## 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)

### 3.c) How to compare two fractions of the same sign (positive or negative) and of the same type (proper, improper) that have unlike denominators and numerators?

#### To compare them, we will build up the fractions to the same denominator, but it's also possible to build them up to the same numerator, if this is easing up the calculations.

#### If necessary, we should start by reducing the fractions to lower terms (simplifying the fractions).

- Reduce (simplify) fractions to the lowest terms here: reduce (simplify) ordinary fractions to the lowest terms, online, with explanations.

#### Calculate the least common multiple, LCM, of the fractions' denominators:

- Factor the fractions' denominators down to their prime factors.
- The fractions' denominators least common multiple, LCM, will contain all their unique prime factors, by the largest exponents.
- Calculate the least common multiple, LCM, here, on numere-prime.ro: two numbers' least common multiple, LCM.

#### Build up the fractions to the same denominator, expanding them = multiply both the numerator and the denominator of each fraction by the same natural number, not zero, also called the expanding number:

- Calculate each fraction's expanding number: that is a non-zero number we get by dividing the least common multiple LCM calculated above by each fraction's denominator.
- Expand each fraction: multiply each fraction's both numerator and denominator by the corresponding expanding number calculated above.
- At this point, fractions are built up to the same denominator, so that it's now only a simple task of comparing fractions' numerators.
- The larger fraction will be the one with the larger numerator if the fractions are positive. If they are negative, the larger fraction will be the one with the smaller numerator.

### Example, compare two positive proper fractions that have different denominators and numerators (unlike denominators and numerators), with explanations: ^{16}/_{24} vs. ^{45}/_{75}

#### Reduce (simplify) the first fraction, ^{16}/_{24}, to the lowest terms:

- Factor both the numerator and the denominator down to prime factors:
- 16 = 2
^{4}; - 24 = 2
^{3}× 3; - Factor numbers down to prime factors, online, on numere-prime.ro: factor numbers down to prime factors.
- Calculate the greatest common factor, GCF (the greatest common divisor, GCD), of the numerator and the denominator of the fraction, take all their common prime factors, by their lowest exponents, if there are:
- GCF (16; 24) = GCF (2
^{4}; 2^{3}× 3) = 2^{3}; - Calculate the greatest common factor, GCF, online, on numere-prime.ro: the greatest common factor GCF of two numbers.
- Divide fraction's both numerator and denominator by their greatest common factor GCF:
^{16}/_{24}=^{24}/_{(23 × 3)}=^{(24 ÷ 23)}/_{((23 × 3) ÷ 23)}=^{2}/_{3}

#### Reduce (simplify) the second fraction, ^{45}/_{75}, to the lowest terms:

- Factor both the numerator and the denominator down to prime factors:
- 45 = 3
^{2}× 5; - 75 = 3 × 5
^{2}; - Calculate the greatest common factor, GCF (the greatest common divisor, GCD), of the numerator and the denominator of the fraction, take all their common prime factors, by their lowest exponents, if there are:
- GCF (45; 75) = GCF (3
^{2}× 5; 3 × 5^{2}) = 3 × 5; - Divide fraction's both numerator and denominator by their greatest common factor GCF:
^{45}/_{75}=^{(32 × 5)}/_{(3 × 52)}=^{((32 × 5) ÷ (3 × 5))}/_{((3 × 52) ÷ (3 × 5))}=^{3}/_{5}

#### The reduced fractions are:

^{16}/_{24}=^{2}/_{3};^{45}/_{75}=^{3}/_{5}.

#### Calculate the least common multiple, LCM, of the reduced fractions' denominators:

- Factor fractions' denominators down to prime factors and then multiply all the unique prime factors, by the largest exponents, if there are.
- The denominator of the first fraction,
^{2}/_{3}is 3, it's already a prime number, it cannot be prime factorized anymore; - The denominator of the second fraction,
^{3}/_{5}is 5, it's a prime number, it cannot be prime factorized anymore. - The fractions' denominators least common multiple LCM must contain all their unique prime factors, by the largest exponents: LCM (3, 5) = 3 × 5 = 15.

#### Build up the fractions to the same denominator, expanding them = multiply both the numerator and the denominator of each fraction by the same natural number, not zero, also called the expanding number:

**Each fraction's expanding number**is calculated by dividing the least common multiple LCM by each fraction's denominator:- The first fraction's expanding number is: 15 ÷ 3 = 5;
- The second fraction's expanding number ÷ 15 ÷ 5 = 3.
- To
**build up the fractions to the same denominator**, expand the fractions: multiply each fraction's both numerator and denominator by their corresponding expanding number: - The first fraction is expanding as
^{2}/_{3}=^{(5 × 2)}/_{(5 × 3)}=^{10}/_{15}; - The second fraction is expanded as
^{3}/_{5}=^{(3 × 3)}/_{(3 × 5)}=^{9}/_{15}. - Our fractions having now the same denominator, all we have to do is to simply compare the numerators: 10 > 9 => the larger positive fraction it's the one with the larger numerator,
^{10}/_{15}>^{9}/_{15}, which means that the initial fraction^{16}/_{24}is larger than the initial fraction^{45}/_{75}.