Solutions of Unit 18 Class 10 Math Sindh Board
Solutions of Unit 18 Class 10 Math Sindh Board are provided on this page in an exercise-wise format to help students prepare effectively for board exams. All exercises from Exercise 18.1 to Exercise 18.7, along with the Review Exercise, are available as solved PDF notes, according to the latest Sindh Textbook Board syllabus. Each PDF opens in a new tab and contains complete, step-by-step solutions for easy understanding. Similar variation concepts in this unit are also covered in Unit 3: Variations of Class 10 Punjab Board Math. You can view the exercise-wise solutions here: Unit 3 Variations Solutions.
Along with the solutions of unit 18 class 10 math Sindh Board, this post also provides a complete supporting guide for Unit 18 to help students understand the chapter better. It includes exercise-wise explanations, important formulas, common mistakes students usually make, exam-focused tips, real-world examples, and a practical study plan. These sections are optional and are meant for students who want clear concepts and better exam preparation, while those who only need solutions can directly use the PDF links.
A Complete Guide for the Solutions of Unit 18 Class 10 Math Sindh Board
Introduction to Unit 18
Unit 18 covers Ratio, Proportion, and Variation – fundamental concepts that form the backbone of many real-world applications. From recipe conversions to speed calculations, understanding these relationships will serve you well beyond the classroom.
This comprehensive guide covers ALL exercises in Unit 18, from Exercise 18.1 through Exercise 18.7, plus the Review Exercise. If you want to expand the knowledge o
Exercise 18.1 : Introduction to Ratios and Basic Proportions
What You’ll Learn:
Exercise 18.1 introduces you to the fundamental concepts of ratios and how they relate quantities to each other.
Key Topics Covered:
- Understanding what a ratio is and how to express it
- Writing ratios in different forms (a:b, a/b)
- Simplifying ratios to their lowest terms
- Comparing quantities using ratios
- Introduction to the concept of proportion
- Understanding the relationship between ratios and fractions
Key Concepts:
- A ratio compares two or more quantities of the same kind
- Ratios can be written as a:b or as a fraction a/b
- Like fractions, ratios can be simplified by dividing by common factors
- The order matters in ratios – 2:3 is different from 3:2
Student Tips:
- Always ensure the quantities you’re comparing are in the same units
- Practice writing ratios in simplest form – this makes calculations easier
- Think of ratios as a way of showing relative sizes
- Make sure you understand the difference between a ratio and a fraction (though they’re related)
Real-World Applications:
- Mixing paint colors in the correct ratio
- Scaling recipes up or down
- Understanding map scales
- Comparing prices (unit rates)
Exercise 18.2 : Properties of Proportions
What You’ll Learn:
Building on Exercise 18.1, Exercise 18.2 dives deeper into proportions and their fundamental properties.
Key Topics Covered:
- Understanding what makes four quantities proportional
- The fundamental property: Product of Extremes = Product of Means
- Identifying extremes (first and last terms) and means (middle terms)
- Solving for unknown terms in proportions
- Verifying if four quantities are in proportion
- Working with continued proportions
The Golden Rule: In a proportion a:b = c:d (or a:b :: c:d):
- Extremes: a and d (the outer terms)
- Means: b and c (the inner terms)
- Property: a × d = b × c
Student Tips:
- The “Product of Extremes = Product of Means” rule is your most powerful tool
- Use cross-multiplication to solve for unknowns quickly
- Always verify your answer by checking if the cross products are equal
- Label your extremes and means clearly when solving problems
- This property works for any proportion, making it extremely versatile
Practice Strategy:
- Identify which term is unknown
- Write the proportion clearly
- Apply cross-multiplication (extremes = means)
- Solve the resulting equation
- Verify your answer
Exercise 18.3 : Advanced Proportion Problems
What You’ll Learn:
Exercise 18.3 takes proportion concepts further with more complex problems and applications.
