Unitary method is a problem-solving technique used in mathematics to find the value of a single unit (one item) and then use that value to calculate the answer for any number of units.
It’s a simple and effective way to tackle problems involving ratios and proportions, especially when dealing with real-world scenarios like shopping, travel, and others.
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What is Unitary Method?
Unitary Method is a fundamental approach in mathematics used to solve problems related to finding the value of a single unit and then the value of multiple units. It’s based on the concept of proportionality, which means if one quantity increases or decreases, the other does so in a direct or inverse proportion.
This is explained with examples, suppose a car runs 15 km in one litre of petrol then it will run 150 km in 10 litres of petrol. Here, the distance covered by car directly increases with increase in petrol consumption, (assuming initial condition are same).
Unitary method is named so because it focuses on finding the value of one unit first.
Examples of Unitary Method
Few examples where unitary method are used are added below
- If 5 apples cost $10, what is the cost of 1 apple?
- If a car travels 60 km in 1 hour, how far will it travel in 3 hours?
How to Use the Unitary Method?
Here’s a step-by-step approach to using the unitary method:
- Step 1: (Identify the Unit) Recognize what one unit represents in the problem. It could be one apple, one hour of travel, or one liter of paint.
- Step 2: (Relate the Unit to a Known Value) Look for the information given about a specific number of units. This could be the cost of 5 apples, the distance traveled in 2 hours, or the amount of paint needed to cover a certain area.
- Step 3: (Find the Unit Value) Divide the known value (from step 2) by the number of units it represents. This gives you the cost of one apple, the speed traveled per hour, or the amount of paint needed per unit area.
- Step 4: (Calculate the Desired Value) Once you have the unit value, multiply it by the desired number of units to find the answer.
Types of Unitary Method
There are two main types of unitary method problems:
- Direct Variation
- Inverse Variation
Direct Variation
In mathematics, direct variation refers to a relationship between two quantities where one quantity changes in direct proportion to the other. Specifically:
Definition: Two quantities are said to be in direct proportion if an increase in one quantity leads to an increase in the other quantity, provided their respective ratios remain the same.
Equation: In direct variation, we express the relationship as y = kx, where:
- y represents the dependent variable (the one that changes).
- x represents the independent variable (the one that remains constant or is controlled).
- k is the constant of proportionality.
Example 1: Cost of Apples
Suppose you’re buying apples at a grocery store. The cost of apples varies directly with the number of apples you purchase. If the cost of 5 apples is $10, we can set up a direct variation equation:
(10/5) = k
Solving for k, we find that k = 2. Therefore, the cost of x apples can be expressed as
Cost = 2x
Example 2: Work Completion
Suppose two workers, A and B, can complete a particular job together. Their work rates vary directly with the number of workers. If A and B together can complete the job in 72 days, we can set up a direct variation equation:
(1/72) = (1/x) + (1/y)
where x represents the number of days A alone can complete the job, and y represents the number of days B alone can complete the job. Solving for x, we find that x = 120. Therefore, A alone can complete the job in 120 days.
Inverse Variation
Inverse variation, on the other hand, describes a relationship where one quantity increases as the other decreases, and vice versa. Specifically:
Definition: Two quantities are said to be in inverse proportion if an increase in one quantity leads to a decrease in the other quantity, provided their product remains constant.
Equation: In inverse variation, we express the relationship as xy = k, where:
- x and y are the variables.
- k is the constant of proportionality.
Example 1: Pressure and Volume
Consider a gas in a container. The pressure of the gas varies inversely with the volume of the container. If the pressure is 10 atm when the volume is 2 liters, we can set up an inverse variation equation:
10 . 2 = k
Solving for k, we find that k = 20. Therefore, the pressure (P) when the volume (V) is x liters can be expressed as:
P = (20/x)
Remember that direct variation involves a constant ratio, while inverse variation involves a constant product.
Applications of Unitary Method
Various application of Unitary Methods are added below as:
Unitary Method in Ratio and Proportion
Unitary Method helps in solving problems where the ratios of two quantities are given, and we need to find the value of one of the quantities. It forms the foundation for understanding ratios and proportions, where you compare quantities of different units.
Unitary Method in Speed Time and Distance
It is used to calculate the speed, time, or distance when any two of these three variables are known. The unitary method helps solve problems involving speed, time, and distance, all interrelated concepts.
Unitary Method in Rates and Percentages
Unitary method is used to find rates (cost per unit) and calculate percentages based on unit values.
