Question

How do you evaluate `\frac { \frac { 1} { 2} - \frac { 2} { 3} } { \frac { 5} { 8} - \frac { 2} { 3} }`?

Answer 1

`(1/2-2/3)/(5/8-2/3)=4`

Explanation:

First of all let us simplify numerator and denominator separately.

Numerator is `1/2-2/3` and as LCD of denominators is `6`

`1/2-2/3=(1xx3)/(2xx3)-(2xx2)/(3xx2)=3/6-4/6=(3-4)/6=-1/6`

Similarly in denominator `5/8-2/3`, as GCD of denominators is `24`

`5/8-2/3=(5xx3)/(8xx3)-(2xx8)/(3xx8)=15/24-16/24=(15-16)/24=-1/24`

Dividing a fraction by a fraction, whether as in `(a/b)/(c/d)` or `a/b-:c/d`, is equivalent to multiplying by the reciprocal or multiplicative inverse of denominator and hence

`(a/b)/color(red)(c/d)=a/bcolor(red)(xx(d/c))`

`(1/2-2/3)/(5/8-2/3)=(-1/6)/(-1/24)=-1/6xx24/(-1)=1/(cancel6^1)xx(cancel24^4)/1=4`

Answer 2

`4`

Explanation:

`(1/2-2/3)/(5/8-2/3)`

L.C.D. of 2,3=6
L.C.D. of 8,3=24

`:.=((3-4)/6)/((15-16)/24)`

`:.=((-1)/6)/((-1)/24)`

`:.=-1/cancel6^1 xx -cancel24^4/1`

`:.=4`