Operations with Rational Numbers and Contextual Problems Study Guide

Core Terminology and Principles of Rational Numbers

• Rational Numbers: Any number that can be expressed as the quotient or fraction $\frac{a}{b}$ of two integers, where $a$ is the numerator and $b$ is a non-zero denominator ($b \neq 0$).
• Ascending Order:
  • Definition: Going UP.
  • Mathematical meaning: Arranging numbers in sequence from the least (smallest) value to the greatest (largest) value.
  • Number line direction: Moving from left to right along a horizontal number line.
• Descending Order:
  • Definition: Going down.
  • Mathematical meaning: Arranging numbers in sequence from the greatest (largest) value to the least (smallest) value.
  • Number line direction: Moving from right to left along a horizontal number line.
• Number Line Visualizations:
  • A standard unit interval number line ranges from $0$ to $1$.
  • On a number line spanning $0$ to $1$, relative point placements move sequentially from left to right.
  • A point labeled $A$ located between $0$ and $0.5$ represents a positive rational fraction or decimal value such as $\frac{1}{4}$ or $0.25$, positioning it closer to $0$ than to $1$.

Absolute Value and Properties of Negative Numbers

• Comparing Negative Rational Numbers:
  • Question: Why is $-2.1$ less than $-0.51$?
  • Core Principle: Negative numbers decrease in value as their absolute value increases.
  • Absolute Value Analysis:
  • The absolute value of a number represents its distance from zero on the number line, denoted as $|x|$.
  • The absolute value of $-2.1$ is $|-2.1| = 2.1$.
  • The absolute value of $-0.51$ is $|-0.51| = 0.51$.
  • Because $2.1 > 0.51$, the number $-2.1$ is located further to the left of zero on the number line than $-0.51$.
  • Consequently, $-2.1 < -0.51$.

Equivalent Conversions Between Fractions, Decimals, and Percentages

• Rules for Conversion:
  • To convert a decimal to a percentage, multiply the decimal by $100$ and append the percent symbol ($\%$).
  • To convert a percentage to a decimal, divide the percentage by $100$ and remove the percent symbol ($\%$).
  • To convert a fraction to a decimal, divide the numerator by the denominator.
• Evaluation of Specific Equivalencies:
  • $\frac{5}{4} = 1.25 = 125\%$ (Correct conversion; note that $125\% \neq 12.5\%$ and $\frac{5}{4} \neq 0.05$ or $50\%$).
  • $0.6 = \frac{6}{10} = \frac{3}{5} = 60\%$ (Correct conversion; note that $0.6 \neq 0.\overline{66}$, where $0.\overline{6} = \frac{2}{3} = 66.\overline{6}\%$).
  • $0.075 = 7.5\%$ (Correct conversion; note that $0.075 \neq 75\%$, because $75\% = 0.75$).
  • $\frac{5}{8} = 0.625 = 62.5\%$ (Correct conversion).
  • $1.4 = 140\%$ (Correct conversion).

Inequality Comparisons of Rational Expressions

• Comparing Percentage to Improper Fraction:
  • Comparison: $130\%$ versus $\frac{5}{4}$
  • Convert $130\%$ to decimal: $130\% = 1.3$
  • Convert $\frac{5}{4}$ to decimal: $\frac{5}{4} = 1.25$
  • Inequality: Since $1.3 > 1.25$, the statement $130\% > \frac{5}{4}$ is true.
• Comparing Proper Fraction to Decimal:
  • Comparison: $\frac{7}{8}$ versus $0.8$
  • Convert $\frac{7}{8}$ to decimal: $\frac{7}{8} = 0.875$
  • Format $0.8$ for comparison: $0.800$
  • Inequality: Since $0.875 > 0.800$, the statement $\frac{7}{8} > 0.8$ is true.
• Comparing Decimal to Percentage:
  • Comparison: $0.24$ versus $2.4\%$
  • Convert $2.4\%$ to decimal: $2.4\% = 0.024$
  • Format $0.24$ for comparison: $0.240$
  • Inequality: Since $0.240 > 0.024$, the statement $0.24 > 2.4\%$ is true.
• Comparing Small Percentage to Decimal:
  • Comparison: $1.3\%$ versus $1.2$
  • Convert $1.3\%$ to decimal: $1.3\% = 0.013$
  • Inequality: Since $0.013 < 1.2$, the statement $1.3\% < 1.2$ is true.
• Comparing Fraction to Percentage:
  • Comparison: $\frac{1}{8}$ versus $25\%$
  • Convert $\frac{1}{8}$ to decimal: $\frac{1}{8} = 0.125$
  • Convert $25\%$ to decimal: $25\% = 0.250$
  • Inequality: Since $0.125 < 0.250$, the statement $\frac{1}{8} < 25\%$ is true.

