Percentages and fractions in context: change, GST and points
What this lesson covers
- A percentage is always a percentage of something, so the base must be identified first.
- Percentage change compares the change with the original amount, not with the new one.
What this exam looks like
- A booklet of real-looking information.
- Questions about what it says and how good it is.
- Percentages are where most marks are lost.
| Resource booklet | infographics, graphs, articles |
| Three questions | each with several parts |
| You | identify, relate, critically engage |
| Top lost marks | percentage and fraction working |
A percentage of what?
- "40% said yes." 40% of whom?
- All 480 students, or the 25 who replied?
- Find the base before anything else.
Finding and writing percentages
- A percentage of an amount: multiply.
- A fraction of an amount: divide, then multiply.
- Part out of whole, times 100: a percentage.
| Working | Answer | |
|---|---|---|
| 35% of 480 | 0.35 × 480 | 168 students |
| 3/8 of 480 | 480 ÷ 8 × 3 | 180 students |
| 72 out of 480 | 72 ÷ 480 × 100 | 15% |
Parts of different wholes
- School B has more students who walk.
- But School A has the higher rate.
- Different wholes: compare percentages.
| Walkers | Rate | |
|---|---|---|
| School A | 45 of 180 | 25% |
| School B | 60 of 300 | 20% |
| More students | School B | 60 > 45 |
| Higher rate | School A | 25% > 20% |
Percentage increase
- Find the change first: $0.36.
- Divide by the ORIGINAL, $2.40.
- A 15% increase.
| Value | Result | |
|---|---|---|
| Old fare | $2.40 | |
| New fare | $2.76 | |
| Change | $0.36 | |
| ÷ original | 0.36 ÷ 2.40 | 15% increase |
Divide by the original, always
- Dividing by the new fare gives 13%.
- That answers a different question.
- Change is compared with where you started.
Percentage decrease
- The same method, going down.
- $35 off a $250 bill.
- A 14% decrease.
| Value | Result | |
|---|---|---|
| Old bill | $250 | |
| New bill | $215 | |
| Change | $35 | |
| ÷ original | 35 ÷ 250 | 14% decrease |
Taking GST back off
- A price with GST is 115% of the original.
- So divide by 1.15, to get $60.
- Taking 15% off gives $58.65: wrong base.
| Working | Result | |
|---|---|---|
| Price with GST | 115% of the original | $69 |
| Undo it | $69 ÷ 1.15 | $60 |
| Wrong way | $69 × 0.85 | $58.65 |
| Check | $60 × 1.15 | $69 |
Percent or percentage points?
- From 20% to 25% is up 5 percentage points.
- As a percentage change, it is up 25%.
- Articles pick whichever sounds bigger.
| Was | 20% of students | |
| Now | 25% of students | |
| Rose by | 5 percentage points | the difference |
| Rose by | 25% | 5 ÷ 20 |
Up 10%, then down 10%
- The second change has a different base.
- $100 becomes $110, then $99.
- Not back to where it started.
| Working | Price | |
|---|---|---|
| Start | $100 | |
| Up 10% | $100 × 1.1 | $110 |
| Down 10% | $110 × 0.9 | $99 |
Say what the number means
- "15%" alone earns little.
- "The bus fare rose by 15%, from $2.40 to $2.76."
- The context goes in every answer.
How the grades step up
- Achieved: accurate, relevant calculations.
- Merit: in context, clearly communicated.
- Excellence: used to justify a decision.
What to take away
- Every percentage has a base. Name it first.
| First | a percentage of what? |
| Compare wholes | use percentages, not counts |
| Change | change ÷ original × 100 |
| Remove GST | divide by 1.15 |
| Points | difference of two percentages |
| Up then down | does not return to the start |
| Answer | a sentence in the context |
Your turn: four questions
- Pause here and work each one out.
- Name the base before you divide.
- Question 4 has two answers.
| Question | |
|---|---|
| 1 | 28 of 350 students cycle to school. What percentage is that? |
| 2 | A jacket went from $80 to $92. Find the percentage increase. |
| 3 | A price including 15% GST is $46. What was it before GST? |
| 4 | Support went from 40% to 50%. By how many points, and by what %? |
Show answer
1. 28 ÷ 350 × 100 = 8%. 2. Change $12; 12 ÷ 80 × 100 = 15% increase. 3. $46 ÷ 1.15 = $40. Check: 40 × 1.15 = 46. 4. Up 10 percentage points; 10 ÷ 40 × 100 = a 25% increase.
The answers
- 28 ÷ 350 × 100 = 8%.
- $12 ÷ $80 × 100 = 15% increase.
- $46 ÷ 1.15 = $40.
- 10 percentage points, which is a 25% increase.
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