Mathematics Notes: Fractions, Decimal Numbers, and Percentages PDF

Mathematics Notes: Fractions, Decimal Numbers, and Percentages Introduction: Welcome to the world of numbers! In this note, we're going to explore some fundamental concepts that are crucial in mathematics and even in our everyday lives: fractions, decimal numbers, and percentages. These are differe...

Mathematics Notes: Fractions, Decimal Numbers, and Percentages Introduction: Welcome to the world of numbers! In this note, we're going to explore some fundamental concepts that are crucial in mathematics and even in our everyday lives: fractions, decimal numbers, and percentages. These are different ways of representing parts of a whole or comparing quantities. Understanding these concepts will unlock a whole new level of mathematical thinking and problem- solving. 1. Fractions: Parts of a Whole Imagine you have a delicious pizza cut into equal slices. A fraction represents a part of that pizza. A fraction is written in the form a/b, where: a is the numerator. It tells you how many parts you have. b is the denominator. It tells you the total number of equal parts the whole is divided into. Example: If a pizza is cut into 8 slices and you take 3 slices, the fraction representing the amount of pizza you have is 3/8. Here, 3 is the numerator and 8 is the denominator. Types of Fractions: Proper Fraction: The numerator is smaller than the denominator (e.g., 1/4, 5/7). This represents a part that is less than the whole. Improper Fraction: The numerator is greater than or equal to the denominator (e.g., 7/3, 4/4). This represents a part that is equal to or greater than the whole. Mixed Number: A combination of a whole number and a proper fraction (e.g., 2 1/2). This also represents a quantity greater than one. The mixed number 2 1/2 is the same as saying you have two whole pizzas and half of another pizza. Equivalent Fractions: Different fractions can represent the same amount. These are called equivalent fractions. You can find equivalent fractions by multiplying or dividing both the numerator and the denominator by the same non-zero number. Example: 1/2 is equivalent to 2/4, 3/6, 4/8, and so on. Imagine taking half a pizza – whether it's one big slice or two smaller slices (where the pizza was cut into four), you still have the same amount. Simplifying Fractions: Simplifying a fraction means finding an equivalent fraction with the smallest possible numerator and denominator. To do this, you divide both the numerator and the denominator by their greatest common factor (GCF). Example: To simplify 6/8, the GCF of 6 and 8 is 2. Dividing both by 2 gives us 3/4. Comparing Fractions: To compare fractions, it's helpful to have them with the same denominator (a common denominator). Once they have a common denominator, you can compare the numerators directly. The fraction with the larger numerator is the larger fraction. Example: To compare 1/3 and 2/5, find a common denominator (15). 1/3 = (1 × 5)/(3 × 5) = 5/15 2/5 = (2 × 3)/(5 × 3) = 6/15 Since 6 is greater than 5, 2/5 is greater than 1/3 ( 2/5 > 1/3 ). Operations with Fractions: Adding and Subtracting: You need a common denominator. Add or subtract the numerators and keep the denominator the same. o Example: 1/4 + 2/4 = (1+2)/4 = 3/4 o Example: 3/5 - 1/5 = (3-1)/5 = 2/5 Multiplying: Multiply the numerators together and multiply the denominators together. o Example: 1/2 × 3/4 = (1 × 3)/(2 × 4) = 3/8 Dividing: Multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping its numerator and denominator. o Example: 1/2 ÷ 3/4 = 1/2 × 4/3 = (1 × 4)/(2 × 3) = 4/6, which can be simplified to 2/3. Real-life Example: Sharing a chocolate bar. If you have a chocolate bar divided into 6 equal pieces and you share 2 pieces with a friend, you gave away 2/6 (or 1/3) of the chocolate bar. Graphical Interpretation: Imagine a circle divided into equal segments. A fraction like 3/8 can be visually represented by shading 3 out of the 8 segments. 