Grade 7
  1. Number System  1.1. Integers  1.1.1. Properties of Integers  1.1.2. Operations on Integers  1.1.3. Additive and Multiplicative Inverses  1.1.4. Applications of Integers  1.2. Rational Numbers  1.2.1. Definition of Rational Numbers  1.2.2. Properties of Rational Numbers  1.2.3. Operations on Rational Numbers  1.2.4. Standard Form of Rational Numbers  1.3. Decimals  1.3.1. Operations on Decimals  1.3.2. Converting Between Fractions and Decimals  1.4. Powers and Exponents  1.4.1. Laws of Exponents  1.4.2. Simplifying Expressions with Exponents  1.4.3. Scientific Notation  1.5. Roots  1.5.1. Square Roots  1.5.2. Cube Roots
  2. Algebra  2.1. Expressions  2.1.1. Algebraic Expressions  2.1.2. Simplifying Expressions  2.1.3. Evaluating Expressions  2.2. Linear Equations  2.2.1. Solving Simple Equations  2.2.2. Applications of Linear Equations  2.3. Algebraic Identities  2.3.1. Expanding Using Identities  2.3.2. Simplifying Using Identities
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Simplifying Ratios  3.1.3. Equivalent Ratios  3.2. Proportion  3.2.1. Concept of Proportion  3.2.2. Direct Proportion  3.2.3. Inverse Proportion  3.3. Unitary Method  3.3.1. Solving Problems Using the Unitary Method
  4. Geometry  4.1. Lines and Angles  4.1.1. Types of Angles  4.1.2. Pairs of Angles  4.1.3. Properties of Parallel Lines and a Transversal  4.1.4. Angle Bisectors  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Angle Sum Property  4.2.3. Exterior Angle Property  4.2.4. Pythagoras Theorem  4.3. Congruence of Triangles  4.3.1. Criteria for Congruence  4.3.2. Applications of Congruence  4.4. Quadrilaterals  4.4.1. Types of Quadrilaterals  4.4.2. Properties of Parallelograms  4.4.3. Special Quadrilaterals  4.4.4. Properties of Rectangles and Rhombuses
  5. Mensuration  5.1. Perimeter and Area  5.1.1. Perimeter of Plane Figures  5.1.2. Area of Plane Figures  5.1.3. Area of a Trapezium  5.1.4. Area of a Parallelogram  5.1.5. Area of a Circle  5.1.6. Area of Composite Figures  5.2. Surface Area and Volume  5.2.1. Cubes and Cuboids  5.2.2. Surface Area of Cylinders  5.2.3. Volume of Prisms and Cylinders  5.2.4. Volume and Surface Area of Cones
  6. Data Handling  6.1. Data Collection  6.1.1. Organizing Data  6.1.2. Frequency Distribution  6.2. Graphical Representation of Data  6.2.1. Bar Graphs  6.2.2. Double Bar Graphs  6.2.3. Pie Charts  6.2.4. Histograms  6.2.5. Line Graphs  6.3. Probability  6.3.1. Simple Events  6.3.2. Experimental Probability  6.3.3. Theoretical Probability
  7. Practical Geometry  7.1. Construction of Triangles  7.1.1. Constructing Triangles Given Side Lengths  7.1.2. Constructing Triangles Given Side and Angle  7.1.3. Constructing Right-Angled Triangles  7.2. Construction of Quadrilaterals  7.2.1. Quadrilaterals Given Side Lengths and Angles  7.2.2. Constructing Parallelograms and Rectangles

Grade 7 → Ratio and Proportion → Unitary Method ↓


Solving Problems Using the Unitary Method


The unitary method is a powerful mathematical tool used to solve problems involving ratios and proportions. It simplifies complex problems by breaking them down into manageable steps, where the value of a unit is first calculated and then used to find the required value. This method is widely used in everyday problems, such as calculating prices, determining quantities, and many other situations. Let's dive deeper into the unitary method and explore its applications.

What is unitary method?

The unitary method is a technique used in mathematics that involves finding the value of a single unit from a given set of values and then using that value to solve various kinds of problems. It mainly revolves around first determining the value of a unit and then multiplying it to find the value of another quantity.

For example, if 5 apples cost $10, we would first find the price of one apple and then use that to find the price of any number of apples.

    Cost of 5 apples = $10 Cost of 1 apple = $10 / 5 = $2 Therefore, the cost of 8 apples = $2 * 8 = $16
    Cost of 5 apples = $10 Cost of 1 apple = $10 / 5 = $2 Therefore, the cost of 8 apples = $2 * 8 = $16

Steps involved in unitary method

To solve a problem using the unitary method, follow these steps:

  1. Identify known values: Determine the values given in the problem, such as cost, quantity, or measurement.
  2. Find the price of one unit: Divide the total price by the number of units to find the price of one unit.
  3. Find the required value: Multiply the value of one unit by the number of units to find the value of the desired quantity.

