3 Questions
Explain the difference between LCM (Least Common Multiple) and HCF (Highest Common Factor) and provide an example of calculating LCM and HCF for two numbers.
The LCM of two or more numbers is the smallest number that is a multiple of each of the numbers. The HCF of two or more numbers is the largest number that divides each of the numbers without leaving a remainder. For example, the LCM of 12 and 15 is 60, and the HCF of 12 and 15 is 3.
Simplify the expression (3x^2 - 5x + 2) / (x - 2) and state any restrictions on the variable x.
The simplified expression is 3x - 1, with the restriction that x cannot equal 2.
Solve the linear equation 2(3x + 1) = 5(x - 2) and state the value of x.
The solution to the linear equation is x = 3.
Study Notes
LCM (Least Common Multiple) and HCF (Highest Common Factor)
- LCM is the smallest number that is a multiple of two or more numbers, while HCF is the largest number that divides two or more numbers exactly without leaving a remainder.
- LCM is used to add fractions with different denominators, whereas HCF is used to simplify fractions.
Calculating LCM and HCF
- To calculate the LCM of two numbers, list the multiples of each number and find the smallest common multiple.
- To calculate the HCF of two numbers, list the factors of each number and find the largest common factor.
Example: Calculating LCM and HCF
- Find the LCM and HCF of 12 and 15:
- Multiples of 12: 12, 24, 36, 48, ...
- Multiples of 15: 15, 30, 45, 60, ...
- LCM of 12 and 15 = 60
- Factors of 12: 1, 2, 3, 4, 6, 12
- Factors of 15: 1, 3, 5, 15
- HCF of 12 and 15 = 3
Simplifying Expressions
- To simplify an expression, cancel out any common factors between the numerator and denominator.
- Simplify the expression (3x^2 - 5x + 2) / (x - 2):
- Factor the numerator: (3x - 1)(x - 2)
- Cancel out the common factor (x - 2): 3x - 1
- Simplified expression: 3x - 1
- Restriction on x: x ≠ 2
Solving Linear Equations
- To solve a linear equation, isolate the variable x on one side of the equation.
- Solve the linear equation 2(3x + 1) = 5(x - 2):
- Expand the equation: 6x + 2 = 5x - 10
- Rearrange the equation: x = 12
- Value of x: x = 12
Test your knowledge of factors, multiples, integers, estimation, approximation, and basic algebraic manipulation with this comprehensive quiz. Challenge yourself with questions on LCM, HCF, prime factorization, square roots, cube roots, BODMAS, fractions, ratios, negative numbers, rounding off, significant figures, currency conversions, and basic algebraic manipulation techniques. Sharpen your math skills and master these fundamental concepts.
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