What is a proportion?

A proportion is an equality of two ratios, for example 3 : 4 = 6 : 8. It is also written with fractions: 3/4 = 6/8. A proportion lets you find one unknown quantity when the other three are known.

What is cross-multiplication?

In the proportion a/b = c/d the products of the diagonal terms are equal: a · d = b · c. So from x : 3 = 20 : 4 we get 4x = 60, that is x = 15. The multiplication goes across, not along — pairing a · b with c · d is the most common mistake in this calculation.

What does a scale of 1 : 50 000 mean?

That one unit on the map corresponds to 50,000 identical units on the ground. One centimetre on the map is 50,000 cm, that is 500 m, or half a kilometre. A scale of 1 : 50 000 is a reducing scale; a scale of 2 : 1 would be enlarging, and is used on drawings of small parts.

How do you divide an amount in the ratio 3 : 4?

Add the numbers of the ratio: 3 + 4 = 7 parts. Divide the whole by 7 to get the value of one part, then multiply it by 3 and by 4. So 350 splits into 150 and 200. The most common mistake is dividing by 3 or by 4 instead of by their sum.

When are two quantities directly proportional?

When their ratio is constant, that is y = k · x for some number k. Doubling one doubles the other: twice the petrol costs twice as much. Not every relationship works like that — a higher speed shortens the journey time rather than lengthening it, so those two quantities are not directly proportional.

Basic level

Proportions and scale

A proportion is an equality of two ratios — one equation that rescales a recipe from four people to six and turns centimetres on a map into kilometres on the ground. Learn cross-multiplication, direct proportionality, dividing a quantity in a given ratio, and scale in both directions.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

FractionsA fraction writes part of a whole as a numerator and a denominator. Learn equivalent fractions, all four operations on fractions, mixed numbers, comparing fractions and reducing to lowest terms.
DivisionDivision is the inverse of multiplication — divide the dividend by the divisor to get the quotient. Learn the names, the link to multiplication, division with a remainder and why you must never divide by zero.

Where this is used

Real situations where you count exactly the way this lesson teaches:

Cooking
A cake recipe for 4 people calls for 300 g of flour and 6 guests turn up. The proportion 300 : 4 = x : 6 gives x = 450 g. Every other ingredient scales the same way — one multiplier of 1.5 for the whole recipe.
Hiking and maps
On a 1 : 50 000 map a route measures 7 cm. On the ground that is 7 · 50 000 = 350,000 cm, so 3.5 km — about 53 minutes of walking at 4 km/h.
Convert it here: Length
Architecture and technical drawing
On a 1 : 100 floor plan a wall measures 4.2 cm, so in the building it is 4.2 m. Before ordering a 1.5 m window the designer measures 1.5 cm on the drawing — the whole conversation with the builder rests on that one conversion factor.
Construction
Mortar is mixed 1 : 4, one part cement to four parts sand. From 60 kg of mix that is 60 : 5 = 12 kg of cement and 48 kg of sand — divided into five parts, not four.

All formulas

Property of a proportion
$\frac{a}{b} = \frac{c}{d} \iff a \cdot d = b \cdot c$
the diagonal products are equal — hence cross-multiplication
The unknown in a proportion
$x = \frac{b \cdot c}{d}$
from x : b = c : d, after cross-multiplying
Direct proportionality
$y = k \cdot x, \quad k = \frac{y}{x}$
the ratio of the two quantities stays constant
Dividing in a ratio
$a : b \ \Rightarrow \ \text{one part} = \frac{S}{a + b}$
the whole splits into a + b equal parts
Reducing scale
$\text{scale } 1 : n \ \Rightarrow \ \text{real} = n \cdot \text{on the drawing}$
on a 1 : 50 000 map one centimetre is 500 metres

A ratio of two quantities says how many times one fits into the other. It is written with a colon or as a fraction: $3 : 4$ is the same as $\tfrac{3}{4}$ . On its own a ratio computes nothing — it is setting two ratios equal that produces the tool which rescales a recipe from four people to six and turns centimetres on a map into kilometres on the ground.

Proportions and cross-multiplication

A proportion is an equality of two ratios:

\frac{a}{b} = \frac{c}{d} \qquad \text{or} \qquad a : b = c : d

Its key property comes from multiplying both sides by $b \cdot d$ :

\frac{a}{b} = \frac{c}{d} \iff a \cdot d = b \cdot c

The products of the terms on the diagonals are equal — hence the name cross-multiplication. That one step turns a proportion into an ordinary equation, from which the unknown follows wherever it stands:

x : b = c : d \quad\Rightarrow\quad x \cdot d = b \cdot c \quad\Rightarrow\quad x = \frac{b \cdot c}{d}

Solve the proportion x : 3 = 20 : 4.

