Understanding Rational Numbers and Percentages

• Rational numbers, percentages, and related mathematical concepts.

Rational Numbers

• A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, where p is the numerator, q is the denominator, and q ≠ 0.
• Every integer is a rational number as it can be expressed as a fraction with a denominator of 1 (e.g., 5 = 5/1).
• Rational numbers can be positive or negative.
• These numbers can be expressed in decimal form, either terminating after a finite number of digits or repeating the same finite sequence of digits.
• Examples of rational numbers include 2/3, -5/7, 6 (since it can be written as 6/1), and 0.75 (since it can be written as 3/4).
• Irrational numbers, such as π or √2, cannot be expressed as a fraction of integers, and their decimal representations neither terminate nor repeat.
• The set of rational numbers is denoted by the symbol ℚ.
• Rational numbers can be represented on a number line.
• Between any two distinct rational numbers, infinitely many other rational numbers exist.

Operations with Rational Numbers

• Addition requires finding a common denominator, then adding the numerators.
• Subtraction is similar to addition, involving finding a common denominator and subtracting the numerators.
• Multiplication involves multiplying the numerators to get the new numerator and multiplying the denominators to get the new denominator.
• Division is performed by multiplying the first rational number by the reciprocal of the second rational number.
• When performing operations, always simplify the resulting fraction to its lowest terms.

Properties of Rational Numbers

• Closure Property: The sum, difference, and product of two rational numbers invariably results in a rational number.
• Commutative Property: The order of addition or multiplication does not change the result (a + b = b + a, a * b = b * a).
• Associative Property: The grouping of rational numbers in addition or multiplication does not affect the result ((a + b) + c = a + (b + c), (a * b) * c = a * (b * c)).
• Distributive Property: Multiplication distributes over addition (a * (b + c) = a * b + a * c).
• Identity Property: The additive identity is 0 (a + 0 = a), and the multiplicative identity is 1 (a * 1 = a).
• Inverse Property: For every rational number a, there’s an additive inverse -a where a + (-a) = 0; for every non-zero rational number a, there’s a multiplicative inverse 1/a where a * (1/a) = 1.

Percentages

• A percentage expresses a number as a fraction of 100, indicated by the percent sign "%."
• "Percent" signifies "per hundred" or "out of one hundred."
• Percentages denote the size of one quantity relative to another.
• The base or total amount is considered 100% when calculating percentages.

Converting Between Percentages, Fractions, and Decimals

• Percentage to Fraction: Divide the percentage by 100 and then express the result as a fraction, simplifying if possible.
• Percentage to Decimal: Divide the percentage by 100.
• Fraction to Percentage: Multiply the fraction by 100.
• Decimal to Percentage: Multiply the decimal by 100.

Calculating Percentages

• To find a percentage of a quantity, multiply the quantity by the percentage expressed as a decimal or fraction.
• Percentage Increase: Calculated as [(New Value - Original Value) / Original Value] * 100%.
• Percentage Decrease: Calculated as [(Original Value - New Value) / Original Value] * 100%.

Applications of Percentages

• Discounts and Sales: Used in calculating sale prices and discounts on products.
• Interest Rates: Essential for determining interest earned on savings or owed on loans.
• Taxes: Applied in computing tax amounts on purchases or income.
• Statistics: Used to express proportions and ratios in various data forms.
• Financial Analysis: Utilized in analyzing financial data and performance metrics.

Working with Percentage Change

• Percentage change quantifies the extent of gain or loss in a quantity's value.
• Increase denotes a gain or growth, whereas decrease indicates a loss or reduction.
• The formula is: Percentage Change = [(Final Value - Initial Value) / Initial Value] × 100.
• A positive value indicates a percentage increase; conversely, a negative value indicates a percentage decrease.

Simple Interest

• Simple interest computes the interest charge on a sum.
• Formula: Simple Interest = P × R × T, where P is the principal amount, R is the interest rate, and T is the time period.
• The interest is directly proportional to the principal amount, the interest rate, and the duration of time.

Compound Interest

• Compound interest is computed on the principal amount, including the accumulated interest from prior periods.
• Formula: A = P (1 + r/n)^(nt), where A is the amount after n years, including interest, P is the principal amount, r is the annual interest rate (as a decimal), n is the number of times interest is compounded per year, and t is the number of years the money is invested or borrowed.
• The interest earned is reinvested, leading to earning interest on previously earned interest.