Date: 17-18 February '25 Task: Formal Assignment Marks: 50 Time: 2 Hours QUESTION 2 [22]2.1 Evaluate: 25 − 4 × ( 7 + 5 ) ÷ 4 + 3 25 - 4 \times (7 + 5) \div 4 + 3 25 − 4 × ( 7 + 5 ) ÷ 4 + 3 2.2 Determine The LCM And The HCF Of 306 And 657 Using Prime Factorization. (4) 2.3 The Ratio Of

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Introduction

Mathematics is a subject that deals with numbers, quantities, and shapes. It is a fundamental subject that is used in various fields such as science, technology, engineering, and mathematics (STEM). In this article, we will evaluate a mathematical expression, find the least common multiple (LCM) and the highest common factor (HCF) of two numbers using prime factorization.

Evaluating the Mathematical Expression

The given mathematical expression is 25−4×(7+5)÷4+325 - 4 \times (7 + 5) \div 4 + 3. To evaluate this expression, we need to follow the order of operations (PEMDAS):

  1. Evaluate the expression inside the parentheses: (7+5)=12(7 + 5) = 12
  2. Multiply 4 by the result: 4×12=484 \times 12 = 48
  3. Divide 48 by 4: 48÷4=1248 \div 4 = 12
  4. Subtract 12 from 25: 25−12=1325 - 12 = 13
  5. Add 3 to the result: 13+3=1613 + 3 = 16

Therefore, the value of the given mathematical expression is 16.

Prime Factorization

Prime factorization is a method of expressing a number as a product of its prime factors. The prime factors of a number are the prime numbers that divide the number exactly.

Finding the LCM and HCF of 306 and 657

To find the LCM and HCF of 306 and 657, we need to find their prime factorization.

Prime Factorization of 306

The prime factorization of 306 is:

306=2×3×3×17306 = 2 \times 3 \times 3 \times 17

Prime Factorization of 657

The prime factorization of 657 is:

657=3×3×73657 = 3 \times 3 \times 73

Finding the LCM

The LCM of two numbers is the product of the highest power of each prime factor that appears in either number.

The LCM of 306 and 657 is:

LCM(306,657)=2×32×17×73=2×9×17×73=4×9×17×73=4×153×73=4×11109=44436LCM(306, 657) = 2 \times 3^2 \times 17 \times 73 = 2 \times 9 \times 17 \times 73 = 4 \times 9 \times 17 \times 73 = 4 \times 153 \times 73 = 4 \times 11109 = 44436

Finding the HCF

The HCF of two numbers is the product of the lowest power of each prime factor that appears in both numbers.

The HCF of 306 and 657 is:

HCF(306,657)=32=9HCF(306, 657) = 3^2 = 9

Conclusion

In this article, we evaluated a mathematical expression, found the LCM and HCF of two numbers using prime factorization, and discussed the importance of mathematics in various fields. We also learned how to follow the order of operations (PEMDAS) and how to find the prime factorization of a number.

Importance of Mathematics

Mathematics is a fundamental subject that is used in various fields such as science, technology, engineering, and mathematics (STEM). It is used to describe the world around us, to make predictions, and to solve problems. Mathematics is also used in finance, economics, and medicine.

Real-World Applications of Mathematics

Mathematics is used in various real-world applications such as:

  • Science: Mathematics is used to describe the laws of physics, to model the behavior of complex systems, and to make predictions.
  • Technology: Mathematics is used in computer science, engineering, and data analysis.
  • Finance: Mathematics is used in finance to model the behavior of financial markets, to make predictions, and to optimize investment strategies.
  • Economics: Mathematics is used in economics to model the behavior of economic systems, to make predictions, and to optimize economic policies.
  • Medicine: Mathematics is used in medicine to model the behavior of complex systems, to make predictions, and to optimize treatment strategies.

Conclusion

Introduction

Mathematics is a subject that deals with numbers, quantities, and shapes. It is a fundamental subject that is used in various fields such as science, technology, engineering, and mathematics (STEM). In this article, we will answer some frequently asked questions about mathematics.

Q&A

Q: What is the difference between a number and a quantity?

A: A number is a mathematical value that represents a quantity. A quantity is a measure of the amount or size of something.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The acronym PEMDAS stands for:

  • P: Parentheses (evaluate expressions inside parentheses first)
  • E: Exponents (evaluate any exponential expressions next)
  • M: Multiplication and Division (evaluate multiplication and division operations from left to right)
  • A: Addition and Subtraction (evaluate any addition and subtraction operations from left to right)

Q: What is the difference between a prime number and a composite number?

A: A prime number is a positive integer that is divisible only by itself and 1. A composite number is a positive integer that is divisible by at least one other number besides itself and 1.

Q: What is the least common multiple (LCM) and the highest common factor (HCF)?

A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. The highest common factor (HCF) of two numbers is the largest number that divides both numbers exactly.

Q: How do I find the prime factorization of a number?

A: To find the prime factorization of a number, you need to find the prime factors of the number and multiply them together. For example, the prime factorization of 12 is:

12=2×2×312 = 2 \times 2 \times 3

Q: What is the difference between a rational number and an irrational number?

A: A rational number is a number that can be expressed as a fraction of two integers. An irrational number is a number that cannot be expressed as a fraction of two integers.

Q: How do I evaluate an expression with multiple operations?

A: To evaluate an expression with multiple operations, you need to follow the order of operations (PEMDAS). First, evaluate any expressions inside parentheses. Then, evaluate any exponential expressions. Next, evaluate any multiplication and division operations from left to right. Finally, evaluate any addition and subtraction operations from left to right.

Q: What is the difference between a function and a relation?

A: A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). A relation is a set of ordered pairs that satisfy a certain condition.

Q: How do I find the domain and range of a function?

A: To find the domain and range of a function, you need to identify the set of possible inputs (the domain) and the set of possible outputs (the range).

Conclusion

In conclusion, mathematics is a subject that deals with numbers, quantities, and shapes. It is a fundamental subject that is used in various fields such as science, technology, engineering, and mathematics (STEM). We also answered some frequently asked questions about mathematics, including questions about numbers, quantities, prime numbers, composite numbers, LCM and HCF, prime factorization, rational numbers, irrational numbers, functions, and relations.

Real-World Applications of Mathematics

Mathematics is used in various real-world applications such as:

  • Science: Mathematics is used to describe the laws of physics, to model the behavior of complex systems, and to make predictions.
  • Technology: Mathematics is used in computer science, engineering, and data analysis.
  • Finance: Mathematics is used in finance to model the behavior of financial markets, to make predictions, and to optimize investment strategies.
  • Economics: Mathematics is used in economics to model the behavior of economic systems, to make predictions, and to optimize economic policies.
  • Medicine: Mathematics is used in medicine to model the behavior of complex systems, to make predictions, and to optimize treatment strategies.

Conclusion

In conclusion, mathematics is a fundamental subject that is used in various fields such as science, technology, engineering, and mathematics (STEM). It is used to describe the world around us, to make predictions, and to solve problems. We also answered some frequently asked questions about mathematics, including questions about numbers, quantities, prime numbers, composite numbers, LCM and HCF, prime factorization, rational numbers, irrational numbers, functions, and relations.