#### Topics

##### Integers

- Concept for Natural Numbers
- Concept for Whole Numbers
- Negative and Positive Numbers
- Concept of Integers
- Representation of Integers on the Number Line
- Concept for Ordering of Integers
- Addition of Integers
- Addition of Integers on Number line
- Subtraction of Integers
- Properties of Addition and Subtraction of Integers
- Multiplication of a Positive and a Negative Integers
- Multiplication of Two Negative Integers
- Product of Three Or More Negative Integers
- Closure Property of Multiplication of Integers
- Commutative Property of Multiplication of Integers
- Multiplication of Integers with Zero
- Multiplicative Identity of Integers
- Associative Property of Multiplication of Integers
- Distributive Property of Multiplication of Integers
- Making Multiplication Easier of Integers
- Division of Integers
- Properties of Division of Integers

##### Fractions and Decimals

- Concept of Fractions
- Types of Fraction
- Concept of Proper Fractions
- Improper Fraction and Mixed Fraction
- Concept for Equivalent Fractions
- Like and Unlike Fraction
- Comparing Fractions
- Addition of Fraction
- Subtraction of Fraction
- Multiplication of a Fraction by a Whole Number
- Fraction as an Operator 'Of'
- Multiplication of a Fraction by a Fraction
- Division of Fractions
- Concept for Reciprocal of a Fraction
- Problems Based on Fraction
- Concept of Decimal Numbers
- Comparing Decimal Numbers
- Addition of Decimal Numbers
- Subtraction of Decimal Numbers
- Multiplication of Decimal Numbers
- Multiplication of Decimal Numbers by 10, 100 and 1000
- Division of Decimal Numbers by 10, 100 and 1000
- Division of a Decimal Number by a Whole Number
- Division of a Decimal Number by Another Decimal Number
- Problems Based on Decimal Numbers

##### Data Handling

##### Simple Equations

##### Lines and Angles

- Concept of Points
- Concept of Line
- Concept of Line Segment
- Concept of Angle - Arms, Vertex, Interior and Exterior Region
- Complementary Angles
- Supplementary Angles
- Adjacent Angles
- Concept of Linear Pair
- Concept of Vertically Opposite Angles
- Concept of Intersecting Lines
- Parallel Lines
- Pairs of Lines - Transversal
- Pairs of Lines - Angles Made by a Transversal
- Pairs of Lines - Transversal of Parallel Lines
- Checking Parallel Lines

##### The Triangle and Its Properties

- Concept of Triangles - Sides, Angles, Vertices, Interior and Exterior of Triangle
- Classification of Triangles (On the Basis of Sides, and of Angles)
- Equilateral Triangle
- Isosceles Triangles
- Scalene Triangle
- Acute Angled Triangle
- Obtuse Angled Triangle
- Right Angled Triangle
- Median of a Triangle
- Altitudes of a Triangle
- Exterior Angle of a Triangle and Its Property
- Angle Sum Property of a Triangle
- Some Special Types of Triangles - Equilateral and Isosceles Triangles
- Sum of the Lengths of Two Sides of a Triangle
- Right-angled Triangles and Pythagoras Property

##### Comparing Quantities

- Concept of Ratio
- Concept of Equivalent Ratios
- Concept of Proportion
- Concept of Unitary Method
- Concept of Percent and Percentage
- Converting Fractional Numbers to Percentage
- Converting Decimals to Percentage
- Converting Percentages to Fractions
- Converting Percentages to Decimals
- Estimation in Percentages
- Interpreting Percentages
- Converting Percentages to “How Many”
- Ratios to Percents
- Increase Or Decrease as Percent
- Concepts of Cost Price, Selling Price, Total Cost Price, and Profit and Loss, Discount, Overhead Expenses and GST
- Profit or Loss as a Percentage
- Concept of Principal, Interest, Amount, and Simple Interest

##### Congruence of Triangles

##### Rational Numbers

- Rational Numbers
- Equivalent Rational Number
- Positive and Negative Rational Numbers
- Rational Numbers on a Number Line
- Rational Numbers in Standard Form
- Comparison of Rational Numbers
- Rational Numbers Between Two Rational Numbers
- Addition of Rational Number
- Subtraction of Rational Number
- Multiplication of Rational Numbers
- Division of Rational Numbers

