# Learn how to compare fractions

## 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 = 24;
• 24 = 23 × 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 (24; 23 × 3) = 23;
• 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 = 32 × 5;
• 75 = 3 × 52;
• 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 (32 × 5; 3 × 52) = 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;
• #### 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.
• 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.
• 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.