KS3 · Maths

Using ratio for exchange rates and unit conversions

You hand over £100 and get €117. Your friend hands over £75. Your brain wants to subtract. Here is why multiplying is the move that works.

Maths · Exchange rates

£100 buys €117. What does £75 buy?
£0£10£20£30£40£50£60£70£80£90£100drag the pounds down to £75 →= 1/1

Share of £100: 100/100. Pounds £100. Euros €117.00

Fraction100/100Pounds£100Euros€117.00

Drag the pounds down from £100 to £75 and watch the euros follow. The fraction is the multiplier, and it does the same job to both currencies.

Watch out: Don't slide, scale. £25 fewer pounds does not mean €25 fewer euros, so £75 is not €92. Multiply both amounts by the same number.

The same exchange, drawn as a graph

025507510004080120160Pounds (£)Euros (€)
Exchange rate (€ for each £1) 1.17 €

Exchange rate (€ for each £1): 1.17 €. Euros for £100: 117 €

Steepness of the line. On axes with the same scale, the higher the exchange rate, the steeper the conversion graph.

At €1.17 per £1 (£100 = €117), read up from £75 on the pounds axis to the line, then across to the euros axis. Your eye lands at about €87, and that is only an approximation. Now slide the rate and watch the line.

Watch out: Start on the axis of the currency you have. With dollars on the vertical axis and $50 in your hand, find 50 on the dollars axis and read across to pounds. Starting on the wrong scale is a really common mistake.

Which deal wins?

Commit to an answer first. The pounds are different in the two offers, so think about how you would compare them fairly.

Two counters offer to swap your pounds. One gives 120 euros for £100. The other gives 30 euros for £20. Which is the better deal for you?

Maths · Ratio tables

Fill in the missing moves

£1 = A$1.92. Use a ratio table to change £120 into Australian dollars, then A$48 back into pounds.

  1. Table set-up: pounds in one column, Australian dollars in the other. The known pair is £1 → A$1.92.
  2. missing step
Which line is step 2?

Maps are multipliers too

Problem

Map A shows a road 2.5 cm long where 1 cm = 30 km. Map B shows a road 4 cm long where 1 cm = 18 km. Which real road is longer? Then, on a map with scale 1 : 50,000, find the real distance for 7 cm and the map distance for 5.3 km.

Maths · Map scales

Which line goes wrong?

On a map, 10 cm represents 6,000 m. Write the scale as 1 : n.

A student's working — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Money, units and map scales all work the same way.

What you need to know

  • An exchange rate says how much of one currency you get for another, and it is always changing.
  • Two currencies are in proportion: they have a constant multiplicative relationship, so when one amount doubles, the other doubles too.
  • Have a go£100 buys €117. How many euros should £200 buy?

    €234

    Doubling the pounds doubles the euros (117 × 2 = 234). Both amounts were multiplied by 2, and no fixed amount was added.

  • A conversion graph is a straight line from (0, 0), because £0 is €0, through a known exchange.
  • To read the graph, start on the axis of the currency you have, go to the line, then across to the other axis.
  • A value read from a graph is only an approximation. A double number line gives the exact conversion.
  • On axes with the same scale, the higher the exchange rate, the steeper the conversion graph.
  • Exchange rates can be written as ratios. Compare deals by making one currency the same in every ratio.
  • A ratio in the form 1 : n is the exchange rate. Pounds to euros 1 : 1.5 means each pound gets you 1.5 euros.
  • Look from the other side by dividing by the euros: euros to pounds 120 : 100 becomes 1 : 5/6.
  • At 1 : 5/6 each euro costs about 83p, and at 1 : 2/3 about 67p. The cheaper euro is the better deal.
  • Have a goOne counter charges 70p for each euro, another charges 75p. Which counter is the better deal, and why?

    The 70p counter.

    Each euro costs you less there, so your pounds buy more euros. A bigger number of pence is the worse deal here.

  • A ratio table converts too. Put each value in the right column, use the multiplier, and use the inverse to go back.
  • Use a ratio table horizontally or vertically and you get the same answer, so pick the easier multiplier.
  • Have a go3 zings = 18 zangs. How many zangs are 20 zings? Which multiplier is the easy one to use?

    120 zangs. Use × 6, from zings to zangs.

    From 3 to 18 is a whole-number multiplier of 6, so 20 × 6 = 120. Going across from 3 to 20 would need the awkward 20/3.

