How do you solve `\frac { 31} { b } = \frac { 124} { 208}`?

Divide the top and bottom of the right hand side by the 'obvious' divisor 2 to get `62/104`. Do it again, to get `31/52`. Hey now we have `31/b = 31/52`. We match them off to get `b = 52`.

Since the fraction on the right side of the equation is quite large, my first thought is to simplify it by `color(blue)"cancelling"`

Both 124 and 208 have a `color(blue)"common factors"` of 2 and 4

Using the ' highest common factor' of 4

`"Then " cancel(124)^31/cancel(208)^(52)=31/52`

`"We now have " 31/b=31/52`

`"Since the numerators are equal then the denominators "`
`"will be equal"`

`rArrb=52" is the solution"`

If the above simplification had not been ' obvious' to us then we would have proceeded as follows.

Multiplying both sides by 208. to ' eliminate' the fraction on the right side.

`208xx31/b=cancel(208)^1xx124/cancel(208)^1`

`rArr(208xx31)/b=124`

`"That is " 6448/b=124`

Repeat this step by multiplying both sides by b

`cancel(b)^1xx6448/cancel(b)^1=bxx124`

`rArr124b=6448`

To solve for b, divide both sides by 124

`(cancel(124) b)/cancel(124)=6448/124`

`rArrb=52`