Question

Prove the sum of two rational numbers is rational where a, b, c, and d are integers and b and d cannot be zero. Steps Reasons 1. A over b plus c over d Given 2. Ad over bd plus cb over bd Multiply to get a common denominator 3. Simplify Fill in the missing step in the proof. ad times cb all over bd ad plus cb all over bd ad minus cb all over bd ad plus cb all over two times bd

Answer 1

`(ad+bc)/(bd)`

Explanation:

Let
`(a)/(b)` and
`(c)/(d)` be two rational numbers, where b and d are not zero and a, b, c and d are integers.

1. Given:

`(a)/(b)+(c)/(d)`

2. Multiply to get a common denominator :

`(a)/(b)+(c)/(d)=(ad)/(bd)+(cb)/(db)`

3. Simplify:

`(a)/(b)+(c)/(d)=(ad)/(bd)+(cb)/(db)=(ad+bc)/(bd)`

4. Since
`b\\eq 0,\ d\\eq 0,` then
`bd\\eq 0.`

If
`a,b,c,d` are integers, then
`bd, ad,bc, ad+bc` are integers too. So the fraction

`(ad+bc)/(bd)`

is a rational number