Grade 6
  1. Number System  1.1. Whole Numbers  1.1.1. Place Value and Face Value  1.1.2. Comparing and Ordering Numbers  1.1.3. Rounding Numbers  1.1.4. Properties of Whole Numbers (Associative, Commutative, Distributive)  1.2. Integers  1.2.1. Introduction to Integers  1.2.2. Addition and Subtraction of Integers  1.2.3. Multiplication and Division of Integers  1.2.4. Properties of Integer Operations  1.3. Fractions  1.3.1. Types of Fractions  1.3.2. Equivalent Fractions  1.3.3. Simplification of Fractions  1.3.4. Addition and Subtraction of Fractions  1.3.5. Multiplication and Division of Fractions  1.3.6. Comparing and Ordering Fractions  1.4. Decimals  1.4.1. Understanding Decimals  1.4.2. Conversion between Fractions and Decimals  1.4.3. Addition and Subtraction of Decimals  1.4.4. Multiplication and Division of Decimals  1.4.5. Comparing and Ordering Decimals
  2. Algebra  2.1. Basics of Algebra  2.1.1. Variables and Constants  2.1.2. Expressions and Equations  2.1.3. Algebraic Terms and Coefficients  2.2. Solving Simple Equations  2.2.1. Balancing Equations  2.2.2. Solving One-Step Equations  2.2.3. Solving Two-Step Equations  2.3. Patterns and Sequences  2.3.1. Identifying Patterns  2.3.2. Arithmetic Sequences
  3. Ratio and Proportion  3.1. Ratios  3.1.1. Understanding Ratios  3.1.2. Equivalent Ratios  3.1.3. Simplifying Ratios  3.2. Proportion  3.2.1. Introduction to Proportion  3.2.2. Direct and Inverse Proportions  3.3. Percentages  3.3.1. Converting Ratios to Percentages  3.3.2. Percentage Increase and Decrease  3.3.3. Discount and Profit Calculations  3.3.4. Simple Interest Calculations in Percentages in Ratio and Proportion
  4. Geometry  4.1. Basic Geometric Shapes  4.1.1. Points, Lines, and Angles  4.1.2. Types of Angles  4.1.3. Intersecting and Parallel Lines  4.2. Triangles  4.2.1. Types of Triangles  4.2.2. Properties of Triangles  4.2.3. Sum of Angles in a Triangle  4.2.4. Congruence of Triangles  4.3. Quadrilaterals  4.3.1. Types of Quadrilaterals  4.3.2. Properties of Quadrilaterals  4.4. Circles  4.4.1. Radius and Diameter  4.4.2. Circumference and Area  4.4.3. Arcs and Chords  4.4.4. Sector and Segment
  5. Mensuration  5.1. Perimeter  5.1.1. Perimeter of Simple Shapes  5.2. Area  5.2.1. Area of Rectangles and Squares  5.2.2. Area of Triangles and Parallelograms  5.2.3. Area of Composite Figures  5.2.4. Area of Trapezium  5.3. Volume  5.3.1. Volume of Cubes and Cuboids  5.3.2. Volume of Cylinders  5.3.3. Volume of Cones  5.3.4. Surface Area of Simple Solids
  6. Data Handling  6.1. Introduction to Data  6.1.1. Types of Data  6.1.2. Organizing Data  6.1.3. Frequency Distribution  6.2. Graphs and Charts  6.2.1. Bar Graphs  6.2.2. Line Graphs  6.2.3. Pictographs  6.2.4. Pie Charts  6.3. Mean, Median, and Mode  6.3.1. Calculating Mean  6.3.2. Finding Median  6.3.3. Identifying Mode  6.3.4. Range of Data in Mean, Median, and Mode
  7. Probability  7.1. Basics of Probability  7.1.1. Understanding Probability  7.1.2. Probability of Simple Events  7.1.3. Experimental and Theoretical Probability
  8. Practical Geometry  8.1. Construction of Shapes  8.1.1. Constructing Triangles  8.1.2. Constructing Quadrilaterals  8.1.3. Constructing Circles and Arcs  8.2. Symmetry  8.2.1. Line of Symmetry  8.2.2. Rotational Symmetry  8.2.3. Translational Symmetry

Grade 6 → Number System → Fractions ↓


Multiplication and Division of Fractions


Understanding fractions

A fraction is a way of representing a part of a whole. It consists of two numbers: the numerator and the denominator. The numerator is the number on top and it shows how many parts we are considering. The denominator is the number on the bottom and it tells us how many equal parts the whole is divided into.

