## How to compare two fractions?

### 1. Fractions that have different signs:

- Any positive fraction is larger than any negative fraction:
- ie:
^{4}/_{25}> -^{19}/_{2}

### 2. A proper and an improper fraction:

- Any positive improper fraction is larger than any positive proper fraction:
- ie:
^{44}/_{25}> 1 >^{19}/_{200} - Any negative improper fraction is smaller than any negative proper fraction:
- ie: -
^{44}/_{25}< -1 < -^{19}/_{200}

### 3. Fractions that have both like numerators and denominators:

- The fractions are equal:
- ie:
^{89}/_{50}=^{89}/_{50}

### 4. Fractions that have unlike (different) numerators but like (equal) denominators.

**Positive fractions**: compare the numerators, the larger fraction is the one with the larger numerator:- ie:
^{24}/_{25}>^{19}/_{25} **Negative fractions**: compare the numerators, the larger fraction is the one with the smaller numerator:- ie: -
^{19}/_{25}< -^{17}/_{25}

### 5. Fractions that have unlike (different) denominators but like (equal) numerators.

**Positive fractions**: compare the denominators, the larger fraction is the one with the smaller denominator:- ie:
^{24}/_{25}>^{24}/_{26} **Negative fractions**: compare the denominators, the larger fraction is the one with the larger denominator:- ie: -
^{17}/_{25}< -^{17}/_{29}

### 6. Fractions that have different denominators and numerators (unlike denominators and numerators).

- To compare them, fractions should be built up to the same denominator (or if it's easier, to the same numerator).
#### 1) If necessary, start by reducing the fractions to the lowest terms (simplify the fractions to their simplest form equivalents).

- Factor the numerator and the denominator of each fraction as products of primes, in exponential form.
- Factor numbers as products of primes, in exponentional form, online
- Calculate the greatest common factor, GCF, of the numerator and the denominator of each fraction: multiply all their unique common prime factors, by the lower exponents.
- We will calculate a GCF for each fraction.
- Each GCF calculated will be used to divide both the numerator and the denominator of each fraction, in order to simplify that fraction.
- Calculate the greatest common factor, GCF, online
- Divide both the numerator and the denominator of each fraction by their greatest common factor, GCF.
- At this moment, the fractions are reduced to their lowest terms (or, if you like, simplified to their simplest form equivalent, irreducible).
- By reducing a fraction its value does not change, only an equivalent fraction is produced, which is easier to work with.
- Reduce fractions to their lowest terms, online, with explanations.

#### 2) Calculate the least common multiple, LCM, of all the fractions' denominators (the least common denominator):

- LCM will be the new denominator of the equivalent fractions to be compared.
- In this particular case, LCM is also called the
**least common denominator**. - Factor all the fractions' denominators as products of primes in exponential form.
- To calculate LCM multiply all the unique prime factors of the denominators, by the largest exponents.
- Calculate the least common multiple of two numbers, LCM, online.

#### 3) Build up the fractions to the same denominator, by expanding them.

**Expanding a fraction**= multiply both the numerator and the denominator of a fraction by the same natural number, not zero, called the**expanding number**, in order to build an equivalent fraction.- Calculate each fraction's
**expanding number**: that is a non-zero number we get by dividing the least common multiple, LCM, by each fraction's denominator. - Expand each fraction: multiply each fraction's both numerator and denominator by the corresponding expanding number calculated above.
- The value of a fraction is not changing by expanding it, only an equivalent fraction that is easier to work with is being produced.

#### 4) Compare the numerators of the new equivalent fractions.

- At this point, the fractions are built up to the same denominator, so it is 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.

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

#### 1) Start by reducing the fractions to the lowest terms (simplify the fractions to their simplest form equivalents).

- Fraction
^{16}/_{24}:- Factor both the numerator and the denominator as products of primes in exponential form:
- 16 = 2
^{4}; - 24 = 2
^{3}× 3; - Calculate the greatest common factor, GCF, of the numerator and the denominator, multiply all their unique common prime factors, by their lower exponents, if there are:
- GCF (16; 24) = GCF (2
^{4}; 2^{3}× 3) = 2^{3}; - Divide both the numerator and the denominator by GCF:
^{16}/_{24}=^{24}/_{(23 × 3)}=^{(24 ÷ 23)}/_{((23 × 3) ÷ 23)}=^{2}/_{3}.

- Fraction
^{45}/_{75}:- Factor both the numerator and the denominator as products of primes in exponential form:
- 45 = 3
^{2}× 5; - 75 = 3 × 5
^{2}; - Calculate the greatest common factor, GCF, of the numerator and the denominator, multiply all their unique common prime factors, by their lower exponents, if there are:
- GCF (45; 75) = GCF (3
^{2}× 5; 3 × 5^{2}) = 3 × 5; - Divide both the numerator and the denominator by GCF:
^{45}/_{75}=^{(32 × 5)}/_{(3 × 52)}=^{((32 × 5) ÷ (3 × 5))}/_{((3 × 52) ÷ (3 × 5))}=^{3}/_{5}.

- The reduced (simplified) fractions are:
^{16}/_{24}=^{2}/_{3};^{45}/_{75}=^{3}/_{5}.

- The simplified fractions are equivalent to the original ones, having the same values as the original ones:
^{16}/_{24}≈ 0.67;^{2}/_{3}≈ 0.67;^{45}/_{75}= 0.6;^{3}/_{5}= 0.6;

- Fraction
#### 2) Calculate the least common multiple, LCM, of all the denominators of the reduced fractions (the least common denominator):

- LCM will be the new denominator of the equivalent fractions to be compared.
- LCM is also called the
**least common denominator**. - To calculate LCM, factor all the denominators as products of primes in exponential notation and then multiply all their unique prime factors by the larger exponents.
- The denominator of the fraction
^{2}/_{3}is 3, it's already a prime number, it cannot be broken down to any other primes; - The denominator of the fraction
^{3}/_{5}is 5, it's also a prime number, it also cannot be broken down to other primes.

- The denominator of the fraction
- LCM (3, 5) = 3 × 5 = 15.

#### 3) Build up the fractions to the same denominator, by expanding them.

**Expanding a fraction**= multiply both the numerator and the denominator of a fraction by the same natural number, not zero, called the**expanding number**, in order to build an equivalent fraction.- Each fraction's
**expanding number**is calculated by dividing the least common multiple LCM by each fraction's denominator:- For the first fraction: 15 ÷ 3 = 5;
- For the second fraction: 15 ÷ 5 = 3.

- Expand each fraction by its expanding number:
- The first fraction:
^{2}/_{3}=^{(5 × 2)}/_{(5 × 3)}=^{10}/_{15}; - The second fraction:
^{3}/_{5}=^{(3 × 3)}/_{(3 × 5)}=^{9}/_{15}.

- The first fraction:
- As in the case of simplifying, expanding fractions does not change their values, it only produces some equivalent fractions which are easier to work with:
^{2}/_{3}≈ 0.67;^{10}/_{15}≈ 0.67;^{3}/_{5}= 0.6;^{9}/_{15}= 0.6.

#### 4) Compare the numerators of the new equivalent fractions.

- Our fractions now have the same denominator, all we have to do is to compare the numerators:
- 10 > 9 =>
^{10}/_{15}>^{9}/_{15}=>^{16}/_{24}>^{45}/_{75}.