# Ratios in the form `1:n` or `n:1`

If you are ever asked to express a ratio in the form `1:n` or `n:1` just treat it like a directly proportional ratio.

**Example 1**

Express `6:10` in the form `1:n`

So the ratio is

`6:10`

and we need to work out

`1:n`

(see our section on percentages for full explanation)

`6:10`

`1:n`

Now put a divide sign in

`6/1=10/n`

Rearranging the formula

`n=(10times1)/6`

`n=10/6`

simplifying the fraction

`n=5/3`

So the ratio in the form `1:n` is

`1:5/3`

**Example 2**

The ratio of girls to boys in a school is:

`800` boys to `1200` girls

What is the ratio in the form `n:1`?

So the ratio is

`800:1200`

and we need to work out

`n:1`

(see our section on percentages for full explanation)

`800:1200`

`n:1`

Now put a divide sign in

`800/n=1200/1`

Rearranging the formula

`(800times1)/1200=n` or

`n=(800times1)/1200`

`n=800/1200`

simplifying the fraction

`n=(8cancel00)/(12cancel00)`

`n=8/12`

simplify the fraction further

`n=4/6`

simplify the fraction further

`n=2/3`

This will not simplify any further so the answer is

`2/3:1`

**Example 3**

A chief is cooking a meal for many and she knows that she has to buy `3` carrots to `8` parsnips. She also knows that the ratio of parsnips to broccoli is `6` parsnips to `1` broccoli. Write out the ratio in the form `A:B:C`

The ratio is `3` carrots `:8` parsnips and

`6` parsnips `:1` broccoli

Line these up we would have

`3` carrots `:8` parsnips and `6` parsnips `:1` broccoli

We need to make the second ratio have 8 parsnips i.e. the same as the first ratio.

Therefore

`6` parsnips `:1`

and we need to work out

`8` parsnips `:n`

(see our section on percentages for full explanation)

`6:1`

`8:n`

Now put a divide sign in

`6/8=1/n`

rearranging the formula

`n=(1times8)/6`

`n=8/6`

Simplify the fraction

`n=4/3`

So the new ratio is

`8` parsnips `:4/3` broccoli

So the full ratio is

`3` carrots `:8` parsnips and `8` parsnips `:4/3` broccoli

Which is the same as

`3` carrots `:8` parsnips `:4/3` broccoli

# Ratios in the form `1:n` or `n:1`

In order that a ratio is written in the form `1:n` we must make the left hand side equal to one.

In order that a ratio is written in the form `n:1` we must make the right hand side equal to one.

**NOTE:**

`n` can be expressed as a decimal or a fraction

**Example 1**

Express `6:10` in the form `1:n`

So the ratio is

`6:10`

and we need to work out

`1:n`

(see our section on percentages for full explanation)

`6:10`

`1:n`

Now put a divide sign in

`6/1=10/n`

Rearranging the formula

`n=(10times1)/6`

`n=10/6`

simplifying the fraction

`n=5/3`

So the ratio in the form `1:n` is

`1:5/3`

**Example 2**

Express `6:10` in the form `n:1`

So the ratio is

`6:10`

and we need to work out

`n:1`

(see our section on percentages for full explanation)

`6:10`

`n:1`

Now put a divide sign in

`6/n=10/1`

Rearranging the formula

`(6times1)/10=n` or

`n=(6times1)/10`

`n=6/10`

simplifying the fraction

`n=3/5`

So the ratio in the form `n:1` is

`3/5:1`

**Example 3**

The ratio of girls to boys in a school is:

`800` boys to `1200` girls

What is the ratio in the form `n:1`?

So the ratio is

`800:1200`

and we need to work out

`n:1`

(see our section on percentages for full explanation)

`800:1200`

`n:1`

Now put a divide sign in

`800/n=1200/1`

Rearranging the formula

`(800times1)/1200=n` or

`n=(800times1)/1200`

`n=800/1200`

simplifying the fraction

`n=(8cancel00)/(12cancel00)`

`n=8/12`

simplify the fraction further

`n=4/6`

simplify the fraction further

`n=2/3`

This will not simplify any further so the answer is

`2/3:1`

**Example 4**

A chief is cooking a meal for many and she knows that she has to buy `3` carrots to `8` parsnips. She also knows that the ratio of parsnips to broccoli is `6` parsnips to `1` broccoli. Write out the ratio in the form `A:B:C`

The ratio is `3` carrots `:8` parsnips and

`6` parsnips `:1` broccoli

Line these up we would have

`3` carrots `:8` parsnips and `6` parsnips `:1` broccoli

We need to make the second ratio have 8 parsnips i.e. the same as the first ratio.

Therefore

`6` parsnips `:1`

and we need to work out

`8` parsnips `:n`

(see our section on percentages for full explanation)

`6:1`

`8:n`

Now put a divide sign in

`6/8=1/n`

rearranging the formula

`n=(1times8)/6`

`n=8/6`

Simplify the fraction

`n=4/3`

So the new ratio is

`8` parsnips `:4/3` broccoli

So the full ratio is

`3` carrots `:8` parsnips and `8` parsnips `:4/3` broccoli

Which is the same as

`3` carrots `:8` parsnips `:4/3` broccoli