42. BODMAS rule-based calculations problems

42. BODMAS rule-based calculations problems - step by step solution procedure and detailed explanation for each problem.

Below are 10 BODMAS rule-based calculation problems, each accompanied by a step-by-step solution procedure and a detailed explanation. BODMAS stands for Brackets, Orders (exponents, roots), Division, Multiplication, Addition, Subtraction, which dictates the order of operations in mathematical expressions. Each problem is designed to illustrate different aspects of applying the BODMAS rule, progressing from simpler to more complex expressions.
Problem 1:
5 + 3 \times 2
Step-by-Step Solution:
  1. Identify operations: The expression has addition (
    +
    ) and multiplication (
    \times
    ).
  2. Apply BODMAS: Multiplication comes before addition.
  3. Perform multiplication:
    3 \times 2 = 6
    .
  4. Perform addition:
    5 + 6 = 11
    .
Final Answer: ( 11 )
Detailed Explanation:
  • BODMAS Rule: According to BODMAS, multiplication has higher precedence than addition. Thus, we compute
    3 \times 2
    before adding 5.
  • Common Mistake: A common error is to perform operations left to right without considering precedence (e.g.,
    5 + 3 = 8
    , then
    8 \times 2 = 16
    ), which is incorrect.
  • Verification: To confirm, rewrite the expression with parentheses to reflect BODMAS:
    5 + (3 \times 2)
    . This yields
    5 + 6 = 11
    .
Problem 2:
12 \div 4 + 7
Step-by-Step Solution:
  1. Identify operations: The expression has division (
    \div
    ) and addition (
    +
    ).
  2. Apply BODMAS: Division comes before addition.
  3. Perform division:
    12 \div 4 = 3
    .
  4. Perform addition:
    3 + 7 = 10
    .
Final Answer: ( 10 )
Detailed Explanation:
  • BODMAS Rule: Division precedes addition in the order of operations. Thus,
    12 \div 4
    is computed first.
  • Common Mistake: Calculating left to right (e.g.,
    12 \div (4 + 7) = 12 \div 11
    ) is incorrect because there are no brackets to group
    4 + 7
    .
  • Verification: The expression can be written as
    (12 \div 4) + 7
    , which is
    3 + 7 = 10
    .
Problem 3:
(6 + 4) \times 3
Step-by-Step Solution:
  1. Identify operations: The expression has brackets, addition (
    +
    ), and multiplication (
    \times
    ).
  2. Apply BODMAS: Brackets are solved first.
  3. Solve inside brackets:
    6 + 4 = 10
    .
  4. Perform multiplication:
    10 \times 3 = 30
    .
Final Answer: ( 30 )
Detailed Explanation:
  • BODMAS Rule: Brackets have the highest priority. The expression inside the brackets (
    6 + 4
    ) must be resolved before multiplication.
  • Common Mistake: Ignoring brackets and multiplying first (e.g.,
    6 + (4 \times 3) = 6 + 12 = 18
    ) violates BODMAS.
  • Verification: The expression is equivalent to
    (6 + 4) \times 3 = 10 \times 3 = 30
    .
Problem 4:
15 - 8 \div 2
Step-by-Step Solution:
  1. Identify operations: The expression has subtraction (
    -
    ) and division (
    \div
    ).
  2. Apply BODMAS: Division comes before subtraction.
  3. Perform division:
    8 \div 2 = 4
    .
  4. Perform subtraction:
    15 - 4 = 11
    .
Final Answer: ( 11 )
Detailed Explanation:
  • BODMAS Rule: Division takes precedence over subtraction. Thus,
    8 \div 2
    is calculated before subtracting from 15.
  • Common Mistake: Performing subtraction first (e.g.,
    (15 - 8) \div 2 = 7 \div 2 = 3.5
    ) is incorrect.
  • Verification: Rewrite as
    15 - (8 \div 2) = 15 - 4 = 11
    .
Problem 5:
2^3 + 5 \times 4
Step-by-Step Solution:
  1. Identify operations: The expression has an exponent (
    ^3
    ), addition (
    +
    ), and multiplication (
    \times
    ).
  2. Apply BODMAS: Orders (exponents) come before multiplication, which comes before addition.
  3. Solve exponent:
    2^3 = 8
    .
  4. Perform multiplication:
    5 \times 4 = 20
    .
  5. Perform addition:
    8 + 20 = 28
    .
Final Answer: ( 28 )
Detailed Explanation:
  • BODMAS Rule: Exponents are resolved after brackets (none here) and before multiplication and addition. Multiplication precedes addition.
  • Common Mistake: Adding before multiplying (e.g.,
    2^3 + 5 = 8 + 5 = 13
    , then
    13 \times 4 = 52
    ) is incorrect.
  • Verification: The expression is
    2^3 + (5 \times 4) = 8 + 20 = 28
    .
