Glossary

5.3 Decimals and Fractions

2 min read•june 24, 2024

Decimals and fractions are two ways to represent parts of a whole. Converting between them lets us compare values easily. This skill is crucial for everyday math, from cooking to finances.

Ordering decimals and fractions helps us rank things by size or value. It's useful in many situations, like comparing prices or test scores. Mastering these skills opens doors to more advanced math concepts.

Converting and Comparing Decimals and Fractions

Fraction and decimal conversions

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  • Divide by to convert to
    • 34=3÷4=0.75\frac{3}{4} = 3 \div 4 = 0.75
  • Write decimal as fraction over power of 10, then simplify to convert decimal to fraction
    • 0.6=610=350.6 = \frac{6}{10} = \frac{3}{5}
  • Convert repeating decimals to fractions using algebraic method
    • Let xx equal
    • Multiply both sides of equation by power of 10 to shift decimal point
    • Subtract original equation from new equation
    • Solve for xx and simplify resulting fraction
  • Convert fractions to percentages by multiplying by 100 ()

Ordering of decimals and fractions

  • Compare decimals digit by digit from left to right
    • Decimal with more digits is larger
    • Equal decimals have same digits
  • Convert fractions to decimals or find common denominator to compare
    • 34\frac{3}{4} and 56\frac{5}{6} converted to decimals: 0.75<0.8330.75 < 0.833
  • Convert all values to same form (decimals or fractions) to order from least to greatest or greatest to least
  • Use to compare approximate values when exact comparison is not necessary

Simplifying Expressions and Calculating with Circles

Order of operations with decimals

  • Follow : Parentheses, Exponents, and (left to right), and Subtraction (left to right)
  • Simplify expressions within parentheses first
  • Evaluate exponents (powers, roots, etc.)
  • Perform multiplication and division from left to right
  • Perform addition and subtraction from left to right
  • 2+(0.5×3)−14=2+1.5−0.25=3.252 + (0.5 \times 3) - \frac{1}{4} = 2 + 1.5 - 0.25 = 3.25

Circle calculations using decimals

  • Calculate using formula C=2πrC = 2\pi r, where rr is
    • If r=2.5r = 2.5, then C=2π(2.5)=5π≈15.71C = 2\pi(2.5) = 5\pi \approx 15.71
  • Calculate area using formula A=πr2A = \pi r^2
    • If r=32r = \frac{3}{2}, then A=π(32)2=94π≈7.07A = \pi(\frac{3}{2})^2 = \frac{9}{4}\pi \approx 7.07
  • Approximate π\pi as 3.14 or 227\frac{22}{7} when needed
  • Express final answer in terms of π\pi if exact value required
  • Use to express appropriate level of precision in calculations

Ratios and Proportions

  • A ratio compares two quantities and can be written as a fraction
  • A is an equation stating that two ratios are equal
  • Use cross multiplication to solve proportions
  • Apply ratios and proportions to solve real-world problems involving scaling, rates, and comparisons

Key Terms to Review (28)