Direct proportionality

Two quantities are directly proportional when their ratio is constant:

y = k \cdot x, \qquad k = \frac{y}{x}

The number $k$ is the constant of proportionality. However many times one quantity grows, the other grows by the same factor: twice the petrol costs twice as much, a board three times as long weighs three times as much.

Litres of fuel	10	20	35	50
Cost	64	128	224	320

In every column the cost divided by the number of litres gives the same value: $k = 6.4$ per litre. That is the test for proportionality — not that both quantities grow, but that they grow at the same rate.

Not every relationship qualifies. Over a fixed route a higher speed gives a shorter journey time, so those two quantities are not directly proportional — it is their product, not their ratio, that stays constant.

Dividing in a ratio

Dividing 350 in the ratio $3 : 4$ does not mean dividing by $3$ or by $4$ . The numbers of the ratio say how many parts each share consists of, so they get added first:

3 + 4 = 7 \quad\Rightarrow\quad \text{one part} = \frac{350}{7} = 50

Then multiply by the numbers of the ratio: $3 \cdot 50 = 150$ and $4 \cdot 50 = 200$ . The check is immediate — $150 + 200 = 350$ , and $150 : 200 = 3 : 4$ .

Divide 84 sweets among three children in the ratio 2 : 3 : 7.

Scale

Scale is the ratio of a length on a drawing to the length it stands for in reality. The notation $1 : n$ means a reduction by a factor of $n$ :

\text{real length} = n \cdot \text{length on the drawing}

The notation $n : 1$ means an enlargement — that is how small machine parts are drawn, since at natural size there would be no room to label them.

A room plan at 1 : 200 — the same shape as the real room, every dimension 200 times smaller.

Scale does not change the shape; it changes every dimension at once and in the same ratio. That is why a $1 : 200$ plan looks exactly like the room seen from above rather than like a room squashed on one side.

In map calculations the trouble usually comes not from the scale itself but from the units: a scale is a pure number, so $1 : 50\,000$ means one centimetre on the map is $50\,000$ centimetres on the ground. Converting to friendlier units happens only after the multiplication — a length unit converter does that part.

On a 1 : 25 000 map a segment measures 6 cm. How many kilometres is that on the ground?

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

5 : 4 = 15 : x

Common mistakes

Multiplying along instead of across — in $\tfrac{a}{b} = \tfrac{c}{d}$ the equal products are $a \cdot d$ and $b \cdot c$ , not $a \cdot b$ and $c \cdot d$ .
Dividing by one of the ratio numbers — a division in the ratio $3 : 4$ means $7$ parts, so the whole is divided by $7$ .
Inverting the scale — at $1 : 50\,000$ the real length comes from multiplying the map dimension, not from dividing it.
Losing the units in a scale calculation — multiplying centimetres by the denominator gives centimetres; converting to kilometres is a separate step.
Calling every growing relationship proportional — proportionality requires a constant ratio, not merely growth.

Formula card

Topic: Proportions and scale

Property of a proportion
$\frac{a}{b} = \frac{c}{d} \iff a \cdot d = b \cdot c$
the diagonal products are equal — hence cross-multiplication
The unknown in a proportion
$x = \frac{b \cdot c}{d}$
from x : b = c : d, after cross-multiplying
Direct proportionality
$y = k \cdot x, \quad k = \frac{y}{x}$
the ratio of the two quantities stays constant
Dividing in a ratio
$a : b \ \Rightarrow \ \text{one part} = \frac{S}{a + b}$
the whole splits into a + b equal parts
Reducing scale
$\text{scale } 1 : n \ \Rightarrow \ \text{real} = n \cdot \text{on the drawing}$
on a 1 : 50 000 map one centimetre is 500 metres

A room plan at 1 : 200 — the same shape as the real room, with every dimension 200 times smaller.

Frequently asked questions

$A fraction bar and a percent sign rendered as geometric forms, with a circle split into equal orbital slices, against a cosmic nebula in violet and magenta$

Numbers

A percentage is a fraction in disguise — three ways to write one number

1/2, 0.5 and 50% are not three different things but three ways of writing the same number. Here is where the fraction bar, the decimal point and the % sign came from, why 1/3 never ends, why +20% and −20% do not return you to the start, and why fractions are the first serious wall in school mathematics.

11 min