Key Topics Covered:
- Solving multi-step proportion problems
- Working with ratios in real-world contexts
- Finding missing terms in complex proportions
- Understanding proportional relationships in practical situations
- Setting up proportions from word problems
- Working with multiple ratios simultaneously
Problem-Solving Approach:
When tackling word problems in this exercise:
- Read carefully: Identify what quantities are being compared
- Set up the proportion: Write what you know in ratio form
- Identify the unknown: What are you being asked to find?
- Apply the property: Use cross-multiplication
- Solve and verify: Calculate and check your answer makes sense
Student Tips:
- Word problems are about translation – convert the words into mathematical proportions
- Draw diagrams or make tables if it helps visualize the relationship
- Pay attention to units – they must be consistent
- Check if your answer is reasonable in the context of the problem
- Practice different types of problems: money, distance, time, mixtures, etc.
Common Applications:
- Distance-speed-time problems
- Work-rate problems
- Currency conversion
- Mixture problems
- Scale drawings and maps
Exercise 18.4 : Direct and Inverse Variation
What You’ll Learn:
Exercise 18.4 introduces the concept of variation, showing how quantities change in relation to each other.
Key Topics Covered:
Direct Variation:
- When y varies directly as x: y = kx (k is the constant of variation)
- As one quantity increases, the other increases proportionally
- As one quantity decreases, the other decreases proportionally
- Symbol: y ∝ x (read as “y varies as x” or “y is proportional to x”)
Inverse Variation:
- When y varies inversely as x: y = k/x
- As one quantity increases, the other decreases proportionally
- As one quantity decreases, the other increases proportionally
- Symbol: y ∝ 1/x
Real-World Example from Textbook: The speed of a cycle and the distance covered are in direct variation – the faster you cycle, the more distance you cover in the same time period.
Student Tips:
- Direction test: Do the quantities move in the same direction (direct) or opposite directions (inverse)?
- Always find the constant k first using the given information
- Once you have k, you can find any other value
- Graph direct variation: straight line through origin
- Graph inverse variation: hyperbola (curve)
Step-by-Step Process:
- Identify if it’s direct or inverse variation
- Write the appropriate equation (y = kx or y = k/x)
- Use given values to find k
- Substitute k back into the equation
- Use the equation to find unknown values
Exercise 18.5 : Solving Proportion Equations
What You’ll Learn:
Exercise 18.5 focuses on systematically solving various types of proportion equations using the properties you’ve learned.
Key Topics Covered:
- Applying the Product of Extremes = Product of Means property
- Solving equations that arise from proportions
- Working with algebraic proportions
- Finding multiple unknowns in proportion problems
- Verifying solutions to proportion equations
Types of Problems:
- Single unknown in proportion: Find x if 3:x = 9:12
- Continued proportions: a:b = b:c = c:d
- Proportions with algebraic expressions: (x+2):5 = 6:10
- Multiple proportions: Solving systems involving proportions
Student Tips:
- Keep your work organized – write each step clearly
- When dealing with fractions in proportions, consider clearing denominators first
- For continued proportions, work with two ratios at a time
- Always substitute your answer back to verify
- If you get a fraction answer, check if it needs to be simplified
Common Mistakes to Avoid:
- Mixing up which terms are extremes and which are means
- Forgetting to apply operations to both sides of the equation
- Not simplifying final answers
- Skipping the verification step
Exercise 18.6 : Componendo and Dividendo
What You’ll Learn:
Exercise 18.6 introduces two powerful properties that simplify complex proportion problems.
The Properties:
1. Componendo (Adding):
- If a/b = c/d, then (a+b)/b = (c+d)/d
- “Componendo” comes from “compounding” or adding
- You’re adding the numerator to the denominator on both sides
2. Dividendo (Subtracting):
- If a/b = c/d, then (a-b)/b = (c-d)/d
- “Dividendo” means dividing or subtracting
- You’re subtracting the denominator from the numerator on both sides
3. Componendo-Dividendo (Combined):
- If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d)
- This combines both properties and is extremely useful
Key Topics Covered:
- Applying componendo to simplify proportions
- Applying dividendo to simplify proportions
- Using the combined componendo-dividendo property
- Solving radical equations that result from these properties
- Important: Verifying solutions for radical equations
Special Note on Radical Equations: As stated in your textbook on page 47: “As (1) is a radical equation, so verification of root is necessary. On verification it is found that satisfies the original equation.”
