# Why does 5/6 divided by 1/2 equal 5/6 multiplied by 2/1?

embizze | High School Teacher | (Level 1) Educator Emeritus

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Show that `5/6-: 1/2 = 5/6 * 2/1` :

(1) One definition of division is multiplication by the multiplicative inverse. Thus dividing by 2 is the same as multiplying by `1/2` .

By this definition it is clear that dividing by `1/2` is the same as multiplying by the multiplicative inverse, which in this case is `2/1` .

(The multiplicative inverse for a real number `a` is the number `b` where `a*b=b*a=1` . In general, this is the reciprocal of `a` or `1/a` since `a*1/a=1/a*a=1` for all real numbers except zero. This is one of the reasons that you cannot divide by zero -- it does not have a multiplicative inverse.)

(2) Write the problem as a "compound fraction":

`(5/6)/(1/2)` . In general we avoid compound fractions -- usually we try to eliminate the fraction in the denominator. We can accomplish this by multiplying numerator and denominator by `2/1` :

`(5/6)/(1/2)=(5/6)/(1/2)*(2/1)/(2/1)=(5/6*2/1)/(1/2*2/1)=5/6*2/1`

(3) We can rewrite the division problem as a multiplication problem. For example if `42-:7=6` then `42=6*7` . A closer example is to find `10-:1/2` . Then when we find an answer, say x, then we know that `10=1/2*x` or `x=20` . Thus if `5/6-:1/2=x` then `5/6=1/2*x` and x must be `10/6=5/6*2/1`