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Simplification

Practice MCQs

Mathematical simplification involves reducing complex mathematical expressions to simpler forms through various operations such as combining like terms, applying algebraic rules, and removing unnecessary elements. Simplifying mathematical expressions enhances clarity, facilitates problem-solving, and allows for easier manipulation and analysis of mathematical equations and formulas.

Quantitative Aptitude Simplification Competitive Exams

Simplification is one of the most frequently asked quantitative aptitude topics. It is based on simplifying numerical expressions using arithmetic operations, brackets, fractions, decimals, percentages, powers, roots, and the correct order of operations. These questions are common in banking exams, SSC, railways, police exams, entrance tests, and other competitive exams.

What is Simplification?

Simplification means reducing a mathematical expression into its simplest possible form. In aptitude exams, simplification questions test speed, accuracy, arithmetic skills, and understanding of operation priority.

Most simplification questions are based on the BODMAS rule. It tells us the correct order in which mathematical operations must be performed.

Quick idea: In simplification, do not solve from left to right blindly. Always follow BODMAS.
Letter Operation Meaning
B Brackets Solve expressions inside brackets first
O Order Powers, roots, squares, cubes
D Division Perform division before addition/subtraction
M Multiplication Perform multiplication before addition/subtraction
A Addition Add numbers after higher priority operations
S Subtraction Subtract numbers after higher priority operations

“In simplification, speed comes from knowing operation order and using smart arithmetic.”

Aptitude Tip
Key points
  • Always follow BODMAS.
  • Solve brackets first.
  • Convert mixed fractions if needed.
  • Convert percentages into fractions or decimals.
  • Use cancellation wherever possible.
  • Avoid unnecessary long calculation.
BODMAS fractions decimals percentages calculation

Visual Understanding

These diagrams show how operation order and simplification flow work.

BODMAS Priority Order
1. Brackets 2. Order 3. Division 4. Multiplication 5. Addition / Subtraction
\[ \text{BODMAS} = B \rightarrow O \rightarrow D \rightarrow M \rightarrow A \rightarrow S \]

Follow this order to avoid mistakes in simplification problems.

Simplification Flow
Expression Apply BODMAS Answer 24 ÷ 6 × 2 + 5 First divide and multiply, then add Correct order gives correct answer
\[ 24 \div 6 \times 2 + 5 = 4 \times 2 + 5 = 13 \]

Division and multiplication are performed before addition and subtraction.

Fraction Simplification
24 36 ÷ 12 2 3 Divide numerator and denominator by HCF
\[ \frac{24}{36} = \frac{2}{3} \]

Reduce fractions before performing larger calculations.

Percentage Conversion
25% divide by 100 1/4 Convert percentages into fractions for faster calculation 25% of 80 = 1/4 of 80 = 20
\[ 25\% = \frac{25}{100} = \frac{1}{4} \]

Common percentage-to-fraction conversions make simplification faster.

Important Rules and Formulas

BODMAS Rule
\[ B \rightarrow O \rightarrow D \rightarrow M \rightarrow A \rightarrow S \]

Follow this operation order while simplifying.

Fraction Reduction
\[ \frac{a}{b}=\frac{a \div h}{b \div h} \]

\(h\) is the HCF of \(a\) and \(b\).

Percentage Form
\[ x\%=\frac{x}{100} \]

Convert percentages into fractions or decimals.

Decimal to Fraction
\[ 0.25=\frac{25}{100}=\frac{1}{4} \]

Useful in decimal-based simplification.

Square Rule
\[ a^2=a \times a \]

Used in order-based simplification.

Square Root
\[ \sqrt{a^2}=a \]

Useful in expressions containing roots.

Reciprocal
\[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b}\times\frac{d}{c} \]

Division by a fraction means multiplication by reciprocal.

Mixed Fraction
\[ a\frac{b}{c}=\frac{ac+b}{c} \]

Convert mixed fractions before calculation.

Common Percentages
\[ 50\%=\frac{1}{2},\quad 25\%=\frac{1}{4} \]

Remember common conversions for speed.

More Percentages
\[ 20\%=\frac{1}{5},\quad 10\%=\frac{1}{10} \]

Useful for quick mental calculation.

Approximation Check
\[ 49.8 \approx 50,\quad 101.2 \approx 100 \]

Used only when approximation is allowed.

Cancellation
\[ \frac{24 \times 15}{30}=12 \]

Cancel before multiplying to save time.