For example, the fraction 3/4 has a numerator of 3 and a denominator of 4. This means we have 3 parts out of a total of 4.

Multiplication of fractions

When you multiply fractions, you're finding a part of a fraction. Let's figure out how this works:

Step-by-step process

  1. Multiply fractions: Multiply the top digits (numerators) of the fractions together.
  2. Multiply the denominators: Multiply the bottom digits (denomins) of the fractions together.
  3. Simplify the fraction: If possible, simplify the resulting fraction to its simplest form.

It's very easy! Here's an example:

Example 1

Multiply the fractions 2/3 and 3/4.

Step 1: Multiply the fractions: 2 × 3 = 6

Step 2: Multiply the denominators: 3 × 4 = 12

Result: 6/12

Step 3: Simplify the fraction: 6/12 = 1/2 (by dividing both numerator and denominator by 6)

So, 2/3 × 3/4 = 1/2.

Illustrative examples

2/3 3/4

This visualization shows two fractions: 2/3 represented by the light blue rectangle and 3/4 represented by the dark blue rectangle.

Division of fractions

Dividing fractions involves multiplying the first fraction by the reciprocal of the second fraction. Inverses simply switch the numerator and denominator.

Step-by-step process

  1. Find the inverse. Turn the second fraction upside down. This is the inverse.
  2. Multiply: Multiply as before (numerators together and denominators together).
  3. Simplify: Simplify the resulting fraction, if possible.

Let's look at an example:

Example 2

Divide the fraction 2/3 by 3/4.

Step 1: Find the reciprocal of 3/4, which is 4/3.

Step 2: Multiply 2/3 by 4/3.

2/3 ÷ 3/4 = 2/3 × 4/3 = (2×4) / (3×3) = 8/9

The result is 8/9.

Illustrative examples

2/3 4/3

In this visualization, the fraction 2/3 is multiplied by the inverse 4/3.

Why does this work?

Understanding multiplication of fractions

Multiplying fractions can seem confusing, but it's pretty logical if you consider what the fractions represent. When you multiply, you're scaling one fraction by another, which means you're calculating the numerator of a fraction.

For example, multiplying 1/2 × 1/3 means you are taking half of one-third, which results in a smaller portion: 1/6.

Understanding division of fractions

Dividing by fractions applies the rule for finding how many parts of one fraction fit into another fraction. By using the inverse, you essentially turn a division problem into a multiplication problem, which, as we have seen, is much simpler.

For example: dividing by 1/2 means you're asking "How many halves fit into a certain number?" Alternatively, multiplying by its inverse (which is 2/1) flips the operation to multiplication, which may be easier to handle.

Example 3

How many 1/4 are in 1/2?

Step 1: Dimensional Analysis - Consider cutting a shape into quarters, then figuring out how many of those quarters make up half of the shape.

Step 2: Convert addition to multiplication using the inverse:

1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2

Therefore, there are two quarters in one half.

General loss

Multiplying and dividing fractions is pretty easy once you're familiar with the concepts. However, here are some tips to avoid pitfalls:

  • Always simplify your fractions when possible. Fractions in their simplest form are much easier to understand and work with.
  • Be careful with signs. Multiplying a negative fraction by a positive fraction will give a negative fraction.
  • When finding the inverse for division, make sure you flip the correct fraction. An incorrect inverse gives the wrong result.

Practice problems

To master these operations, practice is a great skill. Here are some problems to solve:

  1. Multiply 1/5 by 2/3. Simplify the answer.
  2. Divide 3/7 by 9/14. What do you get?
  3. Multiply 4/5 by 1/2 and divide the result by 2/3.
  4. Is 6/8 the same as 3/4 through multiplication or division? Perform.
  5. What will be the result of multiplying 1/3 by 9/10 and dividing by 6/15? Simplify completely.

Solve these problems and test your understanding of multiplying and dividing fractions. By practicing with these step-by-step methods, mastering fraction operations is within your reach!


Grade 6 → 1.3.5


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Addition and Subtraction of Fractions
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Multiplication and Division of Fractions

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