Ratio - Basic Question

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

please can some one explain

Answer 1

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 $B$.

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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, $\dfrac{A}B$.

Answer 2

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.