Sample Problems on Unitary Method
Problem 1: If 3 oranges cost $2.10, how much does 1 orange cost?
Solution:
Identify Unit: One orange
Relate Unit to a Known Value: We know the cost of 3 oranges ($2.10).
Find Unit Value: Unit value (cost of 1 orange) = $2.10 / 3 oranges = $0.70 per orange.
Calculate Desired Value: The question asks for the cost of 1 orange, which we already found as $0.70.
Problem 2: A car travels 120 km in 2 hours. What is the speed of the car?
Solution:
Identify Unit: Speed is measured in kilometers per hour (km/h). So, our unit is 1 hour.
Relate Unit to a Known Value: We know the distance traveled in 2 hours (120 km).
Find Unit Value: Speed (per hour) = Total distance / Time taken = 120 km / 2 hours = 60 km/hour.
Calculate Desired Value: Question asks for the speed, which we found as 60 km/hour.
Problem 3: A recipe requires 2 cups of flour for 8 cupcakes. How many cups of flour are needed for 12 cupcakes?
Solution:
Identify Unit: One cupcake
Relate Unit to a Known Value: We know the amount of flour required for 8 cupcakes (2 cups).
Find Unit Value: Flour per cupcake = Total flour / Number of cupcakes = 2 cups / 8 cupcakes = 0.25 cups per cupcake.
Calculate Desired Value: We need to find the flour for 12 cupcakes.
Flour Required = Unit value (flour per cupcake) × Number of cupcakes
Flour Required = 0.25 cups/cupcake × 12 cupcakes = 3 cups.
Problem 4: If 7 meters of cloth cost $14, what is the cost of 3 meters of cloth?
Solution:
Identify Unit: One meter of cloth
Relate Unit to a Known Value: We know the cost of 7 meters of cloth ($14).
Find Unit Value: Cost per meter = Total cost / Number of meters = $14 / 7 meters = $2 per meter.
Calculate Desired Value: Cost of 3 meters = Unit value (cost per meter) × Number of meters
= $2/meter × 3 meters
= $6.
Problem 5: A painter needs 5 liters of paint to cover 20 square meters of wall. How much paint is needed to cover 10 square meters?
Solution:
Identify Unit: Paint needed per square meter
Relate Unit to a Known Value: We know the paint needed for 20 square meters (5 liters).
Find Unit Value: Paint per square meter = Total paint / Area covered = 5 liters / 20 square meters = 0.25 liters per square meter.
Calculate Desired Value: Paint needed for 10 square meters = Unit value (paint per square meter) × Area to be covered
= 0.25 liters/square meter × 10 square meters
= 2.5 liters.
Problem 6: A train travels 360 km in 6 hours. At what speed will it cover 240 km?
Solution:
Identify Unit: Speed (km/h) – We can find the speed in 1 hour and then use it for any time duration.
Relate Unit to a Known Value: We know the distance traveled in 6 hours (360 km).
Find Unit Value: Speed (per hour) = Total distance / Time taken = 360 km / 6 hours = 60 km/hour.
Calculate Desired Value: Since we already found the speed as 60 km/h, this speed will also apply to cover 240 km. The train will cover 240 km at 60 km/hour.
Note: Unitary method is useful for both direct and inverse proportion problems. In problem 6, even though distance reduces (inverse proportion to time), the speed (unit value per hour) remains constant.
Practice Questions on Unitary Method
Q1. If 3 kg of rice costs $27, find the cost of 5 kg of rice.
Q2. A cyclist covers a distance of 45 km in 3 hours. Calculate the speed of the cyclist.
Q3. 7 meters of cloth cost $14. What is the cost of 3 meters of cloth?
Q4. A recipe requires 2 cups of flour for 8 cupcakes. How many cups of flour are needed for 12 cupcakes?
Q5. A bus travels 480 km in 8 hours. How long would it take to travel 360 km?
FAQs on Unitary Method
What is the Unitary Method?
Unitary Method is a technique used to find the value of a single unit from the value of a multiple units and vice versa.
When do we use the Unitary Method?
It is used in problems involving ratios, proportions, and relationships between two quantities.
Can the Unitary Method be used for complex problems?
Yes, it can be applied to more complex problems as long as the relationship between the quantities is proportional.
What is an example of the Unitary Method in daily life?
Calculating the price per unit when shopping or determining the cost of fuel per mile are common examples.
Are there limitations to the unitary method?
For complex problems with multiple variables, algebraic methods might be more efficient.