Simple Direct Proportion and Currency

What direct proportion means

Two quantities are in direct proportion when they are linked by a constant multiplier. If one quantity is doubled, the other doubles; if one is divided by five, the other is divided by five.

Suppose four identical notebooks cost £6. If the number bought doubles from four to eight, the cost also doubles from £6 to £12. If the number is halved to two, the cost is halved to £3. This works because every notebook has the same price.

A useful way to picture the relationship is to keep the pairs aligned:

The multiplier between corresponding entries must be the same in both rows. Moving from 4 to 8 means multiplying by 2, so £6 must also be multiplied by 2.

Not every relationship that increases is proportional. A taxi fare containing a fixed starting charge is not in direct proportion to distance: travelling twice as far does not necessarily make the whole fare twice as large.

Finding one unit, then scaling up

The unitary method is often the clearest approach:

1. Divide to find the amount for one unit.
2. Multiply to find the amount for the required number of units.

If six cinema tickets cost £51, one ticket costs

$£51 \div 6 = £8.50.$

Therefore eleven tickets cost

$11 \times £8.50 = £93.50.$

The units help explain each step: dividing a total cost by a number of tickets gives a cost per ticket, and multiplying that unit price by eleven gives the new total cost.

You do not always have to pass through one. If five identical packs contain 60 cards, fifteen packs are three times as many, so they contain $60\times3=180$ cards. Choose a scale factor that makes the arithmetic efficient.

Expressing proportion as equal ratios

Proportional pairs form equal ratios. If $x_1$ corresponds to $y_1$, and $x_2$ corresponds to , then the quantities may be compared as

$\frac{y_1}{y_2}=\frac{x_1}{x_2}$

or as

$\frac{y_1}{x_1}=\frac{y_2}{x_2}.$

The order matters. Keep like with like: if the first fraction is cost divided by quantity, the second must also be cost divided by quantity.

For example, five metres of fabric cost £37.50. If $c$ is the cost of eight metres, write

$\frac{37.50}{5}=\frac{c}{8}.$

Cross-multiplication gives

$5c=37.50\times8,$

so

$c=£60.$

This is an algebraic proportion because a letter represents the unknown quantity. Cross-multiplication is not a separate rule about proportion: it follows from multiplying both sides of an equation by both denominators.

An alternative arrangement is

$\frac{5}{8}=\frac{37.50}{c}.$

It gives the same answer because both fractions compare the first situation with the second. What would be wrong is to reverse only one of the two ratios.

Choosing a method and checking the result

A ratio table, a scale factor, the unitary method and an equality of ratios all describe the same constant multiplicative relationship. Choose the method that makes the numbers easiest.

After calculating, check the answer against the context. Eight metres is more than five metres, so its cost must exceed £37.50. The scale factor is $8\div5=1.6$, and $£37.50\times1.6=£60$, confirming the result.

You can also substitute the answer into the original proportion:

$\frac{37.50}{5}=7.50 \quad\text{and}\quad \frac{60}{8}=7.50.$

The equal values show that the cost per metre is constant.

Currency conversion as direct proportion

An exchange rate gives a constant multiplicative relationship between two currencies at a stated time. For example,

$£1=€1.18$

means that every pound corresponds to €1.18.

To convert pounds into euros, multiply by 1.18. Thus

$£250\times1.18=€295.$

To reverse the conversion, divide by 1.18. Thus

$€472\div1.18=£400.$

A ratio table makes the direction visible:

Exchange rates may instead be quoted with the other currency equal to one, such as $€1=£0.85$. Use the rate exactly as stated rather than assuming that conversion always means multiplying. Label each currency throughout so that the direction remains clear.

Currency answers normally need two decimal places because the main units divide into 100 smaller units. Perform the calculation using the full value and round only the final monetary answer when required.

Direct proportion

• A constant multiplier connects the two quantities.
• Multiplying one quantity by a scale factor multiplies the other by the same factor.
• A fixed extra charge breaks direct proportion.

Core methods

Unitary method: divide to find one unit, then multiply to find the required amount.

Equality of ratios: keep corresponding quantities in the same order.