Rational Mastery Worksheet v3 Rational Mastery Blueprint
Quiz Review: Units 9-10
Analytical Solving & Modeling
Name: ____________________________________
Date: ________
1
Solving by Cross-Multiplication
Comprehensive Guide
Solve for x: \( \frac{x}{x-2} - \frac{x+3}{x} = 0 \)
1. Form Proportion \( \frac{x}{x-2} = \frac{x+3}{x} \)
2. Cross-Multiply \( x(x) = (x-2)(x+3) \)
3. Solve: \( x = 6 \)
Practice Problem: Solve for x: \( \frac{x}{x+1} - \frac{x-4}{x} = 0 \)
Show all work
2
Modeling with Ratios
Step-by-Step Logic
Shelter has 3x cats and x dogs. Add 5 cats. Now cats are 80% (0.8) of population.
1. Identify Totals
Cats: \( 3x+5 \), Total: \( 4x+5 \)
2. Build Proportion \( \frac{3x+5}{4x+5} = 0.8 \)
3. Solve Equation
\( 3x + 5 = 0.8(4x + 5) \)
\( x = 5 \)
Practice Problem: Club has 2 boys for 1 girl. Let x = girls. Add 6 boys. Boys now 75%. Solve for x.
Show all work
Rational Mastery Blueprint (cont.)
Part 2 Review
3
Conductance & Resistance
Step-by-Step Guide
R1 = 3x. R2 = x. Total R = \( 3 \Omega \). Find x.
1. Set Up Equation \( \frac{1}{3} = \frac{1}{3x} + \frac{1}{x} \)
2. Combine Terms \( \frac{1}{3} = \frac{4}{3x} \)
3. Solution: \( x = 4 \Omega \)
Practice Problem: Resistor A has 3 times resistance of B. Total R = \( 4 \Omega \). Find Resistor B (x).
Show all work
4
Winning Streak Modeling
Logical Breakdown
Wins: 70% of 40 (28 wins). Reach 80%.
1. Target Ratio \( \frac{28 + x}{40 + x} = 0.80 \)
2. Solve the Change
\( x = 20 \) games
Practice Problem: Made 12/20 free throws (60%). How many wins in a row reach 75%?
Show all work
Practice leads to mastery • Units 9-10 Review blueprint
Rational Mastery Answer Key Answer Key
Rational Mastery Blueprint
1 Solving by Cross-Multiplication
Equation: \( \frac{x}{x+1} - \frac{x-4}{x} = 0 \)
1. Proportion: \( \frac{x}{x+1} = \frac{x-4}{x} \)
2. Cross-multiply: \( x^2 = (x+1)(x-4) \)
3. Expand: \( x^2 = x^2 - 3x - 4 \)
4. Solve: \( 0 = -3x - 4 \implies 3x = -4 \)
Answer: \( x = -\frac{4}{3} \)
2 Modeling with Ratios
Model: \( \frac{2n + 6}{3n + 6} = 0.75 \)
1. Proportion: \( 2n + 6 = 0.75(3n + 6) \)
2. Distribute: \( 2n + 6 = 2.25n + 4.5 \)
3. Solve: \( 1.5 = 0.25n \implies n = 6 \)
Note: Subtract \( 2n \) and \( 4.5 \) from both sides to isolate the variable \( n \).
Answer: \( 6 \) girls initially
3 Conductance & Resistance
Equation: \( \frac{1}{4} = \frac{1}{3x} + \frac{1}{x} \)
1. Combine: \( \frac{1}{4} = \frac{1}{3x} + \frac{3}{3x} = \frac{4}{3x} \)
2. Cross: \( 1(3x) = 4(4) \implies 3x = 16 \)
Note: Cross-multiply to clear fractions by setting diagonal products equal.
Answer: \( \frac{16}{3} \text{ } \Omega \approx 5.33 \text{ } \Omega \)
4 Winning Streak Modeling
Model: \( \frac{12 + x}{20 + x} = 0.75 \)
1. Cross: \( 12 + x = 0.75(20 + x) \)
2. Distribute: \( 12 + x = 15 + 0.75x \)
3. Solve: \( 0.25x = 3 \implies x = 12 \)
Answer: \( 12 \) wins
Verified algebraic solutions for Rational Mastery Quiz Review. Grayscale design for optimal black and white printing.
