# Ratio - Basic Question

If ratio of A:B = 1:2

if it is doubled , should it be not 2:4

i see many problems where they are simply multiplying numerator by 2

-

## migrated from mathoverflow.netJun 29 '13 at 6:12

This question came from our site for professional mathematicians.

A ratio remains unaffected/unchanged if we multiply the numerator & the denominator by the same non-zero number –  lab bhattacharjee Jun 29 '13 at 6:19

The statement that $A:B=m:n$ means that $\dfrac{A}B=\dfrac{m}n$. Thus, $A:B=1:2$ means that $\dfrac{A}B=\dfrac12$: $A$ is half is big as $B$. Similarly, $A:B=2:4$ means that $\dfrac{A}B=\dfrac24$: $A$ is two-fourths as big as $B$. And since $\dfrac12=\dfrac24$, so these are the same statement.

If $A:B=m:n$, so that $\dfrac{A}B=\dfrac{m}n$, and you want to double the ratio of $A$ to $B$, you need to double the fraction $\dfrac{m}n$, i.e., multiply it by $2$, in which case you get $$2\cdot\frac{m}n=\frac{2m}n\;,$$ and the statement that $\dfrac{A}B=\dfrac{2m}n$ can be written in ratio notation as $A:B=2m:n$.

More generally, if $k$ is any positive number, $\dfrac{m}n=\dfrac{km}{kn}$, so the statements $A:B=m:N$ and $A:B=km:kn$ say exactly the same thing about the relative sizes of $A$ and $B$.

It may help to look at a very concrete example. Suppose that a certain group consists of $1$ man ($A$) and $2$ women ($B$); clearly $A:B=1:2$. Now suppose that I double the numbers of men and women, getting a group with $2$ men and $4$ women; now $A:B=2:4$. But the actual ratio of men to women is unchanged: the group still has half as many men as women, or one man for every two women. And this would still be the case if I multiplied the numbers of men and women by $100$, to get $A:B=100:200$.

Now go back to the original group of one man and two women. If I double just the number of men, I get a group of $2$ men and $2$ women: $A:B=2:2$. And this really does represent a change in the ratio: instead of $1$ man for every $2$ women, I now have $1$ man for each woman. In fact, it’s correct to write $A:B=1:1$ for this group of $4$, reducing the ratio to lowest term.

Just remember: a ratio written in the $A:B$ notation is really just a fraction, $\dfrac{A}B$.

-
Brian excellent , now when they say ratio of A:B is doubled and if you multiply 2 with numerator , you are doubling the odds in favour of A ? why not A / 2B rather then 2A / B –  abhinav Jun 29 '13 at 6:56
@abhinav The ratio of boys to the girls in a class room is 1:2 .ie there is 1 boy for every two girls.It is the same as 2 boys for every 4 girls or3 boys the for every 6 girls.(In other words multiplying both numerator and denominator doesn't change the ratio. Now examine the expression doubling the ratio. The ratio of boys to the girls has been doubled. This means there are 2 boys for every 2 girls instead of 1 boy for every 2 girl. So, 1:2 is now 2:2(or 1:1). So why not A/2B? The equivalent of A/2B is "the ratio of girls to the boys is doubled" or "ratio of boys to the girls is halved" –  jaseem Jul 1 '13 at 13:20

The ratio is a numerator and denominator: ie a number of denominations, eg of coin.

In the ratio $1:2$, you have one coin, where two make the dollar, ie 50c. When you double both numbers to $2:4$, you now have two coins, where four make the dollar, ie 50c.

If you want to double the amount of money you have, you double the ratio, to $2:2$, ie two 50-cent coins, or $1:1$, one 1 dollar piece.

You can halve the ratio by having the same number of coins, where twice as many make the dollar, so $1:4$ means you have one coin, where four make the dollar.

-