of eighty. Forty.
| Step | Reasoning | Result |
|---|---|---|
| 80 km per hour | for every 1 hour, 80 km | |
| So in 5 hours | 5 × 80 | 400 km |
| And in half an hour | 0.5 × 80 | 40 km |
What this is actually for
- Two bottles, different sizes, different prices.
- You cannot compare them until both are per litre.
- Two twenty a litre against two thirty. The big one wins.
| Option | Working | Unit price |
|---|---|---|
| 2 L for $4.40 | 4.40 ÷ 2 | $2.20 per litre |
| 1.5 L for $3.45 | 3.45 ÷ 1.5 | $2.30 per litre |
| Better value | the larger bottle | by 10c a litre |
What to take away
- Six things, and the third one is where marks get lost.
| A ratio | is a recipe: so many for every so many |
| Simplify or scale | the same move, in two directions |
| Sharing | count the total parts first, always |
| One part | is the total divided by the number of parts |
| A rate | compares different units, so it keeps them |
| Per | means for every |
Your turn
- Pause here and try all four.
- Answers are in the description.
| Question | |
|---|---|
| 1 | Simplify 24:36. |
| 2 | Share $45 in the ratio 4:5. |
| 3 | A car travels 180 km in 2.5 hours. What is its speed? |
| 4 | Scale 3:7 so the first number is 12. |
Show answer
1. 2:3. Both parts divide by 12. 2. $20 and $25. Four plus five is nine parts, and $45 ÷ 9 is $5 a part. Check: 20 + 25 = 45. 3. 72 km per hour, from 180 ÷ 2.5. 4. 12:28. Going from 3 to 12 means multiplying by 4, so the 7 becomes 28.
The answers
- Both parts of 24:36 divide by 12.
- Nine parts, so one part is five dollars.
- 3 to 12 is times four, so 7 becomes 28.
| Answer | |
|---|---|
| 1 | 2:3 |
| 2 | $20 and $25 |
| 3 | 72 km/h |
| 4 | 12:28 |
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