Addition: Addition is a fundamental mathematical operation that combines two or more numbers or quantities to find their sum. It is a way of combining values to obtain a total or overall amount. This key term is essential in understanding various mathematical concepts and operations within the context of pre-algebra.
Circumference: Circumference is the distance around the outer edge or boundary of a circular object, such as a circle or sphere. It is a fundamental concept in geometry that is closely related to the measurement and understanding of circles.
Decimal: A decimal is a numerical representation that uses a decimal point to express a value that is not a whole number. Decimals allow for the precise expression of quantities and enable the representation of fractional values, which is essential in various mathematical and scientific applications.
Decimal to Fraction Conversion: Decimal to fraction conversion is the process of expressing a decimal number as an equivalent fraction. This is an important skill in mathematics, as it allows for the representation of numerical values in different forms, facilitating various calculations and problem-solving techniques.
Denominator: The denominator is the bottom number in a fraction that indicates the total number of equal parts the whole has been divided into. It represents the divisor and determines the size or value of each fractional part.
Division: Division is a fundamental mathematical operation that involves partitioning a quantity into equal parts or groups. It represents the inverse of multiplication, allowing us to find how many times one number is contained within another. This key term is essential in understanding various mathematical concepts, from whole numbers to exponents and scientific notation.
Equivalent Fractions: Equivalent fractions are different fractions that represent the same value or amount. They have different numerators and denominators, but the ratio between the numerator and denominator is the same, resulting in the same fractional value.
Fraction: A fraction is a numerical quantity that represents a part of a whole. It is expressed as a ratio of two integers, where the numerator represents the part and the denominator represents the whole.
Fraction to Decimal Conversion: Fraction to decimal conversion is the process of transforming a fraction, which represents a part of a whole, into its equivalent decimal representation. This conversion allows for more precise and convenient handling of fractional quantities in various mathematical and real-world applications.
Improper Fraction: An improper fraction is a fraction where the numerator is greater than the denominator. It represents a value greater than 1 and can be expressed as a mixed number or a decimal.
Least Common Multiple: The least common multiple (LCM) is the smallest positive integer that is divisible by two or more given integers. It is a fundamental concept in mathematics that is closely related to finding factors and multiples of numbers, as well as understanding the relationships between fractions and decimals.
Mixed Number: A mixed number is a representation of a quantity that combines a whole number and a proper fraction. It is used to express values that cannot be fully represented by a whole number alone.
Multiplication: Multiplication is a mathematical operation that involves the repeated addition of a number to itself. It is one of the four basic arithmetic operations, along with addition, subtraction, and division. Multiplication is used to find the total number of items or the area of a rectangle, and it is a fundamental concept in various mathematical contexts, including algebra, geometry, and statistics.
Numerator: The numerator is the top number in a fraction, representing the number of equal parts being considered or the number of units being counted. It is a crucial component in the visual representation and mathematical operations involving fractions.
Order of Operations: The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed to evaluate an expression. This term is crucial in the context of evaluating, simplifying, and translating expressions, as well as solving equations using various properties of equality.
PEMDAS: PEMDAS is an acronym that represents the order of operations in mathematics, which is the sequence in which mathematical operations should be performed to evaluate an expression. The term stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Understanding and applying the PEMDAS rule is crucial for accurately evaluating and simplifying mathematical expressions.
Percent: Percent is a way of expressing a quantity as a fraction of 100. It is a widely used concept in mathematics, finance, and various other fields to represent proportions, rates, and relationships between quantities.
Pi: Pi (π) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number, meaning its decimal representation never ends or repeats, and it is widely used in various mathematical and scientific applications, particularly in the study of geometry and trigonometry.
Place Value: Place value is a fundamental concept in mathematics that describes the value of a digit based on its position within a number. It is the foundation for understanding and working with whole numbers, decimals, and other numerical representations.
Proper Fraction: A proper fraction is a fractional representation where the numerator is less than the denominator, resulting in a value that is less than one. Proper fractions are a fundamental concept in understanding fractions and their operations within the context of 4.1 Visualize Fractions, 4.2 Multiply and Divide Fractions, 4.6 Add and Subtract Mixed Numbers, and 5.3 Decimals and Fractions.
Proportion: Proportion is the relationship between two or more quantities where the ratio between them remains constant. It is a fundamental concept that underlies various mathematical topics, including decimals, fractions, ratios, rates, and percent calculations.
Radius: The radius is the distance from the center of a circle to its perimeter. It is a fundamental measurement in geometry that describes the size and shape of circular objects and is essential for calculating various properties related to circles, such as circumference, area, and volume.
Reciprocal: The reciprocal of a number is the value obtained by dividing 1 by that number. It represents the inverse or opposite of the original value, and is often denoted by the exponent -1. The reciprocal is a fundamental concept in mathematics that has applications across various topics, including the operations of multiplication, division, and solving equations.
Repeating Decimal: A repeating decimal is a decimal number in which one or more digits in the decimal part repeat infinitely. This pattern of repeating digits is a characteristic of certain fractions when expressed as a decimal.
Rounding: Rounding is the process of approximating a numerical value to a simpler value, typically to a specified number of decimal places or significant figures. It is a fundamental concept in mathematics and is particularly relevant in the context of whole numbers and decimals.
Significant Figures: Significant figures refer to the meaningful digits in a measurement or calculation, providing an indication of the precision and accuracy of the value. This concept is crucial in the context of decimals, fractions, and scientific notation, as it helps determine the appropriate level of precision when representing and manipulating numerical data.
Simplification: Simplification is the process of reducing an expression or equation to its most basic and concise form without changing its underlying meaning or value. This concept is crucial in various mathematical operations, including working with fractions, mixed numbers, decimals, and polynomials, as it helps to make complex expressions easier to understand, manipulate, and perform further calculations on.
Terminating Decimal: A terminating decimal is a decimal number that can be expressed as a fraction with a finite number of digits in the denominator. It is a decimal representation that eventually ends or 'terminates' after a finite number of digits.
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