  • Units convert the same way. Put the equivalence in the top row, such as 1 inch ≈ 2.54 cm or 1 kg = 2.2 pounds.
  • You don't have to find the value of one unit first. Use the multiplier straight from the equivalence.
  • Maps are multiplicative too: the scale factor is the multiplier. Without a scale, you can't compare maps.
  • Write a map scale as 1 : n with no units, so convert both parts into the same units first.
  • For a real distance, multiply the map distance by n, then convert the units. For a map distance, divide by n.
  • Think multiplicatively. Using multiplication as repeated addition leads to incorrect additive strategies with ratios.
  • We assume exchange is free, although companies that exchange money do take a fee.

The big picture

Exchange rates, unit conversions and map scales are all the same thing: a constant multiplier linking two quantities. You get exact answers from a double number line or a ratio table, only approximate ones from a graph, and you compare deals and write map scales using ratios in the form 1 : n.

Key points

1Exchange rates, unit conversions and map scales are constant multipliers between two quantities: change one by a multiplier and the other follows.
2A graph gives an approximate reading, starting on the axis of the currency you have. A double number line or ratio table gives the exact value.
3In the form 1 : n, n is the exchange rate. Compare deals by making one currency the same in every ratio.
4Going the opposite way in a ratio table uses the inverse. Pick the easier of the horizontal or vertical multiplier.
5A map scale written as 1 : n needs both parts in the same units. Multiply by n for real distance, divide by n for map distance.

Worked example

Problem

Using 5 miles = 8 km, work out 4 miles in km. Is that further than 6 km?

⚠ Watch out

Going back the wrong way. If £1 = A$1.92, you turn dollars into pounds by dividing by 1.92, not by multiplying by 1.92 again. Ask yourself which operation undoes the one you used going forwards.

🧠

Memory hook

Lockstep, not slide: the two quantities move together by one multiplier. You never slide one by adding a fixed amount.

✓

Check yourself

A road is 4 cm long on a map with scale 1 : 300,000. How long is the real road, and which units do you convert between to give it in kilometres?

Flashcards

(15)
What is an exchange rate?
The rate at which one currency is exchanged for another. It can be thought of as converting between units of money, and it is always changing.
What does it mean for two quantities to be in proportion?
They have a constant multiplicative relationship: when one amount doubles, so does the other.
Where does a currency conversion graph start, and why?
At (0, 0), because zero pounds is zero euros. The straight line joins it to a known exchange.
Is a value read off a conversion graph exact?
No, it is only an approximation. A double number line gives the exact conversion.
Dollars sit on the vertical axis and you want pounds. Where do you start reading?
On the dollars axis, then across to the pounds axis. Starting on the wrong scale is a really common mistake.
Two conversion graphs share the same axis scales. How do their exchange rates show up?
The higher the exchange rate, the steeper the line.
In a ratio 1 : n, what does n tell you?
It is the exchange rate. Pounds to euros 1 : 1.5 means for every one pound you get 1.5 euros.
How do you compare exchange deals written as ratios?
Create equivalent ratios where one currency is the same in all of them, then compare the other. More euros for each pound is the better deal.
£1 = 182 yen. Convert £400 to yen, and 50,960 yen to pounds.
£400 = 400 × 182 = 72,800 yen. Going back, divide: 50,960 ÷ 182 = £280.
Why can a ratio table be used horizontally or vertically?
A multiplicative relationship gives the same answer either way, so choose the easier multiplier, such as a whole number.
Which two unit equivalences does this lesson use, and where do they go in a ratio table?
1 inch ≈ 2.54 cm and 1 kg = 2.2 pounds. The equivalence goes in the top row.
Why do maps need a scale?
Without a scale, comparisons between maps can't be made. The scale factor is the multiplier.
5 cm on a map represents 2,500 m. How do you write the scale as 1 : n?
Convert 2,500 m to 250,000 cm (1 m = 100 cm), so 5 cm : 250,000 cm is 1 : 50,000, with no units.
How do you turn a real distance into a map distance, and the reverse?
Real distance: multiply the map distance by n, then convert units. Map distance: convert the real distance to the map's units, then divide by n.
What goes wrong if you treat ratios as repeated addition?
It leads to incorrect additive strategies. Focus on the multiplicative relationships between parts and wholes.

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