Workshop Mathematics (DPT-I)

Practice makes Math perfect......🤓🤓

 CHAPTER - 01 

(BASICS OF SIMPLE FRACTIONS)

Simple Fractions-

A fraction represents a part of a whole. It consists of two numbers:

• Numerator: The top number, indicating how many parts you have.
• Denominator: The bottom number, indicating the total number of equal parts the whole is divided into.

Example:

• In the fraction $\frac{3}{4}$  -​
  • Numerator (3): You have 3 parts.
  • Denominator (4): The whole is divided into 4 equal parts

Real Life Example: In real life, we will many examples of fractions, such as: 

• If a pizza is divided into two equal parts, then each part is equal to half of the whole pizza 
• If we divide a slice of watermelon into three equal parts then each part is equal 1/3rd of the whole

Addition and Subtraction of Fractions-

To add or subtract fractions, they need a common denominator:

• Example (Addition): $$\frac{\mathrm{<span\; style="font-size:\; medium;">1</span>}}{\mathrm{<span\; style="font-size:\; medium;">4</span>}}<span\; style="font-size:\; medium;">+</span>\frac{\mathrm{<span\; style="font-size:\; medium;">1</span>}}{\mathrm{<span\; style="font-size:\; medium;">2</span>}}<span\; style="font-size:\; medium;">=</span>\frac{\mathrm{<span\; style="font-size:\; medium;">1</span>}}{\mathrm{<span\; style="font-size:\; medium;">4</span>}}<span\; style="font-size:\; medium;">+</span>\frac{\mathrm{<span\; style="font-size:\; medium;">2</span>}}{\mathrm{<span\; style="font-size:\; medium;">4</span>}}<span\; style="font-size:\; medium;">=</span>\frac{\mathrm{<b><span\; style="font-size:\; medium;">3</span></b>}}{\mathrm{<b><span\; style="font-size:\; medium;">4</span></b>}}$$
• Example (Subtraction): $$\frac{\mathrm{<span\; style="font-size:\; medium;">3</span>}}{\mathrm{<span\; style="font-size:\; medium;">4</span>}}<span\; style="font-size:\; medium;">-</span>\frac{\mathrm{<span\; style="font-size:\; medium;">1</span>}}{\mathrm{<span\; style="font-size:\; medium;">4</span>}}<span\; style="font-size:\; medium;">=</span>\frac{\mathrm{<span\; style="font-size:\; medium;">2</span>}}{\mathrm{<span\; style="font-size:\; medium;">4</span>}}<span\; style="font-size:\; medium;">=</span>\frac{\mathrm{<b><span\; style="font-size:\; medium;">1</span></b>}}{\mathrm{<b><span\; style="font-size:\; medium;">2</span></b>}}$$

Multiplication of Fractions-

To multiply fractions, multiply the numerators together and the denominators together:

• Example: $$\frac{\mathrm{<span\; style="font-size:\; medium;">1</span>}}{\mathrm{<span\; style="font-size:\; medium;">2</span>}}<span\; style="font-size:\; medium;">\times </span>\frac{\mathrm{<span\; style="font-size:\; medium;">3</span>}}{\mathrm{<span\; style="font-size:\; medium;">4</span>}}<span\; style="font-size:\; medium;">=</span>\frac{\mathrm{<span\; style="font-size:\; medium;">1</span>}<span\; style="font-size:\; medium;">\times </span>\mathrm{<span\; style="font-size:\; medium;">3</span>}}{\mathrm{<span\; style="font-size:\; medium;">2</span>}<span\; style="font-size:\; medium;">\times </span>\mathrm{<span\; style="font-size:\; medium;">4</span>}}<span\; style="font-size:\; medium;">=</span>\frac{\mathrm{<span\; style="font-size:\; medium;"><b>3</b></span>}}{\mathrm{<span\; style="font-size:\; medium;"><b>8</b></span>}}$$

Percentage-

A percentage is a way to express a number as a fraction of 100. For example, 25% means $\frac{25}{100}$ or 0.25. To find a percentage of a number:

• Example: Find 20% of 50. $$\mathrm{<span\; style="font-size:\; medium;">20</span>}\mathrm{<span\; style="font-size:\; medium;">\%</span>}\text{<span style="font-size: medium;"> of </span>}\mathrm{<span\; style="font-size:\; medium;">50</span>}<span\; style="font-size:\; medium;">=</span>\frac{\mathrm{<span\; style="font-size:\; medium;">20</span>}}{\mathrm{<span\; style="font-size:\; medium;">100</span>}}<span\; style="font-size:\; medium;">\times </span>\mathrm{<span\; style="font-size:\; medium;">50</span>}<span\; style="font-size:\; medium;">=</span>\mathrm{<span\; style="font-size:\; medium;"><b>10</b></span>}$$

Quadratic Equations-

A quadratic equation is a polynomial equation of the form $ax^2 + bx + c = 0$ where $a$, $b$, and $c$ are constants.

To solve quadratic equations, you can use:

1. Factoring: Expressing it as $(px + q)(rx + s) = 0$
   
2. Quadratic Formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
3. Completing the Square: Rearranging the equation to make one side a perfect square.

Example: For the equation $x^2 - 5x + 6 = 0$

• Factoring: $(x - 2)(x - 3) = 0$ gives solutions $x = 2$ and $x = 3$
  

PRACTICE EXERCISE

Fractions-

Addition of Fractions:

• $\frac{2}{3}+\frac{1}{6}=?$
  
• $\frac{5}{8}+\frac{3}{4}=?$

Subtraction of Fractions:

• $\frac{7}{10} - \frac{2}{5} = ?$
  
• $\frac{9}{12}-\frac{1}{3}=?$

Multiplication of Fractions:

• $\frac{2}{5}\times \frac{3}{7}=?$
  
• $\frac{4}{9}\times \frac{1}{2}=?$

Percentage-

C

onvert Fractions to Percentages:

• Convert $\frac{2}{5}$ to a percentage.
• Convert $\frac{7}{20}$ to a percentage.