2. Decimal Numbers: Another Way to Represent Parts Decimal numbers are another way to represent fractions where the denominator is a power of 10 (like 10, 100, 1000, etc.). They use a decimal point (.) to separate the whole number part from the fractional part. Example: The fraction 1/10 can be written as the decimal number 0.1 (read as "zero point one"). The fraction 35/100 can be written as 0.35 (read as "zero point three five"). Place Value in Decimals: Each digit in a decimal number has a specific place value:... Hundreds Tens Ones. Tenths Hundredths Thousandths... Example: In the number 12.345:  1 is in the tens place (1 × 10) 2 is in the ones place (2 × 1) 3 is in the tenths place (3 × 1/10) 4 is in the hundredths place (4 × 1/100) 5 is in the thousandths place (5 × 1/1000) Converting Fractions to Decimals: To convert a fraction to a decimal, you divide the numerator by the denominator. Example: To convert 3/4 to a decimal, divide 3 by 4: 3 ÷ 4 = 0.75 Types of Decimals: Terminating Decimals: Decimals that end after a finite number of digits (e.g., 0.5, 0.25, 1.75). Repeating Decimals: Decimals where one or more digits repeat infinitely (e.g., 0.333..., 1.666..., 0.142857142857...). We often use a dot above the repeating digit(s) to show the repetition, but since we are avoiding that, we will write out a few repetitions or use ellipses (...). Real-life Example: Measuring length. If a table is 1 and a half meters long, you can write this as 1.5 meters. Graphical Interpretation: Imagine a number line. Decimal numbers can be plotted precisely on this line, showing their exact position between whole numbers. For example, 0.5 would be exactly halfway between 0 and 1. 3. Percentages: Fractions Out of One Hundred A percentage is a way of expressing a number as a fraction of 100. The word "percent" means "out of one hundred" and is represented by the symbol %. Example: 50% means 50 out of 100, which is equivalent to the fraction 50/100 or 1/2. Converting Fractions to Percentages: To convert a fraction to a percentage, you can first convert the fraction to a decimal and then multiply by 100. Example: To convert 3/4 to a percentage: 1. Convert to decimal: 3 ÷ 4 = 0.75 2. Multiply by 100: 0.75 × 100 = 75% Alternatively, if you can make the denominator 100, the numerator will be the percentage. Converting Decimals to Percentages: To convert a decimal to a percentage, multiply the decimal by 100. Example: To convert 0.65 to a percentage: 0.65 × 100 = 65% Calculating Percentages of Amounts: To find a percentage of an amount, convert the percentage to a decimal or a fraction and then multiply it by the amount. Example: To find 25% of 80: 1. Convert 25% to a decimal: 25/100 = 0.25 2. Multiply by 80: 0.25 × 80 = 20 Alternatively: 1. Convert 25% to a fraction: 25/100 = 1/4 2. Multiply by 80: (1/4) × 80 = 80/4 = 20 Real-life Example: Discounts in a shop. If a shirt originally costs $40 and there is a 20% discount, the discount amount is 20% of $40, which is (20/100) × 40 = $8. The new price is $40 - $8 = $32. Graphical Interpretation: Imagine a bar divided into 100 equal parts. A percentage like 60% can be visually represented by shading 60 out of the 100 parts. 4. Different Types of Numbers and Their Definitions: Understanding different types of numbers is essential for building a strong foundation in mathematics. Here are some key categories: Natural Numbers (N): These are the counting numbers: 1, 2, 3, 4,... They are positive whole numbers. Whole Numbers (W): These include all natural numbers plus zero: 0, 1, 2, 3, 4,... Integers (Z): These include all whole numbers and their negative counterparts:..., -3, -2, -1, 0, 1, 2, 3,... Rational Numbers (Q): These are numbers that can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero. This includes all integers, terminating decimals, and repeating decimals. Examples: 1/2, -3/4, 5 (which is 5/1), 0.75, 0.333... Irrational Numbers: These are numbers that cannot be expressed as a simple fraction. Their decimal representations are non-terminating and non-repeating. Famous examples include π (pi) and the square root of 2 (√2). Conclusion: Fractions, decimal numbers, and percentages are powerful tools for representing and understanding quantities. They are interconnected and used extensively in various fields, from baking and cooking to science and finance. By understanding their definitions, how to convert between them, and how to perform basic operations, you will be well-equipped to tackle a wide range of mathematical problems and confidently navigate the numerical aspects of your daily life. Keep practicing and exploring these concepts, and you'll be amazed at how much sense the world of numbers makes!