Fractions

What is a fraction?
A fraction represents a part of a whole or any number of equal parts. It is also a ratio between two integers where the upper part is the numerator and the lower part the denominator.

When an object is divided into a number of equal parts then each part is called a fraction.

• The top number named the numerator says how many parts we have, 1 circle in our case.

•  The bottom number says how many equal parts the circle was divided into.

Similar and Unlike Fractions

Similar fractions are fractions with the same denominator

$\displaystyle \frac{1}{3},\frac{2}{3},\frac{5}{3},\frac{{17}}{3},\frac{{20}}{3}$

Unlike fractions are fractions with different denominator

$\displaystyle \frac{2}{3},\frac{2}{5},\frac{5}{{11}},\frac{{21}}{4},\frac{9}{{12}}$

Equivalent Fractions
Equivalent Fractions are fractions that have the same value but different numerators and denominators.

$\displaystyle \frac{{12}}{{16}}$ is equal to $\displaystyle \frac{3}{4}$

Because if we reduce $\displaystyle \frac{{12}}{{16}}$ to lowest terms we get$\displaystyle \frac{3}{4}$

$\displaystyle \frac{{12}}{{16}}=\frac{{2\times 2\times 3}}{{2\times 2\times 2\times 2}}=\frac{3}{{2\times 2}}=\frac{3}{4}$

How to reduce a fraction to the lowest terms?

A fraction is in its lowest term when the denominator and the numerator have no common factor other then 1.

To reduce the fraction to the lowest term:

Firstly, find all the factors of the numerator and the denominator.

Secondly, find the highest common factor (HCF) of these two numbers.

Thirdly, reduce the fraction by dividing both the denominator and the numerator with their highest common factor.

Example: Reduce the fraction $\displaystyle \frac{{20}}{{55}}$ to its lowest term.

Finding all the factors of 20 and 55.

20 = 2 x 2 x 5 and 55 = 5 x 11.

HCF (20, 55) = 5.

Now we reduce the fraction by dividing both the denominator and the numerator with their highest common factor that is 5.

$\displaystyle \frac{{20}}{{55}}=\frac{4}{{11}}$

$\displaystyle \frac{4}{{11}}$ is its lowest term since 4 and 11 have no common factors other than 1.

Proper, Improper and Mixed Fraction

Proper Fraction

Proper fraction is a fraction where the numerator is less the denominator

$\displaystyle \frac{1}{3},\frac{2}{5},\frac{6}{{11}},\frac{7}{{20}},\frac{{13}}{{14}}$

Improper Fraction

Improper fraction is a fraction where the numerator is greater than denominator

$\displaystyle \frac{5}{3},\frac{7}{4},\frac{{11}}{9},\frac{{23}}{{11}},\frac{{129}}{{89}}$

Mixed Fraction

Mixed fraction is a fraction that includes both a whole number and a proper fraction

$\displaystyle 2\frac{1}{2},3\frac{2}{7},5\frac{1}{6},7\frac{{20}}{{98}}$

How to convert an improper fraction into a mixed fraction and a mixed fraction into improper fraction?

Improper fraction $\displaystyle \to$ Mixed Fraction

• Divide the numerator by the denominator.
•  The whole number we get we write it in front of the fraction.
•  The reminder we get is the numerator of our fraction.
• The denominator stays the same.

Convert $\displaystyle \frac{{17}}{3}$ into a mixed fraction.

$\displaystyle 17\div 3=5$  with a reminder of 2.

We write in front of the fraction the number 5 and the reminder 2 above the denominator which stays the same:  $\displaystyle 5\frac{2}{3}$

$\displaystyle \frac{{17}}{3}=5\frac{2}{3}$

Mixed Fraction $\displaystyle \to$  Improper fraction

• Multiply the whole number by the denominator.

• Add the total to the numerator.

• The denominator will stay the same.

Convert $\displaystyle 5\frac{1}{6}$ into an improper fraction.

The whole number is 5 and the denominator is 6 so: $\displaystyle 5\times 6=30$.
We add the product 30 to the numerator of our fraction: 1 + 30 = 31.
Since the denominator stays the same our improper fraction is: $\displaystyle \frac{{31}}{6}$.
$\displaystyle 5\frac{1}{6}=\frac{{31}}{6}$

Comparing Fractions

Three cases of comparing fractions are:

Fractions with similar DenominatorsThe fraction with the smaller numerator is smaller, meanwhile the fraction with the larger numerator is larger.

Which fraction is smaller, $\displaystyle \frac{3}{7}$ or $\displaystyle \frac{6}{7}$?

We see that our fractions have the same denominator so we compare the numerators.

Since 3 < 6 then  $\displaystyle \frac{3}{7}<\frac{6}{7}$.

Fractions with similar Numerators: The fraction with the smaller denominator is larger and conversely the fraction with the larger denominator is smaller.

Which fraction is larger,$\displaystyle \frac{9}{2}$ or $\displaystyle \frac{9}{7}$?

We see that the fractions have the same numerators so we compare the denominators.

Since 2 < 7 then $\displaystyle \frac{9}{2}>\frac{9}{7}$.

Fractions with unlike Denominators. We can´t compare them as in the cases above so we have to follow this rules:

1) Firstly we find the least common denominator

2) We convert the fraction into equivalent fraction with the LCD.

3) Since they have the same denominator we compare the numerators as in the first case

Which fraction is smaller?  $\displaystyle \frac{3}{5}$ or $\displaystyle \frac{7}{9}$

Since the fractions have different denominators we find the least common denominator.

5 = 1 x 5
9 = 3 x 3
LCD(5, 9) = 1 x 3 x 3 x 5 = 45

We convert our fractions into equivalent one´s with the denominator 45

$\displaystyle \frac{3}{5}\times \frac{9}{9}=\frac{{27}}{{45}}$

$\displaystyle \frac{7}{9}\times \frac{5}{5}=\frac{{35}}{{45}}$

Since the fractions have same denominator then we compare the numerators.

Since 27 < 35 then $\displaystyle \frac{{27}}{{45}}<\frac{{35}}{{45}}$ or $\displaystyle \frac{3}{5}<\frac{7}{9}$