Operations and Priority in Mathematics

Understanding Multiplication and Division

Multiplication is fundamentally defined as repeated addition. If you have multiple instances of the same value, you can multiply the value by the number of instances to find the total.

Example: 10 identical boxes each have a mass of $1.8 \text{ kg}$. What is the total mass of the 10 boxes?

You can solve this by writing down $1.8$ ten times and adding them all together, or you can use multiplication: $1.8 \text{ kg} \times 10 = 18 \text{ kg}$.

Division is the inverse process, representing the repeated subtraction of the same number from a total. It determines how many times a specific value fits into another.

Example: A container of $20 \text{ litres}$ of lemonade is used to fill glasses, each holding $250 \text{ ml}$ when full. How many glasses can be filled from the container?

First, ensure units are consistent: $20 \text{ l} = 20,000 \text{ ml}$. The problem can be solved by seeing how many times $250$ can be subtracted from $20,000$, or by dividing $20,000$ by $250$:

$\frac{20000}{250} = \frac{2000}{25} = \frac{400}{5} = 80$.

Using Indices

Indices are used to represent repeated multiplication or division by the same number. This notation simplifies long expressions into a compact form.

Example: Write $25 \times 10 \times 25 \times 10 \times 25 \times 10 \times 10$ in index form.

First, reorder the terms to group identical numbers together: $25 \times 25 \times 25 \times 10 \times 10 \times 10 \times 10$.

Then, express each group as a power: $25 \times 25 \times 25 = 25^3$ and $10 \times 10 \times 10 \times 10 = 10^4$. The final index form is $25^3 \times 10^4$.

Inverse Operations

Inverse operations are pairs of operations that undo each other. Multiplication is the inverse of division, and addition is the inverse of subtraction. These relationships are essential for solving equations where a value is unknown.

Example: $34$ is added to a number and the result is $78$. What was the number?

To find the original number, perform the inverse operation of addition (subtraction) on the result:

$? \rightarrow +34 \rightarrow 78$

$44 \leftarrow -34 \leftarrow 78$

Algebraically, if $34 + x = 78$, then $x = 78 - 34 = 44$.

Example: A number is divided by $8$ and the result is $104$. What was the number?

To find the original number, perform the inverse operation of division (multiplication) on the result:

$? \rightarrow \div 8 \rightarrow 104$

$832 \leftarrow \times 8 \leftarrow 104$

Algebraically, if $\frac{x}{8} = 104$, then $x = 104 \times 8 = 832$.

Simplification by Cancellation

Calculations and expressions can be simplified using cancellation, which involves dividing both the numerator and the denominator of a fraction by the same number or algebraic term. This should be done before evaluating the final calculation to keep numbers manageable.

Example: Write down a calculation that is equivalent to $\frac{21 \times 55}{63 \times 85}$.

We can cancel common factors to simplify:

$\frac{21 \times 55}{63 \times 85} = \frac{21 \times 55}{(3 \times 21) \times 85} = \frac{1 \times 55}{3 \times 85} = \frac{1 \times (5 \times 11)}{3 \times (5 \times 17)} = \frac{1 \times 11}{3 \times 17}$.

Here, we cancelled the $21$ and $63$ by dividing both by $21$, and cancelled $55$ and $85$ by dividing both by $5$.

Priority of Operations

When a calculation contains multiple operations, they must be performed in a specific order known as the convention for priority of operations. This is often remembered by acronyms like BIDMAS or PEMDAS.

1. Brackets: Perform calculations inside brackets first. If there are nested brackets, work from the innermost to the outermost. Alternatively, multiply out the brackets.
2. Indices: Calculate all powers, roots, and reciprocals. Note that the reciprocal of $x$ is $\frac{1}{x}$.
3. Division and Multiplication: These operations have equal priority. Carry them out as they appear from left to right.
4. Addition and Subtraction: These operations also have equal priority. Carry them out as they appear from left to right.

Examples for the Order of Operations

Calculate: $3 + 6 \times 9$

Multiplication takes priority over addition: $3 + 54 = 57$.

Calculate: $3 - (6 - 9)$

Evaluate the bracket first: $3 - (-3)$. Subtracting a negative is the same as adding: $3 + 3 = 6$.

Calculate: $8 - 3(6 - 8)$

First, evaluate the bracket: $8 - 3(-2)$. The number outside the bracket multiplies the result: $8 - (-6) = 8 + 6 = 14$.

Calculate: $8 - 3(6 - 8)^2$

Evaluate the bracket first: $8 - 3(-2)^2$. Next, apply the index to the bracketed term (not the $3$): $8 - 3 \times 4$. Then perform the multiplication: $8 - 12$. Finally, perform the subtraction to get $-4$.