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 → Number System → Rational Numbers ↓


Properties of Rational Numbers


Rational numbers are numbers that can be expressed as a fraction p/q where p and q are integers and q ≠ 0 In this guide, we are going to discuss various properties of rational numbers that help in organizing and efficiently performing mathematical operations. These properties include:

  • Closing assets
  • Commutative property
  • Associative property
  • Distributive property
  • Identity property
  • Inverse property

Closing assets

The closure property states that the sum, difference, and product of any two rational numbers is always a rational number.

Add

Suppose you have two rational numbers a/b and c/d. Their sum is:

(a/b) + (c/d) = (ad + bc) / bd

Here, ad + bc and bd are also integers, and since bd ≠ 0, the result is a rational number.

Example: Add 2/3 and 4/5.

(2/3) + (4/5) = (2*5 + 4*3) / (3*5) = (10 + 12) / 15 = 22/15

The result 22/15 is also a rational number.

Subtraction

For subtraction, similar to addition, if you subtract a/b from c/d, you get:

(a/b) - (c/d) = (ad - bc) / bd

Since ad - bc and bd are integers, and bd ≠ 0, this is also a rational number.

Example: Subtract 7/8 from 5/6.

(5/6) - (7/8) = (5*8 - 7*6) / (6*8) = (40 - 42) / 48 = -2/48 = -1/24

The result -1/24 is a rational number.

Multiplication

In multiplication, the product of two rational numbers a/b and c/d is:

(a/b) * (c/d) = (ac) / (bd)

Since ac and bd are both integers and bd ≠ 0, the result is a rational number.

Example: Multiply 3/4 by 2/5.

(3/4) * (2/5) = (3*2) / (4*5) = 6/20 = 3/10

The result 3/10 is also a rational number.

Commutative property

The commutative property states that the order of adding or multiplying two rational numbers does not affect the result.

Add

For addition, a/b + c/d = c/d + a/b. The sum remains unchanged.

Example: Using 1/2 + 1/3 would be 1/3 + 1/2.

(1/2) + (1/3) = (3/6) + (2/6) = 5/6
(1/3) + (1/2) = (2/6) + (3/6) = 5/6

In both sequences, the result is 5/6, which demonstrates the commutative property.

Multiplication

For multiplication, a/b * c/d = c/d * a/b. The product remains unchanged.

Example: Using 2/3 * 4/5 equals 4/5 * 2/3.

(2/3) * (4/5) = (8/15)
(4/5) * (2/3) = (8/15)

In both sequences, the result is 8/15, which follows the commutative property.

Associative property

The associative property states that the way rational numbers are grouped in addition or multiplication does not change their sum or product.

Add

When adding, (a/b + c/d) + e/f = a/b + (c/d + e/f)

Example: Add 1/4 + 1/5 + 1/6 with separate groups.

((1/4) + (1/5)) + (1/6) = (9/20) + (1/6) = 29/60
(1/4) + ((1/5) + (1/6)) = (1/4) + (11/30) = 29/60

The total of the two groups is 29/60.

Multiplication

For multiplication, (a/b * c/d) * e/f = a/b * (c/d * e/f)

Example: Multiply 2/3 * 3/4 * 4/5 with different groups.

((2/3) * (3/4)) * (4/5) = (1/2) * (4/5) = 2/5
(2/3) * ((3/4) * (4/5)) = (2/3) * (1/5) = 2/15

Note that even though the intermediate results change with rearrangement, this property remains valid for the entire expression.

Distributive property

The distributive property connects multiplication and addition, showing that multiplying a number by a group of numbers added together is the same as doing each multiplication separately.

As a formula, a/b * (c/d + e/f) = a/b * c/d + a/b * e/f.

Example: Demonstrate the distributive property using 2/3 and (3/4 + 5/6).

(2/3) * ((3/4) + (5/6)) = (2/3) * (19/12) = 38/36 = 19/18
(2/3) * (3/4) + (2/3) * (5/6) = 1/2 + 5/9 = 19/18

Both methods give the same results, which confirms the distributive property.

Identity property

The identity property makes it clear that the value of a rational number will not change if it is added or multiplied by a certain number.

Add

The identity number for sum is 0 Thus, a/b + 0 = a/b.

Example: Add 0 to 7/8.

(7/8) + 0 = 7/8

Multiplication

The identity number for multiplication is 1 Thus, a/b * 1 = a/b.

Example: Multiply 1 by 9/10.

(9/10) * 1 = 9/10

Inverse property

For every rational number, there is another rational number that is its inverse, which brings the operation result to the identity value.

Additive inverse

The additive inverse of a/b is -a/b because a/b + (-a/b) = 0.

Example: Find the additive inverse of 5/7.

(5/7) + (-5/7) = 0

Multiplicative inverse

The multiplicative inverse (or inverse) of a/b is b/a because (a/b) * (b/a) = 1.

Example: Find the multiplicative inverse of 3/4.

(3/4) * (4/3) = 1

Understanding these properties of rational numbers will help solve problems easily and ensure that the calculations are done correctly. When these properties are actively applied in tasks involving rational numbers, mathematics becomes less complicated. Keep practicing these properties with enough examples to gain proficiency.


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