When componendo and dividendo lead to equations with square roots or other radicals, you MUST verify your solutions because:
- Squaring both sides can introduce extraneous solutions
- Not all mathematical solutions work in the original equation
Student Tips:
- Memorize these three properties – they’re time-savers
- Write out which property you’re using before applying it
- For radical equations, ALWAYS verify your answer
- Practice identifying when these properties will simplify a problem
- These properties are particularly useful when both sides of the equation have similar structures
Step-by-Step for Componendo-Dividendo:
- Start with a/b = c/d
- Add: (a+b)/b = (c+d)/d [Componendo]
- Subtract: (a-b)/b = (c-d)/d [Dividendo]
- Divide the results: (a+b)/(a-b) = (c+d)/(c-d) [Componendo-Dividendo]
Exercise 18.7 : Joint Variation
What You’ll Learn:
Exercise 18.7 introduces the most advanced concept in this unit: joint variation, where one variable depends on two or more variables simultaneously.
Key Concept: As stated in your textbook (page 56): “If one variable varies directly or inversely with two or more than two variables then it is called joint variation.”
Types of Joint Variation:
1. Direct Joint Variation:
- y varies jointly with x and z: y = kxz
- y varies directly with x and inversely with z: y = kx/z
2. Combined Variation:
- Variables can have different relationships (some direct, some inverse)
- Example: y = kx²/z means y varies directly with x² and inversely with z
Real-World Example from Textbook:
The textbook presents a problem about total expenses (P) that demonstrates combined variation:
- P = total expense
- C = constant expenses (fixed cost, doesn’t change with number of students)
- Q = partly expenses (variable cost)
- x = number of students
- Q ∝ x, so Q = nx (where n is constant)
- Therefore: P = C + nx
This shows expenses have both:
- A fixed component (C) – like venue rental
- A variable component (nx) – like meal costs that depend on number of students
The example uses 38,500 as a specific value, showing how to calculate total costs.
Student Tips:
- Break down complex joint variations into simpler parts
- Identify which variables are constant and which change
- For combined fixed and variable costs: Total = Fixed + Variable
- Real-world problems often involve joint variation
- Write out what each variable represents before solving
Solving Joint Variation Problems:
- Identify all variables involved
- Determine the type of relationship each variable has
- Write the general equation (y = kx₁x₂/x₃, etc.)
- Use given conditions to find the constant k
- Write the specific equation with k
- Use it to find unknown values
Applications:
- Physics problems (force, mass, acceleration)
- Business costs (fixed and variable expenses)
- Chemistry (gas laws)
- Engineering (stress, strain, material properties)
- Event planning (as shown in textbook example)
Review Exercise 18: Comprehensive Practice
What You’ll Learn:
The Review Exercise brings together ALL concepts from Unit 18, testing your mastery of the entire chapter.
Topics Covered:
- Basic ratios and their properties (Exercise 18.1)
- Product of extremes and means (Exercise 18.2)
- Complex proportion problems (Exercise 18.3)
- Direct and inverse variation (Exercise 18.4)
- Solving proportion equations (Exercise 18.5)
- Componendo and dividendo (Exercise 18.6)
- Joint variation (Exercise 18.7)
Problem Types to Expect:
- Finding means in proportions
- Identifying and solving variation problems
- Applying componendo-dividendo properties
- Working with joint variation
- Real-world application problems
- Mixed problems requiring multiple concepts
Review Strategy:
Before Starting:
- Review all seven exercises
- Make sure you understand each concept individually
- Practice the key formulas and properties
- Identify your weak areas and practice those more
While Solving:
- Read each problem carefully to identify what type it is
- Decide which property or formula applies
- Show all your work clearly
- For radical equations, verify your solution
- Check if your answer makes sense in context
After Completing:
- Review any problems you found difficult
- Verify all your answers
- Make sure you can explain your solution method
- Practice similar problems if you’re still unsure
Master Formula Sheet for Unit 18
Ratios and Proportions:
- Ratio: a:b or a/b
- Proportion: a:b = c:d or a:b :: c:d
- Product Property: a × d = b × c (extremes = means)
Variation:
- Direct: y = kx (y ∝ x)
- Inverse: y = k/x (y ∝ 1/x)
- Joint: y = kxz (y varies jointly with x and z)
- Combined: y = kx/z (direct with x, inverse with z)
Componendo and Dividendo:
- Componendo: If a/b = c/d, then (a+b)/b = (c+d)/d
- Dividendo: If a/b = c/d, then (a-b)/b = (c-d)/d
- Both: If a/b = c/d, then (a+b)/(a-b) = (c+d)/(c-d)
Common Mistakes and How to Avoid Them
Mistake 1: Confusing Direct and Inverse Variation
How to avoid: Ask yourself: “If one increases, does the other increase (direct) or decrease (inverse)?”
