7. Ratio and Proportion

We need to be a bit careful because lots of people use the words "ratio", "fraction" and "proportion" to mean the same thing in everyday speech.

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This makes it difficult when we meet the terms in mathematics, because they are not necessarily used to mean the same thing.

Ratios and Fractions

Example 1

Ethanol or methanol (wood-based methyl alcohol) is sometimes added to gasoline to reduce pollution and cost. Car engines can typically run on a petrol-ethanol mixture in a ratio of `9:1`. The "`9:1`" means that for each nine units of petrol, there is 1 unit of ethanol.

Demonstrating the ratio 9 to 1

For example, if we had 9 L of petrol, we would need 1 L of ethanol.

We can see that altogether we would have 10 L of the mixture.

As fractions, the proportion of each liquid is:

`1/10` ethanol

`9/10` petrol

You can see a more advanced question involving ratios and gasoline in Applied Verbal Problems, in the algebra chapter.

Example 2

Mixing concrete

Concrete is a mixture of gravel, sand and cement, usually in the ratio `3:2:1`.

Demonstrating the ratio 3 to 2 to 1

We can see that there are 3 + 2 + 1 = 6 items altogether. As fractions, the amount of each component of the concrete is:

Gravel: `3/6=1/2`

Sand: `2/6=1/3`

Cement: `1/6`

Example 3

One of the most famous ratios is the ratio of the circumference of a circle to its diameter.

circumference and diameter of a circle

The value of that ratio cannot be determined exactly. It is approximately 3.141592654... We call it:

`pi` (the Greek letter "pi").

See more on Pi.


We can talk about the proportion of one quantity compared to another.

In mathematics, we define proportion as an equation with a ratio on each side.

Example 4

Considering our ethanol/petrol example above, if we have 54 L of petrol, then we need 6 L of ethanol to give us a 9:1 mix.

We could write this as:

`54:6 = 9:1`

We could use fractions to write our proportion, as follows:



Example 5

A normal walking speed is 1 km in 10 min. This is a rate, where we are comparing how far we can go in a certain amount of time.

Our walking rate is equivalent to 6 km/h.

Example 6 - Conversion of Units

A bullet leaves a gun travelling at 500 m/s. Convert this speed into km/h.


If the bullet goes 500 m each second, it means it travels:

60 × 500 = 30,000 m in one minute

60 × 30,000 = 1,800,000 m in one hour

`(1, 800, 000 text(m/h))/(1,000) = 1 800text( km/h)`

So the bullet is travelling 1,800 km/h.

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A famous ratio: `Phi`

Now let's move on to an interesting ratio called `Phi` ("phi"), in Math of Beauty.