Applications in proportion

Ratios are a way of comparing two quantities and expressing the relationship between them. The unitary method is often used to solve problems involving ratios, in which the value of one unit is first calculated and then it is used to find other values.

Suppose the ratio of the number of dogs and cats we have is 2:3. If we know there are 10 dogs, how many cats will there be?

    Ratio of dogs to cats = 2:3 Number of dogs = 10 First, calculate the value of one part using the ratio: Total parts = 2 + 3 = 5 The value of one part (dogs) = 10 / 2 = 5 Now, calculate the number of cats using the value of one part: Number of cats = 5 * 3 = 15
    Ratio of dogs to cats = 2:3 Number of dogs = 10 First, calculate the value of one part using the ratio: Total parts = 2 + 3 = 5 The value of one part (dogs) = 10 / 2 = 5 Now, calculate the number of cats using the value of one part: Number of cats = 5 * 3 = 15

Applications in proportion

Proportions are equations that express two ratios as equals. The unitary method can solve problems involving direct and inverse proportions by finding the value of a unit and then obtaining the required value.

For example, if 5 meters of cloth costs $20, what will be the price of 8 meters of the same cloth?

    Cost of 5 meters = $20 First, find the cost of 1 meter: Cost of 1 meter = $20 / 5 = $4 Now, find the cost of 8 meters: Cost of 8 meters = $4 * 8 = $32
    Cost of 5 meters = $20 First, find the cost of 1 meter: Cost of 1 meter = $20 / 5 = $4 Now, find the cost of 8 meters: Cost of 8 meters = $4 * 8 = $32

Illustrative example: Using visuals

To understand the unitary method better, consider the following example. Suppose you want to find the number of stars, if there are 12 stars at 4 locations.

Given: 12 stars cover 4 places. If you want to know how many stars cover 6 places?

    Stars in 4 spaces = 12 First, calculate stars in 1 space = 12 / 4 = 3 Stars in 6 spaces = 3 * 6 = 18
    Stars in 4 spaces = 12 First, calculate stars in 1 space = 12 / 4 = 3 Stars in 6 spaces = 3 * 6 = 18

Practical example using the unitary method

Example 1: Finding speed

If a car travels 150 kilometres in 3 hours, what is its speed per hour?

    Distance = 150 kilometers Time = 3 hours First, find speed per hour: Speed per hour = 150 km / 3 hours = 50 km/h
    Distance = 150 kilometers Time = 3 hours First, find speed per hour: Speed per hour = 150 km / 3 hours = 50 km/h

Example 2: Determining the total cost

If the cost of 7 notebooks is $21, find the cost of 15 notebooks.

    Cost of 7 notebooks = $21 First, find the cost of 1 notebook: Cost of 1 notebook = $21 / 7 = $3 Now, find the cost of 15 notebooks: Cost of 15 notebooks = $3 * 15 = $45
    Cost of 7 notebooks = $21 First, find the cost of 1 notebook: Cost of 1 notebook = $21 / 7 = $3 Now, find the cost of 15 notebooks: Cost of 15 notebooks = $3 * 15 = $45

Example 3: Calculating quantity

If 12 litres of paint covers 96 square metres of area, how much paint will be required to cover 120 square metres of area?

    Paint for 96 sq meters = 12 liters First, calculate paint needed for 1 sq meter: Paint for 1 sq meter = 12 liters / 96 sq meters = 0.125 liters/sq meter Now, calculate paint required for 120 sq meters: Paint for 120 sq meters = 0.125 liters/sq meter * 120 sq meters = 15 liters
    Paint for 96 sq meters = 12 liters First, calculate paint needed for 1 sq meter: Paint for 1 sq meter = 12 liters / 96 sq meters = 0.125 liters/sq meter Now, calculate paint required for 120 sq meters: Paint for 120 sq meters = 0.125 liters/sq meter * 120 sq meters = 15 liters

Conclusion

The unitary method is an essential mathematical concept that simplifies complex problems into easy-to-manage calculations. By first determining the value of one unit and then using it to find the other required values, this method provides a simple approach to solving a wide range of practical problems involving ratios and proportions. From determining costs and quantities to calculating speed and much more, the unitary method serves as a vital tool in mathematics and real-life applications.


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Solving Problems Using the Unitary Method

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