Rational Mastery Answer Key v3 Answer Key
Rational Mastery Blueprint
1 Solving by Cross-Multiplication
Equation: \( \frac{x}{x+1} - \frac{x-4}{x} = 0 \)
1. Proportion: \( \frac{x}{x+1} = \frac{x-4}{x} \)
2. Cross-multiply: \( x^2 = (x+1)(x-4) \)
3. Expand: \( x^2 = x^2 - 3x - 4 \)
4. Solve: \( 0 = -3x - 4 \implies 3x = -4 \)
Answer: \( x = -4/3 \)
2 Modeling with Ratios
Model: \( \frac{2n + 6}{3n + 6} = 0.75 \)
1. Proportion: \( 2n + 6 = 0.75(3n + 6) \)
2. Distribute: \( 2n + 6 = 2.25n + 4.5 \)
3. Solve: \( 1.5 = 0.25n \implies n = 6 \)
Note: Subtract \( 2n \) and \( 4.5 \) from both sides to isolate the variable \( n \).
Answer: \( n = 6 \) girls
3 Conductance & Resistance
Equation: \( \frac{1}{4} = \frac{1}{3x} + \frac{1}{x} \)
1. Combine: \( \frac{1}{4} = \frac{1}{3x} + \frac{3}{3x} = \frac{4}{3x} \)
2. Cross: \( 1(3x) = 4(4) \implies 3x = 16 \)
Note: Cross-multiplying clears the fractions by equating products of means and extremes.
Answer: \( 16/3 \text{ } \Omega \approx 5.33 \text{ } \Omega \)
4 Winning Streak Modeling
Model: \( \frac{12 + x}{20 + x} = 0.75 \)
1. Cross: \( 12 + x = 0.75(20 + x) \)
2. Distribute: \( 12 + x = 15 + 0.75x \)
3. Solve: \( 0.25x = 3 \implies x = 12 \)
Answer: \( 12 \) wins
Verified algebraic solutions for Rational Mastery Review blueprint. Optimized for black and white grayscale printing.
Rational Mastery Worksheet v3 Final Rational Mastery Blueprint
Quiz Review: Units 9-10
Analytical Solving & Modeling
Student Name: ____________________________________
Date: ________
1
Solving by Cross-Multiplication
Comprehensive Guide
Solve for x:
1. Form Proportion
\[ \frac{x}{x-2} = \frac{x+3}{x} \]
2. Cross-Multiply
\[ x^2 = (x-2)(x+3) \]
3. Final Answer: \( x = 6 \)
Practice Problem:
Show all work
2
Modeling with Ratios
Step-by-Step Logic
Shelter has \( 3x \) cats and \( x \) dogs. Add 5 cats. Now cats are 80% (0.8) of population.
1. Identify New Totals
Cats: \( 3x+5 \)
2. Build Proportion \[ \frac{3x+5}{4x+5} = 0.8 \]
3. Final Answer: \( x = 5 \)
Practice Problem:
Show all work
3
Conductance & Resistance
Physical Modeling Guide
\( R_1 = 3x \). \( R_2 = x \). Total \( R = 3 \Omega \). Find \( x \).
1. Set Up Equation \[ \frac{1}{3} = \frac{1}{3x} + \frac{1}{x} \]
2. Combine Terms
Multiply \( 1/x \) by \( 3/3 \) to get \( 3/3x \).
\[ \frac{1}{3} = \frac{1+3}{3x} = \frac{4}{3x} \]
3. Solution \( x = 4 \Omega \)
Practice Problem:
Show all work
4
Winning Streak Modeling
Logical Breakdown
Player won 70% of 40 games (28 wins). Reach 80%.
1. Target Ratio \[ \frac{28 + x}{40 + x} = 0.80 \]
2. Algebraic Solution
Add streaks to wins and total games.
