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Chapter Analysis
Intermediate30 pages • EnglishQuick Summary
The chapter 'Finding the Unknown' introduces the concept of solving equations and emphasizes understanding the role of operations such as addition, subtraction, multiplication, and division in maintaining balance within equations. It elucidates how equations can be manipulated to isolate unknowns through examples involving real-world applications and ancient mathematical knowledge from Brahmagupta. The reader is guided through systematic strategies to frame and solve equations, enhancing algebraic thinking.
Key Topics
- •Solving simple equations
- •Algebraic expressions
- •Inverse operations
- •Balancing equations
- •Real-life applications of equations
- •History of algebra
- •Systematic problem solving
Learning Objectives
- ✓Understand how to balance equations using basic operations.
- ✓Apply algebraic methods to solve real-world problems.
- ✓Familiarize with historical context and evolution of algebra.
- ✓Develop confidence in solving one-variable equations.
- ✓Analyze and critique various solution strategies.
Questions in Chapter
Fill in the blanks with integers: (a) 5 × ___ – 8 = 37; (b) 37 – (33 – ____ ) = 35; (c) – 3 × (– 11 + ____ ) = 45
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Solve these equations and check the solutions: (a) 3x – 10 = 35; (b) 5s = 3s; (c) 3u – 7 = 2u + 3; (d) 4 (m + 6) – 8 = 2m – 4; (e) u / 15 = 6
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A taxi driver charges a fixed fee of ₹800 per day plus ₹20 for each kilometer traveled. If the total cost for a taxi ride is ₹2200, determine the number of kilometers traveled.
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Additional Practice Questions
If Sarah has twice as many dimes as nickels and the total value of the coins is $3.40, how many nickels does she have?
mediumAnswer: Let x be the number of nickels. Then, 2x will be the number of dimes. Each nickel is worth $0.05 and each dime is worth $0.10. Thus, the equation is: 0.05x + 0.10(2x) = 3.40. Solving this equation gives x = 12 nickels.
The sum of three consecutive integers is 27. What are the integers?
easyAnswer: Let the integers be x, x+1, x+2. The equation is x + (x + 1) + (x + 2) = 27. Solving, x = 8, so the integers are 8, 9, 10.
A company’s revenue is represented as 4x + 5 dollars, where x is the number of units sold. If the revenue is $25 when 5 units are sold, what is the revenue per unit?
mediumAnswer: The revenue per unit is derived from 4x + 5 = 25 when x = 5. Solving, the revenue per unit is $4.
If the cost of production of an item is represented by 7y + 9 and a company spends $100 to produce 4 items, how much does it cost to produce one item?
hardAnswer: The equation is 7y + 9 = 100 for y = 4. So, 7y + 9 simplifies to 28 + 9 = 37 per item.
A rectangle’s length is 3 units more than twice its width. If the perimeter is 54 units, what are the rectangle’s dimensions?
mediumAnswer: Let the width be w. The length is 2w + 3. The perimeter is 2(w + 2w + 3) = 54. Solving gives w = 7, so the dimensions are width 7 and length 17.