##### Perimeter and Area

- Mensuration
- Concept of Perimeter
- Perimeter of a Rectangle
- Perimeter of Squares
- Perimeter of Triangles
- Perimeter of Polygon
- Concept of Area
- Area of Square
- Area of Rectangle
- Triangles as Parts of Rectangles and Square
- Generalising for Other Congruent Parts of Rectangles
- Area of a Parallelogram
- Area of a Triangle
- Circumference of a Circle
- Area of Circle
- Conversion of Units
- Problems based on Perimeter and Area

##### Algebraic Expressions

- Algebraic Expressions
- Terms, Factors and Coefficients of Expression
- Like and Unlike Terms
- Types of Algebraic Expressions as Monomials, Binomials, Trinomials, and Polynomials
- Addition of Algebraic Expressions
- Subtraction of Algebraic Expressions
- Evaluation of Algebraic Expressions by Substituting a Value for the Variable.
- Use of Variables in Common Rules

##### Practical Geometry

- Construction of a Line Parallel to a Given Line, Through a Point Not on the Line
- Construction of Triangles
- Constructing a Triangle When the Length of Its Three Sides Are Known (SSS Criterion)
- Constructing a Triangle When the Lengths of Two Sides and the Measure of the Angle Between Them Are Known. (SAS Criterion)
- Constructing a Triangle When the Measures of Two of Its Angles and the Length of the Side Included Between Them is Given. (ASA Criterion)
- Constructing a Right-angled Triangle When the Length of One Leg and Its Hypotenuse Are Given (RHS Criterion)

##### Exponents and Powers

- Concept of Exponents
- Multiplying Powers with the Same Base
- Dividing Powers with the Same Base
- Taking Power of a Power
- Multiplying Powers with Different Base and Same Exponents
- Dividing Powers with Different Base and Same Exponents
- Numbers with Exponent Zero, One, Negative Exponents
- Miscellaneous Examples Using the Laws of Exponents
- Decimal Number System Using Exponents and Powers
- Expressing Large Numbers in the Standard Form

##### Symmetry

##### Visualizing Solid Shapes

- Plane Figures and Solid Shapes
- Faces, Edges and Vertices
- Nets for Building 3-d Shapes - Cube, Cuboids, Cylinders, Cones, Pyramid, and Prism
- Drawing Solids on a Flat Surface - Oblique Sketches
- Drawing Solids on a Flat Surface - Isometric Sketches
- Visualising Solid Objects
- Viewing Different Sections of a Solid

- Comparing like fractions with same denominators
- Comparing unlike fractions with same numerators
- Comparing unlike fractions with different numerators

## Notes

**Comparing Fractions:**

Consider `1/2 and 1/3`

The portion of the whole corresponding to `1/2` is clearly larger than the portion of the same whole corresponding to `1/3`. But often it is not easy to say which one out of a pair of fractions is larger. We should, therefore, like to have a systematic procedure to compare fractions. It is particularly easy to compare like fractions.

**1. Comparing like fractions with the same denominators:**

Let us compare two like rational numbers: `3/8 and 5/8`.

In both the fraction, the whole is divided into 8 equal parts. For `3/8 and 5/8`,

We take 3 and 5 parts respectively out of the 8 equal parts. Clearly, out of 8 equal parts, the portion corresponding to 5 parts is larger than the portion corresponding to 3 parts.

Hence, `5/8 > 3/8`.

Note the number of the parts taken is given by the numerator. It is, therefore, clear that for two fractions with the same denominator, the fraction with the greater numerator is greater.

Between `4/5 and 3/5, 4/5` is greater.

Between `11/20 and 13/20, 13/20` is greater.

