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#