Unitary Method (VCE SSCE General Mathematics): Revision Notes
Unitary Method
What is the unitary method?
The unitary method is a mathematical technique that helps us solve ratio problems by first finding the value of a single unit. This approach is particularly useful when working out prices, as it allows us to calculate the cost of one item and then use this to find the cost of any number of items.
The word "unitary" refers to finding the value of one unit. Once we know the price of one item, we can easily work out the price for any quantity we need.
This method can be applied to a wide range of ratio problems, making it a versatile tool in mathematics.
How the unitary method works
The unitary method follows a straightforward three-step process that breaks down proportional problems into manageable calculations:
Step 1: Find the unit value Divide the total value by the number of items to find the value of one item.
Step 2: Calculate the new total Multiply the unit value by the desired number of items.
Step 3: State the answer Write your final answer clearly with appropriate units.
This method works because proportional relationships can be scaled up or down reliably once we know the unit value. By finding what one item costs, we maintain the ratio while changing the quantity.
Worked example: golf balls
Let's examine a practical example to see how the unitary method works step by step.
Worked Example: Golf Balls
Problem: If 24 golf balls cost $86.40, how much do 7 golf balls cost?
Solution:
Step 1: Find the cost of one golf ball by dividing the total cost by the number of golf balls.
This tells us that each individual golf ball costs $3.60.
Step 2: Multiply the cost of one golf ball by 7 to find the cost of 7 golf balls.
Step 3: Write the final answer.
7 golf balls cost $25.20.
Understanding the process
The unitary method works by systematically breaking down a proportional relationship. Here's what happens at each stage:
Identify the total: We start by recognising the total number of items involved and their combined value. In our example, this was 24 golf balls costing $86.40.
Determine the unit allocation: We use division to find what one single item is worth. This is the crucial "unitary" step that gives the method its name.
Scale to the required amount: We multiply the unit value by however many items we need. This maintains the proportional relationship while giving us our desired quantity.
This systematic approach ensures accuracy and makes complex ratio problems much simpler to solve.
Another example
Let's reinforce our understanding with a simpler scenario:
Worked Example: Simple Calculation
If 15 items cost $45, we can find the cost of 3 items using the unitary method:
Step 1: Find the cost of one item:
Step 2: Find the cost of 3 items:
Notice how we always calculate the value of one item as an intermediate step, even when our final answer involves multiple items. This "going through one" is the key characteristic of the unitary method.
Exam tips
When using the unitary method in examinations, keep these important points in mind:
Exam Success Tips:
- Always show your working clearly by writing out both the division step (finding the unit value) and the multiplication step (finding the new total)
- Check your answer makes sense - if you're calculating the cost of fewer items than in the original problem, your answer should be less than the original total cost
- Remember to include appropriate units in your final answer (such as $ for currency)
- Be careful to divide and multiply by the correct numbers - it's easy to mix up which number goes where under exam pressure
- Label each step clearly so markers can follow your reasoning
Remember!
Key Points to Remember:
- The unitary method finds the value of one unit first before calculating the value of any number of units
- The process uses two operations: division to find the unit value, then multiplication to find the new total
- Always divide the total value by the original quantity to determine the unit price
- Multiply the unit price by your desired quantity to get the final answer
- This method maintains proportional relationships and works for many types of problems, not just pricing questions