Fractions

Fractions are a way of expressing ratios and are synonymous to division. They are written in the form:

Note

Fractions are used as a nicer way of showing division i.e., \(a \div b = \frac{a}{b}\).

\[ \frac{A}{B} \]

• The expression on the top of the fraction, \(A\), is called the numerator.

• The expression on the bottom, \(B\), is known as the denominator.

Simplifying Fractions

Plenty of fractions can be reduced to simpler forms, for example \(\frac{2}{4}\) is the same as \(\frac{1}{2}\). The best way of simplifying is to find a number that is a factor of the numerator and the denominator, so that it cancels.

Example

Simply \(\frac{350}{1000}\) into its simplest form.

Solution: We first notice that a factor of 10 can be taken out of both the numerator and denominator, leaving us with:

\[ \frac{10 \times 35}{10 \times 100} = \frac{\cancel{10}\times 35}{\cancel{10}\times 100} = \frac{35}{100} \]

the only remaining factors of 35 are 7 and 5. 5 is also a factor of 100, so we can take it out as a factor as well:

\[ \frac{35}{100} = \frac{5 \times 7}{5 \times 20} = \frac{\cancel{5} \times 7}{\cancel{5} \times 20} = \frac{7}{20} \]

We cannot simplify any further because the numerator and denominator share no more factors.

---

We could check this with Python, as the following would return true.

350 / 1000 == 7 / 20

True

---

Note

Don’t fall into the trap when simplifying, shown in this incorrect example:

\[ \frac{a+b}{b+c} = \frac{a+\cancel{b}}{\cancel{b}+c} = \frac{a}{c} \]

This is WRONG! If you are still uncertain, then use \(a = 1\), \(b=2\), \(c=3\) and then be shocked to find that \(\frac{1}{3} = \frac{3}{5}\)!

Example

Simpify \(\frac{2x + 6xy}{4x^2 + 10x^3}\).

Solution: We note first all the terms are even, this means we can take a factor of 2 out. Also all terms have an \(x\) in them, meaning we can take a factor of \(x\) out.

Since we now have a factor of \(2x\) on both numerator and denominator, it will cancel, leaving us with a fraction that can’t be simplified any more. The following steps are illustrated below.

\[ \frac{2x + 6xy}{4x^2 + 10x^3} = \frac{2(x + 3xy)}{2(2x^2 + 5x^3)} = \frac{2x(1 + 3xy)}{2x(2x + 5x^2)} = \frac{\cancel{2x}(1 + 3xy)}{\cancel{2x}(2x + 5x^2)} = \frac{1 + 3y}{2x + 5x^2} \]

---

Python is capable of helping us to simplify these symbolic fractions.

Python Note

Like numpy, sympy is a Python library, this time focused on what is known as symbolic mathematics, i.e., maths with letters as well as numbers.

from sympy import symbols, simplify

x, y = symbols('x y')
numerator = 2 * x + 6 * x * y
denominator = 4 * x ** 2 + 10 * x ** 3
simplify(numerator / denominator)

\[\displaystyle \frac{3 y + 1}{x \left(5 x + 2\right)}\]

---

Fractions in Water

What fraction of water’s mass is hydrogen?

Recall the relative atomic mass of oxygen and hydrogen are 16 and 1 respectively. Remember also that the molecular formula for water is H₂O.

Solution: So the molecular mass of water is equal to \((1 \times 2) + 16 = 18\). The mass of the hydrogen in water is 2, so out of 18 amu (atomic mass units) 2 are hydrogen. Expressed as a fraction this is \(\frac{2}{18}\) which simplifies to \(\frac{1}{9}\). You can check this with some simple Python code.

Multiplying Fractions

Multiplying two fractions together is a simple process, the numerator is the product of the numerators and the denominator is the product of the denominators.

\[ \frac{a}{b}\times\frac{x}{y} = \frac{a \times x}{b \times y}. \]

Generally, when we multiply many fractions together, the numerator is the product of all the numerators and similarly with the denominators. Using the Π notation, we can generalise this as:

Note

If we are multiplying a fraction by a number that isn’t in fraction form, such as an integer or π, then it sill only multiply the numerator. We do this by treating it as a fraction in the form \(\frac{\textrm{number}}{1}\) as shown in the second example here.

\[ \frac{a}{b} \times \frac{c}{d} \times \ldots \times \frac{y}{z} = \frac{a \times c \times \ldots \times y}{b \times d \times \ldots \times z} = \frac{\prod_a^y \textrm{numerators}}{\prod_b^z \textrm{denominators}} \]

Example

Find:

\[ \frac{1}{2} \times \frac{2}{3} \times \frac{4}{5} \times \frac{5}{7} \]

