Grade 8
  1. Number Systems  1.1. Rational Numbers  1.2. Irrational Numbers  1.3. Real Numbers  1.4. Properties of Real Numbers  1.4.1. Commutative Property  1.4.2. Associative Property  1.4.3. Distributive Property  1.5. Representation on the Number Line  1.6. Operations on Real Numbers  1.6.1. Addition and Subtraction  1.6.2. Multiplication and Division  1.7. Rationalization  1.8. Exponents and Powers  1.9. Square Roots and Cube Roots
  2. Algebra  2.1. Algebraic Expressions  2.2. Identities and Simplification  2.2.1. Multiplying Binomials  2.2.2. Using Algebraic Identities  2.3. Factorization  2.3.1. Common Factors  2.3.2. Factorization by Grouping  2.3.3. Factorization of Trinomials  2.3.4. Special Products  2.4. Linear Equations in One Variable  2.4.1. Solving Linear Equations  2.4.2. Word Problems on Linear Equations
  3. Geometry  3.1. Understanding Quadrilaterals  3.1.1. Properties of Quadrilaterals  3.1.2. Types of Quadrilaterals  3.1.2.1. Parallelogram  3.1.2.2. Rectangle  3.1.2.3. Square  3.1.2.4. Rhombus  3.1.2.5. Trapezium  3.2. Constructions  3.2.1. Constructing Quadrilaterals  3.2.2. Constructing Bisectors  3.2.3. Constructing Angles  3.2.4. Constructing Triangles  3.3. Symmetry and Transformations  3.3.1. Reflection  3.3.2. Rotation  3.3.3. Translation  3.3.4. Symmetry in Patterns
  4. Mensuration  4.1. Area and Perimeter  4.1.1. Perimeter of Polygons  4.1.2. Area of Triangles  4.1.3. Area of Quadrilaterals  4.1.4. Area of Circles  4.2. Surface Area and Volume  4.2.1. Cubes  4.2.2. Cuboids  4.2.3. Cylinders  4.2.4. Cones  4.2.5. Spheres
  5. Data Handling  5.1. Organizing Data  5.2. Frequency Distribution Tables  5.3. Graphical Representation of Data  5.3.1. Bar Graphs  5.3.2. Histograms  5.3.3. Pie Charts  5.3.4. Line Graphs (added)  5.4. Probability  5.4.1. Introduction to Probability  5.4.2. Experimental Probability
  6. Coordinate Geometry  6.1. Cartesian Plane  6.2. Plotting Points  6.3. Coordinate System  6.4. Distance Between Points  6.5. Applications of Coordinate Geometry
  7. Introduction to Graphs  7.1. Linear Graphs  7.2. Applications of Graphs  7.3. Distance-Time Graphs  7.4. Speed-Time Graphs
  8. Comparing Quantities  8.1. Ratio and Proportion  8.2. Percentage  8.2.1. Increase and Decrease Percent  8.2.2. Discount in Percentage  8.2.3. Profit and Loss in Percentage  8.2.4. Simple Interest  8.2.5. Compound Interest
  9. Exponents and Powers  9.1. Laws of Exponents  9.2. Negative Exponents  9.3. Standard Form  9.4. Applications of Exponents
  10. Introduction to Squares and Square Roots  10.1. Properties of Square Numbers  10.2. Finding Square Roots  10.2.1. Prime Factorization Method  10.2.2. Division Method  10.3. Estimating Square Roots
  11. Introduction to Cubes and Cube Roots  11.1. Properties of Cube Numbers  11.2. Finding Cube Roots  11.2.1. Prime Factorization Method in Finding Cube Roots

Grade 8 → Comparing Quantities ↓


Ratio and Proportion


In mathematics, comparing quantities helps us understand the world around us. It allows us to determine relationships between quantities, express these relationships simply, and solve real-life problems more efficiently. Two fundamental concepts that help with this are ratio and proportion.

What is the ratio?

A ratio is a way of comparing two or more quantities using division. It tells us how much of one thing is compared to another. A ratio can be expressed in different forms: using the word "to", with a colon, or as a fraction.

For example, to compare 4 apples and 2 oranges, we can write:

  • 4 to 2
  • 4:2
  • 4/2

All of these notations express the same ratio. Ratios can also be simplified just like fractions. For example, the ratio 4:2 can be simplified by dividing both numbers by their greatest common divisor (GCD). In this case, the GCD is 2, so the simplified ratio is 2:1.

Visual example

Consider the scenario with the colored circles below:

Here we are comparing red circles with blue circles. There are 2 red circles and 3 blue circles. The ratio of red circles and blue circles is written 2:3.

Different types of ratios

It is important to note that a ratio can involve more than two quantities. For example, if we have a bag containing 4 red balls, 5 green balls, and 7 blue balls, then the ratio of red, green, and blue balls is 4:5:7.

Properties of proportion

  • Ratios are usually expressed in their simplest form.
  • The order matters in a ratio; 2:3 is not the same as 3:2.
  • Ratios have no units; they are comparisons relative to each other.

What is the ratio?

A ratio is an equation that states that two ratios are equal. Ratios are used to solve problems where we need to find the missing term in a ratio when comparing quantities. If two ratios a:b and c:d are equal, then they are said to be proportional. It is written as:

a:b = c:d
Or 
a/b = c/d

To determine if two ratios form a proportion, you can cross-multiply the terms and verify if the cross-products are equal. For example:

2/3 = 4/6

By cross-multiplying, we get:

2 × 6 = 3 × 4

If we simplify both sides, both will be equal to 12, so the ratios will be proportional.

Understanding proportion with examples

Imagine if you need to paint a fence and the paint is mixed in a specific ratio, 1 part white and 2 parts green. If you have 5 parts white paint, how much green paint do you need?

The ratios can be written as follows:

white/green = 1/2 = 5/x

By cross-multiplying:

1 × x = 2 × 5

We find that x = 10. Therefore, you will need 10 parts green to maintain the same proportion.

Use of ratio and proportion in real life

1. Method of cooking

Ratios are often used in recipes to maintain flavor balance. If a recipe calls for a 3:2 ratio of sugar to flour and you have 6 cups of sugar, you need to figure out how much flour is needed.

sugar/flour = 3/2 = 6/x

Cross-multiplying gives:

3 × x = 2 × 6

Simplifying, x = 4 cups of flour needed.

Practice problems

  1. Determine whether the ratios 8 : 12 and 2 : 3 form a proportion.
  2. If 9 oranges cost $18, how much will 15 oranges cost?
  3. The teacher-student ratio in a school is 1:30. If there are 450 students, how many teachers are there?
  4. The ratio of flour and sugar in a recipe is 4:1. If you have 12 cups of flour, how much sugar do you need?

Solution

1. Checking the ratio:

8:12 = 2:3
Cross-multiplying: 8 × 3 == 12 × 2
24 = 24 (True, so they form a ratio)

2.

9/18 = 15/x
Cross-multiplying: 9x = 15 × 18
9x = 270
x = 30
So, 15 oranges cost $30.

3.

1/30 = x/450
Cross-multiplying: 1 × 450 = 30x
450 = 30x
x = 15
There are 15 teachers there.

4.

4/1 = 12/x
Cross-multiplication: 4x = 12 × 1
4x = 12
x = 3
You need 3 cups of sugar.

Summary

In conclusion, ratio and proportion are essential concepts in math that allow us to compare quantities and find unknown values based on given information. Understanding these concepts helps simplify and solve real-world problems, making it easier to understand the relationships between things. Practicing and becoming familiar with different examples increases our ability to apply these concepts effectively.


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Ratio and Proportion

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