Rule: Simplification questions are not difficult if the order of operations is followed correctly. Always solve brackets and powers before ordinary arithmetic.

Common Types of Questions

BODMAS-Based Questions

Expressions containing brackets, division, multiplication, addition, and subtraction.

  • Bracket simplification
  • Order of operations
  • Multiple operators
  • Stepwise solving
Fraction Questions

Questions involving addition, subtraction, multiplication, or division of fractions.

  • LCM method
  • Cancellation
  • Mixed fractions
  • Reciprocal method
Decimal Questions

Expressions containing decimal numbers and place value operations.

  • Decimal addition
  • Decimal multiplication
  • Decimal division
  • Decimal to fraction
Percentage Questions

Convert percentage into fraction or decimal before solving.

  • Percentage of number
  • Fraction conversion
  • Successive percentage
  • Fast calculation
Exam approach: Before calculating, check whether the expression can be simplified by cancellation, fraction conversion, or percentage conversion.
Method Bank
Use BODMAS

Solve brackets before ordinary operations.

\[ 8+4\times3=8+12=20 \]
Use Cancellation

Cancel common factors before multiplication.

\[ \frac{25\times16}{20}=20 \]
Convert Percentage

Convert percentage into fraction.

\[ 25\%\text{ of }80=\frac{1}{4}\times80=20 \]
Convert Mixed Fraction

Convert before calculation.

\[ 2\frac{1}{3}=\frac{7}{3} \]

Tip: In simplification, avoid writing too many steps when cancellation can reduce the expression quickly.

Simplification Solving Flow
Read Expression Apply BODMAS Simplify Final Answer Brackets → powers → divide/multiply → add/subtract
This flow helps reduce calculation mistakes in exam conditions.
\[ \text{Correct order} \Rightarrow \text{Correct answer} \]
\[ \text{Cancellation} \Rightarrow \text{Faster calculation} \]

Step-by-Step Solving Method

Step What to Do Example
Step 1 Read the full expression carefully. \(18 + 6 \div 3 \times 4\)
Step 2 Solve brackets first, if present. \((12+8)\div5\)
Step 3 Solve powers and roots. \(3^2=9\)
Step 4 Perform division and multiplication from left to right. \(6\div3\times4=8\)
Step 5 Perform addition and subtraction at the end. \(18+8=26\)
Important: Division and multiplication have the same priority, so solve them from left to right. Addition and subtraction also have the same priority.

Solved Examples

Question Method Answer
Simplify: \(18 + 6 \div 3 \times 4\) First solve division and multiplication from left to right.
\[ 6 \div 3 \times 4 = 2 \times 4 = 8 \]
\[ 18+8=26 \]
26
Simplify: \(45 - 5 \times 6 + 12\) First multiply.
\[ 5\times6=30 \]
Then add/subtract from left to right.
\[ 45-30+12=27 \]
27
Simplify: \((24+16)\div 5\) First solve bracket.
\[ 24+16=40 \]
Then divide.
\[ 40\div5=8 \]
8
Simplify: \(25\%\) of \(240\) Convert \(25\%\) into fraction.
\[ 25\%=\frac{1}{4} \]
\[ \frac{1}{4}\times240=60 \]
60
Simplify: \(\frac{3}{4}+\frac{1}{2}\) Convert to common denominator.
\[ \frac{1}{2}=\frac{2}{4} \]
\[ \frac{3}{4}+\frac{2}{4}=\frac{5}{4} \]
\(\frac{5}{4}\)
Simplify: \(2\frac{1}{2}+3\frac{1}{4}\) Convert mixed fractions.
\[ 2\frac{1}{2}=\frac{5}{2},\quad 3\frac{1}{4}=\frac{13}{4} \]
\[ \frac{5}{2}+\frac{13}{4}=\frac{10}{4}+\frac{13}{4}=\frac{23}{4} \]
\(\frac{23}{4}\) or \(5\frac{3}{4}\)
Simplify: \(12^2 - 8^2\) Calculate squares.
\[ 12^2=144,\quad 8^2=64 \]
\[ 144-64=80 \]
80
Simplify: \(\frac{36\times25}{15}\) Cancel before multiplying.
\[ \frac{36\times25}{15}=36\times\frac{5}{3}=12\times5=60 \]
60

Note: In simplification, a small mistake in operation order can change the entire answer.

