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Simplification is the process of making a numerical expression easier to understand or calculate by reducing it to its simplest form. In arithmetic, this often means performing operations in the correct order and reducing fractions to their simplest terms. Simplification helps us solve problems quickly and accurately, especially in exams like the Assam Direct Recruitment Examination (ADRE) for Grade IV posts.
Two important tools in simplification are the Highest Common Factor (HCF) and the Least Common Multiple (LCM). These help us work efficiently with fractions and numerical expressions. Before we dive into these, it is essential to understand the order of operations, which tells us the correct sequence to solve expressions involving multiple operations.
When an expression has more than one arithmetic operation, such as addition, subtraction, multiplication, division, or brackets, it is important to solve it in the right order. This avoids confusion and ensures everyone gets the same answer.
The rule to remember is called BODMAS or BIDMAS, which stands for:
This order ensures clarity and correctness.
graph TD A[Start] --> B[Brackets] B --> C[Orders (powers/roots)] C --> D[Division and Multiplication (left to right)] D --> E[Addition and Subtraction (left to right)] E --> F[End]
Example: Simplify \( 5 + 2 \times (8 - 3)^2 \div 5 \)
Answer: 15
Before working with fractions or complex expressions, it is helpful to understand two key concepts:
The Highest Common Factor of two or more numbers is the largest number that divides all of them exactly without leaving a remainder.
Why is HCF important? When simplifying fractions, dividing both numerator and denominator by their HCF reduces the fraction to its simplest form.
The Least Common Multiple of two or more numbers is the smallest number that is a multiple of all of them.
Why is LCM important? When adding or subtracting fractions with different denominators, finding the LCM of the denominators helps us find a common denominator quickly.
| Numbers | HCF | LCM |
|---|---|---|
| 12 and 18 | 6 | 36 |
| 8 and 20 | 4 | 40 |
| 15 and 25 | 5 | 75 |
To simplify a fraction:
Example: Simplify \(\frac{18}{24}\)
When adding fractions with different denominators, follow these steps:
Example: Add \(\frac{1}{3} + \frac{1}{4}\)
Expressions often combine whole numbers, fractions, and brackets. Use the order of operations along with HCF and LCM to simplify efficiently.
For example, when you see an expression like \(2 \times \left(\frac{3}{4} + \frac{5}{6}\right) - 1\), first simplify inside the brackets by adding fractions using LCM, then multiply and subtract.
By following the BODMAS rule and using HCF and LCM to simplify fractions, you can solve complex arithmetic expressions step-by-step with confidence and accuracy. This approach reduces errors and saves time during exams.
Step 1: Solve inside the brackets separately.
\(12 \div 3 = 4\)
\(8 \div 4 = 2\)
Step 2: Add the results.
\(4 + 2 = 6\)
Answer: 6
Step 1: Simplify each fraction.
HCF of 18 and 24 is 6, so \(\frac{18}{24} = \frac{3}{4}\).
HCF of 6 and 8 is 2, so \(\frac{6}{8} = \frac{3}{4}\).
Step 2: Add the simplified fractions.
\(\frac{3}{4} + \frac{3}{4} = \frac{6}{4}\)
Step 3: Simplify \(\frac{6}{4}\).
HCF of 6 and 4 is 2, so \(\frac{6}{4} = \frac{3}{2}\) or \(1 \frac{1}{2}\).
Answer: \(1 \frac{1}{2}\) or \(\frac{3}{2}\)
Step 1: Find LCM of denominators 4 and 6.
LCM(4,6) = 12.
Step 2: Convert fractions to have denominator 12.
\(\frac{3}{4} = \frac{9}{12}\), \(\frac{5}{6} = \frac{10}{12}\)
Step 3: Add the fractions inside the bracket.
\(\frac{9}{12} + \frac{10}{12} = \frac{19}{12}\)
Step 4: Multiply by 2.
\(2 \times \frac{19}{12} = \frac{38}{12}\)
Step 5: Simplify \(\frac{38}{12}\).
HCF of 38 and 12 is 2, so \(\frac{38}{12} = \frac{19}{6}\)
Step 6: Subtract 1.
Convert 1 to \(\frac{6}{6}\): \(\frac{19}{6} - \frac{6}{6} = \frac{13}{6}\)
Answer: \(\frac{13}{6}\) or \(2 \frac{1}{6}\)
Step 1: Simplify each fraction.
HCF of 45 and 60 is 15, so \(\frac{45}{60} = \frac{3}{4}\).
HCF of 15 and 20 is 5, so \(\frac{15}{20} = \frac{3}{4}\).
Step 2: Subtract the fractions.
\(\frac{3}{4} - \frac{3}{4} = 0\)
Step 3: Calculate \(6 \div 3 = 2\).
Step 4: Add the results.
\(0 + 2 = 2\)
Answer: 2
Step 1: Find LCM of denominators 3 and 4.
LCM(3,4) = 12.
Step 2: Convert fractions to denominator 12.
\(\frac{2}{3} = \frac{8}{12}\), \(\frac{3}{4} = \frac{9}{12}\)
Step 3: Add the fractions inside the bracket.
\(\frac{8}{12} + \frac{9}{12} = \frac{17}{12}\)
Step 4: Multiply by \(\frac{12}{15}\).
\(\frac{17}{12} \times \frac{12}{15} = \frac{17 \times 12}{12 \times 15} = \frac{17}{15}\)
Step 5: Simplify if possible.
17 and 15 have no common factors other than 1, so fraction is already simplified.
Answer: \(\frac{17}{15}\) or \(1 \frac{2}{15}\)
When to use: When adding, subtracting, or comparing fractions to reduce calculation complexity.
When to use: When dealing with addition or subtraction of fractions with different denominators.
When to use: While simplifying expressions with brackets, powers, multiplication, division, addition, and subtraction.
When to use: When expressions have multiple brackets and mixed operations.
When to use: To verify answers quickly during practice or exams.
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