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All courses Math II · N-RN.3 62 of 73
Explain closure properties of rational numbers and irrational results from rational/irrational sums and products.

Classify the sum of two rational numbers as rational

Problem
Is 1/3 + 5/6 rational or irrational?
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Hint

Rewrite the fractions with a common denominator so you can add them directly.

A sum of rational numbers is still rational.

Solution walkthrough

01

Identify both number types

\[1/3~\text{and}~5/6~\text{are}~\text{rational}\]

Each number is a ratio of integers with a nonzero denominator.

02

Use closure under addition

\[\text{rational}+\text{rational}=\text{rational}\]

The rational numbers are closed under addition.

03

Add with a common denominator

\[1/3+5/6=2/6+5/6=7/6\]

Renaming one-third as two-sixths makes the exact sum visible.

04

Classify the result

\[7/6~\text{is}~\text{rational}\]

Seven and six are integers and the denominator is nonzero, so the sum meets the definition.

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Another way

  1. Closure alone is enough once both addends are recognized as rational.

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Common mistake

Do not classify a non-integer fraction as irrational. Rational numbers include every integer ratio with nonzero denominator.