Common Traps and Shortcuts

Common Traps
  • Solving the expression from left to right without BODMAS.
  • Adding before multiplication.
  • Ignoring brackets.
  • Forgetting to convert mixed fractions.
  • Making decimal place mistakes.
  • Using approximation when exact answer is required.
Useful Shortcuts
  • Cancel common factors before multiplication.
  • Convert common percentages into fractions.
  • Use squares and cubes from memory.
  • Convert decimals into fractions when easier.
  • Use LCM for adding fractions.
  • Do not skip brackets.
Exam approach: In simplification questions, accuracy is more important than speed. First apply the correct order, then calculate quickly.

Practice

A) Multiple Choice Questions
  1. Simplify: \(20 + 5 \times 4\)
    40 100 25 30
  2. Simplify: \(36 \div 6 \times 3\)
    2 12 18 24
  3. \(25\%\) of \(160\) is:
    20 30 40 50
  4. Simplify: \(\frac{2}{3}+\frac{1}{3}\)
    \(\frac{1}{3}\) 1 \(\frac{2}{3}\) 2
  5. Simplify: \(9^2 - 7^2\)
    22 30 32 34
B) Solve the Higher-Order Problems
  1. Simplify: \(48 \div 8 \times 6 + 12\) (Hint: Division and multiplication first.)
  2. Simplify: \((36+24)\div 12 + 15\) (Hint: Solve bracket first.)
  3. Simplify: \(40\%\) of \(250 + 25\%\) of \(120\) (Hint: Convert percentages to fractions.)
  4. Simplify: \(\frac{5}{6}+\frac{3}{4}\) (Hint: Use LCM of 6 and 4.)
  5. Simplify: \(\frac{45\times32}{24}\) (Hint: Cancel before multiplying.)
C) Match the Concept with the Correct Meaning
Concept Correct Meaning
BODMAS Correct order of operations
Bracket Solved before other operations
Percentage Number out of 100
Fraction Part of a whole
Cancellation Removing common factors to simplify calculation
Approximation Using nearby values for quick estimation
Aptitude Reminder

Simplification questions test accuracy and calculation speed. Always follow BODMAS, simplify fractions, convert percentages, and use cancellation wherever possible.

Task: Create five questions using BODMAS, fractions, percentages, squares, and cancellation.

Show Suggested Answers
Multiple Choice
  1. 40
    Multiplication first:
    \[ 20+5\times4=20+20=40 \]
  2. 18
    Division and multiplication from left to right:
    \[ 36\div6\times3=6\times3=18 \]
  3. 40
    \[ 25\%\text{ of }160=\frac{1}{4}\times160=40 \]
  4. 1
    \[ \frac{2}{3}+\frac{1}{3}=\frac{3}{3}=1 \]
  5. 32
    \[ 9^2-7^2=81-49=32 \]
Higher-Order Problems
  1. Simplify:
    \[ 48\div8\times6+12 \]
    \[ 6\times6+12=36+12=48 \]
    Answer = 48.
  2. First solve bracket:
    \[ 36+24=60 \]
    Then:
    \[ 60\div12+15=5+15=20 \]
    Answer = 20.
  3. Convert percentages:
    \[ 40\%\text{ of }250=\frac{40}{100}\times250=100 \]
    \[ 25\%\text{ of }120=\frac{1}{4}\times120=30 \]
    \[ 100+30=130 \]
    Answer = 130.
  4. LCM of 6 and 4 is 12.
    \[ \frac{5}{6}+\frac{3}{4} = \frac{10}{12}+\frac{9}{12} = \frac{19}{12} \]
    Answer = \(\frac{19}{12}\).
  5. Simplify by cancellation:
    \[ \frac{45\times32}{24} = 45\times\frac{4}{3} = 15\times4 = 60 \]
    Answer = 60.
Concept Matching
  1. BODMAS → Correct order of operations
  2. Bracket → Solved before other operations
  3. Percentage → Number out of 100
  4. Fraction → Part of a whole
  5. Cancellation → Removing common factors to simplify calculation
  6. Approximation → Using nearby values for quick estimation
Clue Explanation

Simplification becomes easy when you first identify operation priority. Use BODMAS, reduce fractions, convert percentages, and cancel common factors before multiplying.

Exam tips
  • Never ignore brackets.
  • Apply BODMAS correctly.
  • Division and multiplication are solved from left to right.
  • Addition and subtraction are solved from left to right.
  • Convert percentages into fractions for faster calculation.
  • Use cancellation before multiplication.
Practice MCQs