Decimals, Percentages, Mixed Numbers

Introduction

Key Concepts

Decimals

Decimals are a way of expressing fractions in a base-ten system, making it easier to perform arithmetic operations. A decimal number consists of two parts: the whole number and the fractional part, separated by a decimal point. For example, in the number 12.34, 12 is the whole number, and 34 represents the fractional part.

Converting Fractions to Decimals: To convert a fraction to a decimal, divide the numerator by the denominator.

For example, to convert $\frac{3}{4}$ to a decimal: $$ \frac{3}{4} = 3 \div 4 = 0.75 $$

Converting Decimals to Fractions: To convert a decimal to a fraction, write the decimal as a fraction with 1 as the denominator multiplied by the appropriate power of 10, then simplify.

For example, to convert 0.6 to a fraction: $$ 0.6 = \frac{6}{10} = \frac{3}{5} $$

Operations with Decimals:

• Addition and Subtraction: Align decimal points before performing the operations.
• Multiplication: Multiply as whole numbers and then place the decimal point in the product.
• Division: Move the decimal point in the divisor to make it a whole number and adjust the dividend accordingly.

Rounding Decimals: Decimals can be rounded to a specific number of decimal places. To round to the nearest tenth, look at the hundredths place:

For example, to round 3.276 to the nearest tenth: $$ 3.276 \approx 3.3 $$

Percentages

Percentages represent parts per hundred and are a way to express fractions and decimals. They are widely used in various fields such as finance, statistics, and everyday life.

Converting Fractions and Decimals to Percentages:

• To convert a fraction to a percentage, multiply by 100 and add the % symbol. $$ \frac{3}{4} = \frac{3}{4} \times 100\% = 75\% $$
• To convert a decimal to a percentage, multiply by 100 and add the % symbol. $$ 0.85 = 0.85 \times 100\% = 85\% $$

Converting Percentages to Fractions and Decimals:

• To convert a percentage to a fraction, write it over 100 and simplify. $$ 60\% = \frac{60}{100} = \frac{3}{5} $$
• To convert a percentage to a decimal, divide by 100. $$ 45\% = \frac{45}{100} = 0.45 $$

Calculating Percentage Increase and Decrease:

• Increase: $$ \text{New Value} = \text{Original Value} + (\text{Original Value} \times \text{Percentage Increase}) $$
• Decrease: $$ \text{New Value} = \text{Original Value} - (\text{Original Value} \times \text{Percentage Decrease}) $$

For example, a 20% increase on 50: $$ 50 + (50 \times 0.20) = 50 + 10 = 60 $$

Mixed Numbers

Mixed numbers combine whole numbers and fractions, providing a more intuitive way to express quantities greater than one.

Converting Improper Fractions to Mixed Numbers: To convert an improper fraction to a mixed number, divide the numerator by the denominator: $$ \frac{9}{4} = 2 \frac{1}{4} $$

Converting Mixed Numbers to Improper Fractions: Multiply the whole number by the denominator and add the numerator: $$ 2 \frac{1}{4} = \frac{(2 \times 4) + 1}{4} = \frac{9}{4} $$

Operations with Mixed Numbers:

• Addition and Subtraction: Convert to improper fractions, perform the operation, and simplify.
• Multiplication: Convert to improper fractions, multiply numerators and denominators, and simplify.
• Division: Convert to improper fractions, multiply by the reciprocal, and simplify.

For example, adding $1 \frac{1}{2}$ and $2 \frac{2}{3}$: $$ 1 \frac{1}{2} = \frac{3}{2}, \quad 2 \frac{2}{3} = \frac{8}{3} $$ $$ \frac{3}{2} + \frac{8}{3} = \frac{9}{6} + \frac{16}{6} = \frac{25}{6} = 4 \frac{1}{6} $$

Advanced Concepts

Theoretical Foundations of Decimals, Percentages, and Mixed Numbers

Understanding the theoretical underpinnings of decimals, percentages, and mixed numbers involves delving into the base-ten number system and the relationships between different numerical representations. Decimals are an extension of the integer system, allowing for the representation of values between whole numbers. Percentages provide a standardized way to express proportions relative to 100, facilitating comparisons and analysis across varied contexts. Mixed numbers bridge whole numbers and fractions, enhancing the flexibility of numerical expressions in real-life scenarios.

Mathematical Derivations:

• Converting between forms showcases the equivalence of different numerical representations. $$ \frac{a}{b} = a \div b = a \times 0.01 \times 100\% $$
• The properties of real numbers ensure that operations on decimals and fractions adhere to the axioms of arithmetic, maintaining consistency and reliability in calculations.

Complex Problem-Solving

Advanced problem-solving involves applying decimals, percentages, and mixed numbers in multifaceted scenarios, often integrating various mathematical concepts.

Example Problem: A store is offering a 15% discount on a jacket originally priced at \$80. Calculate the discounted price and determine how much additional 5% off would save.

Solution:

• First Discount: $$ 15\% \text{ of } \$80 = 0.15 \times 80 = \$12 $$ $$ \text{Discounted Price} = \$80 - \$12 = \$68 $$
• Additional 5% Discount on \$68: $$ 5\% \text{ of } \$68 = 0.05 \times 68 = \$3.40 $$ $$ \text{Final Price} = \$68 - \$3.40 = \$64.60 $$

Interdisciplinary Connections

Decimals, percentages, and mixed numbers are integral to various fields beyond pure mathematics. In economics, percentages are used to calculate interest rates, inflation, and growth rates. In science, decimals are essential for precision in measurements and data analysis. In everyday life, mixed numbers simplify the representation of quantities in cooking, construction, and financial transactions.

Real-World Application: In finance, understanding percentages is crucial for calculating compound interest, where the amount of interest earned grows over time based on both the initial principal and the accumulated interest. $$ A = P \left(1 + \frac{r}{n}\right)^{nt} $$ where:

• A = the amount of money accumulated after n years, including interest.
• P = principal amount (initial investment).
• r = annual interest rate (decimal).
• n = number of times interest is compounded per year.
• t = time the money is invested for in years.

For example, investing \$1,000 at an annual interest rate of 5% compounded annually for 3 years: $$ A = 1000 \left(1 + \frac{0.05}{1}\right)^{1 \times 3} = 1000 \times 1.157625 = \$1,157.63 $$

Advanced Applications

In statistics, percentages are used to represent data distributions, probabilities, and statistical measures such as mean, median, and mode percentages. Decimals are fundamental in representing continuous data and in performing precise calculations required for complex statistical analyses.

Percentiles and Deciles: These are measures that divide a dataset into equal parts, helping to understand the distribution and variability within the data.

• Percentile: Indicates the value below which a given percentage of observations fall.
• Decile: Divides the data into ten equal parts, each representing 10% of the dataset.

Comparison Table

Summary and Key Takeaways