Solution: We find this as:

\[ \frac{1}{2} \times \frac{2}{3} \times \frac{4}{5} \times \frac{5}{7} = \frac{1 \times 2 \times 4 \times 5}{2 \times 3 \times 5 \times 7} = \frac{40}{210} = \frac{4}{21} \]

---

1/2 * 2/3 * 4/5 * 5/7

0.19047619047619047

---

Example

Now find:

\[ 5\sqrt{2} \times \frac{3}{10} \]

Solution: As highlighted in the note:

\[ 5 \sqrt{2} \times \frac{3}{10} = \frac{5\sqrt{2}}{1}\times {3}{10} = \frac{5\sqrt{2}\times 3}{1\times 10} = \frac{15\sqrt{2}}{10} = \frac{3\sqrt{2}}{2} \]

---

import numpy as np

5 * np.sqrt(2) * 3 / 10

2.121320343559643

---

Moles of Water

A chemist prepares a solution containing \(\frac{1}{50}\) mole of propanol in 1000 ml of water. How many moles of propanol are there in a 250 ml aliquot of this solution?

Solution: To calculate this, we need to know what fraction 250 ml is to 1000 ml. That will be \(\frac{250}{1000}\), which simplifies to \(\frac{1}{4}\).

The number of moles taken out is then the product of two fractions:

\[ \frac{1}{50}\times \frac{1}{4} - \frac{1 \times 1}{50 \times 4} = \frac{1}{200} \]

---

This can be easily computed in Python as:

ratio = 1 / 50 #mol
reduction = 250 / 1000

ratio * reduction

0.005

---

Dividing Fractions

The inverse of the fraction is where the numerator and denominator switch, e.g., fraction \(\frac{a}{b}\) has an inverse of \(\frac{b}{a}\).

Division is the inverse of multiplication, so when we divide a fraction, we are essentially multiplying by the inverse of that fraction. So a general example of division of fractions would look like:

\[ \frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a}{b}\times\frac{d}{c} = \frac{ad}{bc} \]

Example

Find the following:

\[ \frac{9}{16} \div \frac{3}{4} \]

Solution: This can be found as:

\[ \frac{9}{16} \div \frac{3}{4} = \frac{9}{16}\times \frac{4}{3} = \frac{9 \times 4}{16 \times 3} = \frac{36}{48} = \frac{3}{4} \]

---

To get the order of operations right in Python, some brackets are necessary.

(9 / 16) / (3 / 4)

0.75

---

Adding and Subtracting Fractions

To be able to add and subtract two fractions, they both need to have the same denominator, called a common demoninator. Take \(\frac{1}{4} + \frac{1}{5}\), for example. We need a new denominator for both in order to add them. The easiest denominator to pick is almost always the product of the two, so in this case \(4\times 5 = 20\).

\[ \frac{1}{4} + \frac{1}{5} = \left(\frac{1}{4} * \frac{5}{5}\right) + \left(\frac{1}{5} \times \frac{4}{4}\right) = \frac{5}{20} + \frac{4}{20} = \frac{9}{20} \]

Subtracting fractions works exactly the same way, the denominators need to be the same, but you then subtract at the end.

Example

Find \(\frac{1}{3} + \frac{1}{6}\). Express your answer as a simplified fraction.

Solution: The first thing is to check whether the denominators are the same and if not, what can we do to fix that. Notice that in this case, we don’t need to change the second fraction, because 3 is a factor of 6. What we would do instead is change the first fraction to have 6 as the denominator by multiplying by \(\frac{2}{3}\).

\[ \frac{1 \times 2}{3 \times 2} + \frac{1}{6} = \frac{2}{6} + \frac{1}{6} = \frac{3}{6} \]

From here, we can see that the fraction can be simplied further. A factor of 3 can be taken out of the top and bottom leaving us with:

\[ \frac{3\times 1}{3\times 2} = \frac{\cancel{3}\times 1}{\cancel{3}\times 2} = \frac{1}{2} \]

---

This can be a single line of Python.

(1 / 3) + (1 / 6)

0.5

---

Example

Express the following as a single fraction:

\[ \frac{1}{x + 1} - \frac{1}{x - 1} \]

Solution: To do this, we will multiply the denominators to get \((x + 1)(x-1)\) as the new denominator.

\[\begin{split} \begin{aligned} & \left(\frac{1}{x+1} \times \frac{x-1}{x-1}\right) - \left(\frac{1}{x-1} \times \frac{x+1}{x+1}\right) \\ = & \frac{x-1}{(x+1)(x-1)} - \frac{x+1}{(x+1)(x-1)} = \frac{(x-1) - (x+1)}{(x+1)(x-1)} = \frac{-2}{(x+1)(x-1)} \end{aligned} \end{split}\]

---

Again, we can use the sympy package to help with this.

x = symbols('x')

simplify(1 / (x + 1) - 1 / (x - 1))

\[\displaystyle - \frac{2}{x^{2} - 1}\]

---