## Tutoring: Comparing ordinary fractions

## How to compare two fractions?

### 1. EQUAL DENOMINATORS but unlike numerators fractions

- a) To compare two positive fractions that have EQUAL DENOMINATORS but different numerators (like denominators, unlike numerators), simply compare the numerators: the larger fraction is the one with the larger numerator, ie:
^{24}/_{25}>^{19}/_{25} - b) To compare two negative fractions that have EQUAL DENOMINATORS but different numerators (like denominators, unlike numerators), simply compare the numerators: the larger fraction is the one with the smaller numerator, ie: -
^{19}/_{25}< -^{17}/_{25} - c) To compare two fractions of different signs (one positive and one negative) that have EQUAL DENOMINATORS but different numerators (like denominators, unlike numerators), the rule is that any positive fraction is larger than any negative fraction, ie:
^{2}/_{25}> 0 > -^{1}/_{25}

### 2. EQUAL NUMERATORS but unlike denominators fractions

- a) To compare two positive fractions that have EQUAL NUMERATORS but different denominators (like numerators, unlike denominators), simply compare the denominators: the larger fraction is the one with the smaller denominator, ie:
^{24}/_{25}>^{24}/_{26} - b) To compare two negative fractions that have EQUAL NUMERATORS but different denominators (like numerators, unlike denominators), simply compare the denominators: the larger fraction is the one with the larger denominator, ie: -
^{17}/_{25}< -^{17}/_{29} - c) To compare two fractions of different signs (one positive and one negative) that have EQUAL NUMERATORS but different denominators (like numerators, unlike denominators), the rule is that any negative fraction is smaller than any positive fraction, ie: -
^{1}/_{25}< 0 <^{1}/_{200}

### 3. Different denominators and numerators (unlike denominators and numerators) fractions

- a) To compare two fractions of different signs (one positive and one negative) that have different denominators and numerators (unlike denominators and numerators), the rule is that any negative fraction is smaller than any positive fraction, ie: -
^{11}/_{24}< 0 <^{10}/_{13} - b) To compare two fractions of the same sign (both positive, or both negative), with different numerators and denominators (unlike numerators and denominators), we first check what kind of fractions we have, proper of improper ones.
- The rule is that a positive improper fraction is always larger than a positive proper fraction:
^{27}/_{25}> 1 >^{20}/_{24}. - As for the negative fractions, the rule is that a negative proper fraction is always larger than a negative improper fraction: -
^{2}/_{19}> -1 > -^{40}/_{15}

- The rule is that a positive improper fraction is always larger than a positive proper fraction:
- c) To compare two fractions of the same sign (both positive or both negative) and the same type (proper, improper) that have different denominators and numerators (unlike denominators and numerators), fractions should be built to the same denominator (or if it's easier, to the same numerators). Please see the next paragraph, 3.c)