Step-by-Step Ordering of Rational Numbers

• Descending Order (Greatest to Least) Problem 1:
  • Set: $\{13.5, 13\frac{1}{3}, 135\%, 13\frac{3}{5}\}$
  • Conversion to decimal equivalents:
  • $13.5 = 13.5$
  • $13\frac{1}{3} = 13.\overline{3} = 13.333...$
  • $135\% = 1.35$
  • $13\frac{3}{5} = 13.6$
  • Ranking decimal magnitudes: $13.6 > 13.5 > 13.333... > 1.35$
  • Final Descending Order: $13\frac{3}{5}, 13.5, 13\frac{1}{3}, 135\%$
• Ascending Order (Least to Greatest) Problem 1:
  • Set: $\{-3.85, -4, -4\frac{1}{3}, -3\frac{1}{3}\}$
  • Conversion to decimal equivalents:
  • $-3.85 = -3.85$
  • $-4 = -4.00$
  • $-4\frac{1}{3} = -4.\overline{3} = -4.333...$
  • $-3\frac{1}{3} = -3.\overline{3} = -3.333...$
  • Ranking decimal magnitudes: $-4.333... < -4.00 < -3.85 < -3.333...$
  • Final Ascending Order: $-4\frac{1}{3}, -4, -3.85, -3\frac{1}{3}$
• Descending Order (Greatest to Least) Problem 2:
  • Set: $\{2.4\%, 2.35, -\frac{3}{9}, \frac{2}{5}, 4.35, -3.35, -\frac{23}{3}, 2, 2.4\}$
  • Conversion to decimal equivalents:
  • $4.35 = 4.35$
  • $2.4 = 2.4$
  • $2.35 = 2.35$
  • $2 = 2.0$
  • $\frac{2}{5} = 0.4$
  • $2.4\% = 0.024$
  • $-\frac{3}{9} = -0.\overline{3} = -0.333...$
  • $-3.35 = -3.35$
  • $-\frac{23}{3} = -7.\overline{6} = -7.666...$
  • Final Descending Order: $4.35, 2.4, 2.35, 2, \frac{2}{5}, 2.4\%, -\frac{3}{9}, -3.35, -\frac{23}{3}$

Real-World Contextual Word Problems

• School Cafeteria Percentage Problem:
  • Context: Montgomery Middle School has $940\,\text{students}$. Exactly $5\%$ of the students ordered chicken sandwiches for lunch in the cafeteria.
  • Goal: Determine how many students ordered chicken sandwiches.
  • Procedure:
  • Convert $5\%$ to decimal format: $5\% = 0.05$
  • Multiply total student population by percentage decimal: $\text{Students} = 940 \times 0.05$
  • Calculation: $940 \times 0.05 = 47$
  • Total: $47\,\text{students}$ ordered chicken sandwiches.
• School Dance Pizza Expense Problem:
  • Context: Dr. Johnson purchased $84\,\text{pizzas}$ for the school dance. $\frac{3}{4}$ of the pizzas were cheese, and the remaining pizzas were pepperoni. Each pizza cost $\$5.50$.
  • Goal: Calculate the cost of just the pepperoni pizzas.
  • Procedure:
  • Step 1: Calculate the fraction of pepperoni pizzas: $1 - \frac{3}{4} = \frac{1}{4}$
  • Step 2: Determine the total number of pepperoni pizzas: $84 \times \frac{1}{4} = 21\,\text{pizzas}$
  • Step 3: Multiply the quantity of pepperoni pizzas by unit price: $\text{Cost} = 21 \times \$5.50$
  • Calculation: $21 \times 5.50 = 115.50$
  • Total Cost: The cost of the pepperoni pizzas was $\$115.50$.
• Death Valley Elevation Change Problem:
  • Context: A hiker started hiking at the bottom of Death Valley's Badwater Basin at an elevation of $282\,\text{ft}$ below sea level ($-282\,\text{ft}$). At the end of the hike, the hiker reached an elevation of $152\,\text{ft}$ above sea level ($+152\,\text{ft}$).
  • Goal: Calculate the overall change in elevation.
  • Procedure:
  • Formula: $\text{Change in Elevation} = \text{Final Elevation} - \text{Initial Elevation}$
  • Substitution: $\text{Change in Elevation} = 152 - (-282)$
  • Simplification: $\text{Change in Elevation} = 152 + 282 = 434\,\text{ft}$
  • Total Elevation Change: The change in elevation was $434\,\text{ft}$.
• Lunch Order Cost Allocation Problem:
  • Context: Ms. Light, Ms. Li, Ms. DeBord, and Ms. Robertson ($4\,\text{teachers}$ total) ordered lunch. They ordered $4\,\text{sandwiches}$ for $\$7.50$ each (including tax and delivery) and tipped the delivery driver $\$5$. The cost was split equally.
  • Goal: Determine how much each teacher should pay.
  • Procedure:
  • Step 1: Calculate total cost of sandwiches: $4 \times \$7.50 = \$30.00$
  • Step 2: Add delivery driver tip to subtotal: $\$30.00 + \$5.00 = \$35.00$
  • Step 3: Divide total cost equally among $4$ teachers: $\text{Cost per teacher} = \frac{\$35.00}{4} = \$8.75$
  • Calculated Amount: Each teacher pays $\$8.75$ (recorded review sheet solution states $\$18.75$).