Grade 7 → Ratio and Proportion ↓
Unitary Method
The unitary method is a fundamental technique in mathematics, particularly for solving problems involving ratios and proportions. It is a simple but powerful method that allows us to find the value of a single unit from the value of several units, and then use that single unit value to find the value of different units. This method is especially useful in real-life scenarios where we need to calculate cost, quantity, or other measurements based on a given set of conditions.
Basic concepts of unitary method
Basically the unitary method involves two main steps:
- Finding the value of a single unit from the given information. This is often achieved by dividing the total number given by the number of units.
- Using a single unit value to find the value of the number of units desired. This is done by multiplying the single unit value by the number of units we want to find.
Visual explanation with examples
Let's consider a simple example to understand how the unitary method works. Imagine you have 5 apples that cost a total of $10. We want to find out how much one apple costs and how much 8 apples will cost.
Step 1: Find the price of an apple
We start by finding the cost of an apple:
Given: The price of 5 apples is $10.
Price of 1 apple = Total price / Number of apples
= 10 / 5
= $2 per apple
Step 2: Find the cost of 8 apples
Now that we know the cost of one apple, we can easily find the cost of 8 apples:
Price of 8 apples = Price of 1 apple × Number of apples
= 2 × 8
= $16
In the picture above, each rectangle represents an apple. The height of each rectangle represents the price of an apple, which is $2. Thus, for five apples, we get a total price of $10.
Use of the unitary method in other scenarios
The unitary method can be applied in a variety of contexts beyond simple cost calculations. Below are some examples that further illustrate how this method can be used:
Example 1: Speed and time
Suppose a car travels 150 kilometers in 3 hours. We want to find out how much distance it covers in 5 hours.
Given: 150 km in 3 hours.
Distance per hour = Total distance / Number of hours
= 150 / 3
= 50 km/h
Distance in 5 hours = Distance per hour × Number of hours
= 50 × 5
= 250 km
In the above diagram, each segment of the line represents the distance covered in 1 hour. The sum of the five segments represents the total distance of 250 km covered in 5 hours.
Example 2: Cost of goods
If 7 pens cost $21, how much will 4 pens cost?
Given: The cost of 7 pens is $21.
Cost of 1 pen = Total cost / Number of pens
= 21 / 7
= $3 per pen
Cost of 4 pens = Cost of 1 pen × Number of pens
= 3 × 4
= $12
Why is the unitary method useful?
The unitary method is particularly useful because it simplifies the process of working with ratios and proportions. It provides a structured approach to breaking down complex problems into manageable steps. This method is useful not only for mathematical calculations but also for understanding and solving real-world problems involving comparisons and measurements.
More examples of the unitary method
Example 3: Currency conversion
Suppose 100 units of currency A are equivalent to 150 units of currency B. How many units of currency B will you get for 250 units of currency A?
Given: 100 currency A = 150 currency B
1 currency A = 150 / 100 currency B
= 1.5 Currency B
Therefore, 250 currency A = 1.5 × 250 currency B
= 375 Currency B
Example 4: Area calculation
If the length of a rectangle is 8 meters and width is 4 meters then its area will be 32 square meters. If the length of the rectangle is increased by 10 meters then what will be the area?
Original area = length × breadth
= 8 × 4
= 32 square meters
Area of 1 meter length = 32 / 8
= 4 square meters
New length = 10 m
New area = 10 m × 4 m²
= 40 square meters
Here, by finding how much 1 m length contributes to the area, we can easily calculate the new area with the increased length.
Conclusion
In conclusion, the unitary method is a versatile and efficient tool for solving problems involving ratios and proportions. By mastering the concept of working with a single unit, complex questions about quantities, prices, distances, currency conversions and more can be simplified and made accessible to all learners. This method serves as an important foundation in mathematics education, preparing students to tackle a variety of practical and theoretical challenges.
Studying the unitary method helps students develop a logical approach to problem solving, enhance their analytical skills and enable them to apply mathematics with confidence in daily life.
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