\( 28+x = 0.8(40+x) \)
\( x = 20 \text{ wins} \)
Practice Problem:
Show all work
Practice leads to mastery • Units 9-10 Review blueprint finished
Rational Mastery Answer Key v3 Final Answer Key
Rational Mastery Blueprint
1 Solving by Cross-Multiplication
Equation: \( \frac{x}{x+1} - \frac{x-4}{x} = 0 \)
1. Proportion: \( \frac{x}{x+1} = \frac{x-4}{x} \)
2. Cross-multiply: \( x^2 = (x+1)(x-4) \)
3. Expand right side: \( x^2 = x^2 - 3x - 4 \)
4. Solve for x: \( 0 = -3x - 4 \implies 3x = -4 \)
Final Answer: \( x = -4/3 \) (or \( \approx -1.33 \))
2 Modeling with Ratios
Model: \( \frac{2x + 6}{3x + 6} = 0.75 \)
1. Form proportion: \( 2x + 6 = 0.75(3x + 6) \)
2. Distribute: \( 2x + 6 = 2.25x + 4.5 \)
3. Isolate x: \( 1.5 = 0.25x \implies x = 6 \)
Note: Subtract 2x and 4.5 from both sides to isolate the variable x.
Final Answer: \( x = 6 \) girls initially
3 Conductance & Resistance
Equation: \( \frac{1}{4} = \frac{1}{3x} + \frac{1}{x} \)
1. Combine right side: \( \frac{1}{4} = \frac{1}{3x} + \frac{3}{3x} = \frac{4}{3x} \)
2. Cross-multiply: \( 1(3x) = 4(4) \implies 3x = 16 \)
3. Final result: \( x = 16/3 \)
Note: Cross-multiplying clears the fractions by equating products of means and extremes.
Final Answer: \( x = 16/3 \text{ Ohms} \approx 5.33 \text{ Ohms} \)
4 Winning Streak Modeling
Model: \( \frac{12 + x}{20 + x} = 0.75 \)
1. Cross-multiply: \( 12 + x = 0.75(20 + x) \)
2. Distribute: \( 12 + x = 15 + 0.75x \)
3. Solve for x: \( 0.25x = 3 \implies x = 12 \)
Final Answer: \( 12 \) wins in a row
Verified algebraic solutions for Rational Mastery Quiz Review. Optimized for high-contrast grayscale printing.
Rational Mastery Worksheet v3 Final v2 Rational Mastery Blueprint
Quiz Review: Units 9-10
Analytical Solving & Modeling
Name: ____________________________________
Date: ________
1
Solving by Cross-Multiplication
Guide
Solve for x: \[ \frac{x}{x-2} - \frac{x+3}{x} = 0 \]
1. Form Proportion \[ \frac{x}{x-2} = \frac{x+3}{x} \]
2. Cross-Multiply \[ x^{2} = (x-2)(x+3) \]
3. Solve: x = 6
Practice: Solve for x: \[ \frac{x}{x+1} - \frac{x-4}{x} = 0 \]
Show work
2
Modeling with Ratios
Logic
Shelter has 3x cats & x dogs. Add 5 cats. Now cats are 80% (0.8) of pop.
1. Identify Totals
Cats: 3x+5, Total: 4x+5
2. Build Proportion
\[ \frac{3x+5}{4x+5} = 0.8 \]
3. Result: x = 5
Practice: Club has 2 boys for every 1 girl. Let x = girls. Add 6 boys. Boys now 75%. Solve for x.
Show work
Part 2: Circuit Applications
3
Conductance & Resistance
Guide
R1 = 3x. R2 = x. Total R = 3 Ohms. Find x.
1. Set Up Equation \[ \frac{1}{3} = \frac{1}{3x} + \frac{1}{x} \]
2. Combine Terms
Multiply 1/x by 3/3 to get 3/3x (Common Denom).
\[ \frac{1}{3} = \frac{1}{3x} + \frac{3}{3x} \]
\[ \frac{1}{3} = \frac{4}{3x} \]
3. Final Result: x = 4 Ohms
Practice: Resistor A has 3 times the resistance of B. Total R = 4 Ohms. Find Resistor B (x).
Show work
Part 3: Performance Analysis
4
Winning Streak Modeling
Guide
Wins: 70% of 40 (28 wins). Target: 80%.
1. Target Ratio \[ \frac{28+x}{40+x} = 0.80 \]
2. Solve for Streaks (x)
Add streaks to both wins and total games.
28 + x = 0.8(40 + x)
Final Result: x = 20 wins
Practice: Made 12/20 free throws (60%). How many must they make in a row to reach 75%?
Show work
Practice leads to mastery • Units 9-10 Review Blueprint Finished
