Question

Which statement is true about the sum of two rational numbers? It can always be written as a fraction. It can never be written as a fraction. It can always be written as a repeating decimal. It can never be written a terminating decimal. ?

Answer 1

I agree with the other person's response. The sum is always rational (it can always be written as a fraction of integers)

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Here's a proof:

Let x and y be two rational numbers. This means

x = a/b

y = c/d

where a,b,c,d are integers. The denominators b and d cannot be 0.

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Let's add x and y, then simplify

x+y = (a/b) + (c/d)

x+y = (ad/bd) + (bc/bd)

x+y = (ad+bc)/(bd)

This last expression is in the form p/q

p = ad+bc is an integer

q = bd is also an integer

we have a ratio or fraction of integers, therefore x+y is also rational

Answer 2

It can always be written as a fraction.

Explanation:

The importance of this question is to differentiate rational and irrational numbers.

Rational numbers can be written as a ratio, hence a fraction, or a repeating or terminating decimal.

Irrational numbers cannot be written as a ratio and will not have a terminating or repeating decimal.

So the first statement is true since the sum of two rational numbers will be rational.

The second statement is false since the sum of two rational numbers can absolutely be represented by a fractional ratio.

The third statement is false since the sum of two rational numbers is not always a repeating decimal.

The fourth statement is false since the sum of two rational numbers could be written as a terminating decimal.

Cheers.