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.

The unitary method is a fundamental technique in mathematics that is used to solve problems related to finding the value of a single unit and then using it to find the value of multiple units. This method is especially useful in problems involving ratios, proportions, and rates.

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.