**2. Comparing unlike fractions with the same numerators:**

**Which is greater `1/3 or 1/5`?**

- In `1/3`, we divide the whole into 3 equal parts and take one. In `1/5`, we divide the whole into 5 equal parts and take one. Note that in `1/3`, the whole is divided into a smaller number of parts than in `1/5`. The equal part that we get in `1/3` is, therefore, larger than the equal part we get in `1/5`. Since in both cases we take the same number of parts (i.e. one), the portion of the whole showing `1/3` is larger than the portion showing `1/5`, and therefore `1/3 > 1/5`.
- In the same way, we can say `2/3 > 2/5`. In this case, the situation is the same as in the case above, except that the common numerator is 2, not 1. The whole is divided into a large number of equal parts for `2/5` than for `2/3`. Therefore, each equal part of the whole in the case of `2/3` is larger than that in the case of `2/5`. Therefore, the portion of the whole showing `2/3` is larger than the portion showing `2/5` and hence, `2/3 > 2/5`.

We can see from the above example that if the numerator is the same in two fractions, the fraction with the smaller denominator is greater of the two.

Thus, `1/8 > 1/10, 3/5 > 3/7, 4/9 > 4/11` and so on.

**3. Comparing unlike fractions with different numerators:**

Compare `5/6 and 13/15`.

**Solution:**

The fractions are unlike. We should first get their equivalent fractions with a denominator which is a common multiple of 6 and 15.

Now, `(5 × 5)/(6 × 5) = 25/30, (13 × 2)/(15 × 2) = (26)/(30)`

Since, `(26/30) > (25/30) "we have" (13/15) > (5/6)`

**Why LCM?**

The product of 6 and 15 is 90; obviously 90 is also a common multiple of 6 and 15. We may use 90 instead of 30; it will not be wrong. But we know that it is easier and more convenient to work with smaller numbers. So, the common multiple that we take is as small as possible. This is why the LCM of the denominators of the fractions is preferred as the common denominator.

## Example

**Find answers to the following. Write and indicate how you solved them.**

Is `5/9` equal to `4/5`?

`5/9, 4/5`

Converting these into like fractions,

`5/9 = 5/9 xx 5/5 = 25/45`

`4/5 = 4/5 xx 9/9 = 36/45`

As, `36/45 ≠ 25/45`,

Therefore, `5/9 "is not equal to" 4/5`.

## Example

**Find answers to the following. Write and indicate how you solved them.**

Is `9/16` equal to `5/9`?

`9/16, 5/9`

Converting these into like fractions,

`9/16 = 9/16 xx 9/9 = 81/144`

`5/9 = 5/9 xx 16/16 = 80/144`

As, `81/144 ≠ 80/144`,

Therefore, `9/16 "is not equal to" 5/9`.

## Example

**Find answers to the following. Write and indicate how you solved them.**

Is `4/5` equal to `16/20`?

`4/5, 16/20`

`16/20 = (4 xx 4)/(5 xx 4) = 4/5`

Therefore, `4/5 = 16/20`.

## Example

Ila read 25 pages of a book containing 100 pages. Lalita read `2/5` of the same book. Who reads less?

Numbers of pages read by Lalita = `2/5 xx 100` = 40

Number of pages read by Ila = 25

Hence, Ila has read less number of pages.

## Example

Asha and Samuel have bookshelves of the same size partly filled with books. Asha’s shelf is `5/6`th full and Samuel’s shelf is `2/5`th full. Whose bookshelf is more full? By what fraction?

Fraction of Asha’s shelf = `5/6`

Fraction of Samuel’s shelf = `2/5`

Converting these into like fractions,

`5/6 = 5/6 xx 5/5 = 25/30`

`2/5 = 2/5 xx 6/6 = 12/30`

`25/30 > 12/30`

Clearly, Asha’s bookshelf is more full.

Difference = `5/6 - 2/5 = 25/30 - 12/30 = 13/30`.

## Example

Jaidev takes `2 1/5` minutes to walk across the school ground. Rahul takes `7/4` minutes to do the same. Who takes less time and by what fraction?

Time taken by Jaidev = `2 1/5 "minutes" = 11/5` min

Time taken by Rahul = `7/4` min

Converting these into like fractions,

`11/5 = 11/5 xx 4/4 = 44/20`

`7/4 = 7/4 xx 5/5 = 35/20`

As 44 > 35,

`11/5 > 7/4`

Hence, Rahul takes lesser time.

Difference = `11/5 - 7/4`

= `44/20 - 35/20 = 9/20` min.

#### Shaalaa.com | Comparison Of Like Fractions

##### Series: Comparing Fractions

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