Mistake 2: Forgetting to Find the Constant
How to avoid: Always calculate k before solving for unknowns. It’s a necessary step, not optional.
Mistake 3: Mixing Up Extremes and Means
How to avoid: In a:b = c:d, extremes are the “outside” terms (a and d), means are the “inside” terms (b and c).
Mistake 4: Not Verifying Radical Equation Solutions
How to avoid: The textbook specifically warns about this. ALWAYS substitute your answer back when square roots or radicals are involved.
Mistake 5: Wrong Units in Ratios
How to avoid: Convert all quantities to the same unit before forming ratios.
Mistake 6: Skipping Steps in Word Problems
How to avoid: Write down what you know, what you need to find, and set up the equation systematically.
Study Plan for Unit 18 Success
Week 1: Foundation (Exercises 18.1-18.3)
- Day 1-2: Master basic ratios and proportions (18.1)
- Day 3-4: Practice extremes and means property (18.2)
- Day 5-7: Solve complex proportion problems (18.3)
Week 2: Variation (Exercises 18.4-18.5)
- Day 1-3: Understand and practice direct/inverse variation (18.4)
- Day 4-7: Solve proportion equations thoroughly (18.5)
Week 3: Advanced Topics (Exercises 18.6-18.7)
- Day 1-4: Master componendo and dividendo (18.6)
- Day 5-7: Practice joint variation problems (18.7)
Week 4: Review and Practice
- Day 1-3: Complete Review Exercise 18
- Day 4-5: Practice weak areas
- Day 6-7: Full chapter review and mock test
Tips for Exam Success
- Show Your Work: Even if you can do it mentally, write it out. Partial credit matters.
- Use the Right Formula: Clearly state which property or formula you’re using.
- Verify When Required: Especially for radical equations – the textbook emphasizes this.
- Check Reasonableness: Does your answer make sense in the problem context?
- Time Management: Don’t spend too long on one problem. Move on and come back.
- Read Carefully: Many mistakes come from misreading the problem, not miscalculation.
- Practice Past Papers: Familiarize yourself with the question style and difficulty level.
Real-World Applications to Remember
Understanding this unit helps you:
- Cooking: Scale recipes up or down accurately
- Shopping: Compare unit prices and find best deals
- Travel: Calculate speed, distance, and time relationships
- Business: Understand costs, profits, and ratios
- Science: Work with formulas involving multiple variables
- Construction: Use scale drawings and proportions
- Finance: Understand interest rates and currency conversion
Final Thoughts
Unit 18 is comprehensive but logical. Each exercise builds on the previous one:
- Exercises 18.1-18.3 establish the foundation of ratios and proportions
- Exercises 18.4-18.5 introduce variation and systematic solving
- Exercises 18.6-18.7 present advanced techniques for complex problems
Master each exercise before moving to the next. Don’t rush – understanding is more important than speed.
Remember what your textbook emphasizes:
- Product of Extremes = Product of Means is fundamental
- Verify radical equation solutions
- Joint variation combines multiple relationships
With consistent practice and the strategies in this guide, you’ll master Unit 18 and build skills that will serve you throughout your mathematical journey.
Good luck with your studies!
