Exchange rates, unit prices and compound interest
What this lesson covers
- Percentages, ratios, rates, and proportions are often applied in financial situations.
Three money problems, one piece of maths
- Changing currency. Comparing prices. Earning interest.
- Every one of them is a rate: something per something.
- Get the "per" right and all three are the same question.
| The rate | How much | Per what |
|---|---|---|
| Exchange rate | 0.92 AUD | per NZ dollar |
| Unit price | $18.00 | per kilogram |
| Interest rate | 2.5% | per month |
Going with the rate: multiply
- The rate is 0.92 Australian dollars for every New Zealand dollar.
- You have 250 New Zealand dollars to spend.
- 250 times 0.92 is 230 Australian dollars.
| Step | Working | Result |
|---|---|---|
| You have | NZD 250 | |
| The rate | 0.92 AUD per NZD | |
| Going to AUD | 250 × 0.92 | AUD 230.00 |
Now the other direction
- You are on an Australian website and something costs 230 dollars.
- What is that in New Zealand money?
- Most people multiply by 0.92 again. That is the wrong way.
Divide, because you are undoing the rate
- Multiplying again gives 211.60, and it is wrong.
- Something was multiplied by 0.92 to land on 230. So divide.
- 230 over 0.92 is exactly 250 New Zealand dollars.
| Step | What is happening | Value |
|---|---|---|
| A price of | AUD 230 | |
| Tempting | 230 × 0.92 | 211.60, wrong |
| It got here by × 0.92 | so undo it | |
| Divide | 230 ÷ 0.92 | NZD 250.00 |
| Check | 250 × 0.92 | 230, correct |
Again, with a different rate
- 0.60 US dollars per New Zealand dollar this time.
- Going out: 500 New Zealand dollars is 300 US dollars.
- Coming back: a 45 dollar US item is 75 New Zealand dollars.
| Step | Working | Result |
|---|---|---|
| Rate | 0.60 USD per NZD | |
| NZD 500 to US | 500 × 0.60 | USD 300.00 |
| USD 45 back to NZ | 45 ÷ 0.60 | NZD 75.00 |
| The NZ figure | is the bigger one | both times |
Dollars per kilogram is a rate too
- Cheese at eighteen dollars a kilogram. You want 400 grams.
- 400 grams is 0.4 of a kilogram.
- 0.4 times 18 is seven dollars twenty.
| Step | Working | Result |
|---|---|---|
| Cheese | $18.00 per kg | |
| You want | 400 g | that is 0.4 kg |
| So | 0.4 × 18 | $7.20 |
| Going back | 7.20 ÷ 18 | 0.4 kg, correct |
Which pack is actually cheaper?
- Three sizes at three prices. You cannot compare them as they are.
- Put every one on the same basis: dollars per kilogram.
- The medium pack wins, and the biggest one does not.
| Pack | Price | Per kilogram |
|---|---|---|
| Small, 400 g | $7.20 | $18.00 per kg |
| Medium, 750 g | $12.75 | $17.00 per kg |
| Large, 1.2 kg | $21.60 | $18.00 per kg |
| Best value | the medium one | not the largest |
The thing last year promised you
- Simple interest pays on the amount you started with, every time.
- Real accounts pay on whatever is in there now.
- So the interest earns interest. That is compound interest.
One hundred dollars, 2.5% a month, three months
- Month one: 2.5% of a hundred is two fifty. You have 102.50.
- Month two: 2.5% of 102.50, not of a hundred. Two fifty-six.
- Month three: 2.5% of 105.06. Two sixty-three. You finish on 107.69.
| When | Interest | Balance |
|---|---|---|
| Start | $100.00 | |
| Month 1 | + $2.50 | $102.50 |
| Month 2 | + $2.56 | $105.06 |
| Month 3 | + $2.63 | $107.69 |
Why those three numbers are different
- Two fifty, then two fifty-six, then two sixty-three.
- The rate was 2.5% every single month. It never moved.
- What changed is the balance it was applied to.
| Month | Interest | Applied to |
|---|---|---|
| Month 1 | $2.50 | on $100.00 |
| Month 2 | $2.56 | on $102.50 |
| Month 3 | $2.63 | on $105.06 |
| Rate never changed | the balance did |
Honestly, how much did that win?
- Simple interest over the same three months gives 107.50.
- Compounding gives 107.69. Nineteen cents.
- Leave it a year at the same rate and the gap is four forty-nine.
| Method | Working | You end with |
|---|---|---|
| Simple, 3 months | $7.50 interest | $107.50 |
| Compound, 3 months | month by month | $107.69 |
| Difference | 19 cents | small, but real |
| Same rate, 12 months | $134.49 | against $130.00 |
There is a formula, and you do not need it yet
- Doing it month by month is the point at this stage.
- Later you will meet the shortcut for the same thing.
- It does exactly what you just did, twelve times in one line.
The part worth remembering
- 2.5% a month compounded is 34.49% over a year.
- No savings account pays that. Credit cards charge it.
- Compound interest is the same maths pointed the other way.
| 2.5% a month, compounded | 34.49% a year |
| No savings account | pays anything like that |
| Credit cards and loans | charge rates like that |
| Same arithmetic | pointed at you instead |
What to take away
- Six things, and the last two are the ones that cost money.
| A rate | is a multiplier with a unit |
| Going with it | multiply |
| Coming back | divide, never multiply twice |
| To compare packs | use dollars per kilogram |
| Compound interest | recalculates every month |
| Which is why | it bites hardest on debt |
Your turn: four questions
- Pause here and work all four properly.
- Question 2 is the one going against the rate.
- Question 4 wants the month-by-month walk, not a shortcut.
| Question | |
|---|---|
| 1 | Convert NZD 350 to AUD. Rate 0.92. |
| 2 | A jacket is USD 63. Rate 0.60. NZD? |
| 3 | Coffee: 600 g for $21.00. Cost of 250 g? |
| 4 | $200 at 2% a month, for 3 months. |
Show answer
1. AUD 322.00, from 350 × 0.92. 2. NZD 105.00, from 63 ÷ 0.60. Dividing, because you are coming back against the rate. Multiplying gives 37.80, which is the classic wrong answer. 3. $35.00 per kg, from 21.00 ÷ 0.6, and then 0.25 × 35 = $8.75 for 250 g. 4. Month 1 earns $4.00 to reach $204.00. Month 2 earns $4.08 to reach $208.08. Month 3 earns $4.16 to reach $212.24. Simple interest would have given $212.00, so compounding is worth 24 cents over the three months.
The answers
- 350 times 0.92 is 322 Australian dollars.
- 63 divided by 0.60 is 105 New Zealand dollars. Divided, not times.
- Coffee is 35 dollars a kilo, so 250 grams is eight seventy-five.
- And the balances run 204.00, then 208.08, then 212.24.
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