Problem 6:
(10 - 2) \div 4 + 3^2
Step-by-Step Solution:
  1. Identify operations: The expression has brackets, subtraction (
    -
    ), division (
    \div
    ), addition (
    +
    ), and an exponent (
    ^2
    ).
  2. Apply BODMAS: Solve brackets first, then exponents, then division, then addition.
  3. Solve brackets:
    10 - 2 = 8
    .
  4. Solve exponent:
    3^2 = 9
    .
  5. Perform division:
    8 \div 4 = 2
    .
  6. Perform addition:
    2 + 9 = 11
    .
Final Answer: ( 11 )
Detailed Explanation:
  • BODMAS Rule: Brackets are resolved first, followed by exponents. Division comes before addition. Note that the exponent
    3^2
    is computed before the division because it has higher precedence.
  • Common Mistake: Adding before dividing (e.g., computing
    8 \div 4 + 9 = 11
    incorrectly by grouping) or ignoring the exponent order.
  • Verification: Rewrite as
    (10 - 2) \div 4 + 3^2 = 8 \div 4 + 9 = 2 + 9 = 11
    .
Problem 7:
20 \div (5 - 1) \times 2
Step-by-Step Solution:
  1. Identify operations: The expression has division (
    \div
    ), brackets, subtraction (
    -
    ), and multiplication (
    \times
    ).
  2. Apply BODMAS: Solve brackets first, then division and multiplication from left to right (they have equal precedence).
  3. Solve brackets:
    5 - 1 = 4
    .
  4. Perform division:
    20 \div 4 = 5
    .
  5. Perform multiplication:
    5 \times 2 = 10
    .
Final Answer: ( 10 )
Detailed Explanation:
  • BODMAS Rule: Brackets are solved first. Division and multiplication have equal precedence and are performed left to right.
  • Common Mistake: Multiplying before dividing (e.g.,
    20 \div (4 \times 2) = 20 \div 8 = 2.5
    ) is incorrect because operations must proceed left to right.
  • Verification: Rewrite as
    (20 \div (5 - 1)) \times 2 = (20 \div 4) \times 2 = 5 \times 2 = 10
    .
Problem 8:
3 \times (4 + 2^2) - 7
Step-by-Step Solution:
  1. Identify operations: The expression has multiplication (
    \times
    ), brackets, addition (
    +
    ), an exponent (
    ^2
    ), and subtraction (
    -
    ).
  2. Apply BODMAS: Solve brackets first, resolving exponents within brackets, then multiplication, then subtraction.
  3. Solve exponent inside brackets:
    2^2 = 4
    .
  4. Solve brackets:
    4 + 4 = 8
    .
  5. Perform multiplication:
    3 \times 8 = 24
    .
  6. Perform subtraction:
    24 - 7 = 17
    .
Final Answer: ( 17 )
Detailed Explanation:
  • BODMAS Rule: Within brackets, exponents are solved before addition. After resolving brackets, multiplication precedes subtraction.
  • Common Mistake: Ignoring the exponent in brackets or subtracting before multiplying (e.g.,
    3 \times 4 + 2^2 - 7
    ) is incorrect.
  • Verification: Rewrite as
    3 \times (4 + 2^2) - 7 = 3 \times (4 + 4) - 7 = 3 \times 8 - 7 = 24 - 7 = 17
    .
Problem 9:
2 + 6 \div 2 \times 3 - 1
Step-by-Step Solution:
  1. Identify operations: The expression has addition (
    +
    ), division (
    \div
    ), multiplication (
    \times
    ), and subtraction (
    -
    ).
  2. Apply BODMAS: Division and multiplication have equal precedence and are performed left to right, followed by addition and subtraction (also left to right).
  3. Perform division:
    6 \div 2 = 3
    .
  4. Perform multiplication:
    3 \times 3 = 9
    .
  5. Perform addition:
    2 + 9 = 11
    .
  6. Perform subtraction:
    11 - 1 = 10
    .
Final Answer: ( 10 )
Detailed Explanation:
  • BODMAS Rule: Division and multiplication are performed left to right. Here,
    6 \div 2 \times 3
    is computed as
    (6 \div 2) \times 3
    , not
    6 \div (2 \times 3)
    .
  • Common Mistake: Incorrectly grouping multiplication first (e.g.,
    6 \div (2 \times 3) = 6 \div 6 = 1
    ) violates the left-to-right rule.
  • Verification: Rewrite as
    2 + ((6 \div 2) \times 3) - 1 = 2 + (3 \times 3) - 1 = 2 + 9 - 1 = 10
    .
Problem 10:
(5 + 3 \times 2)^2 \div 4
Step-by-Step Solution:
  1. Identify operations: The expression has brackets, addition (
    +
    ), multiplication (
    \times
    ), an exponent (
    ^2
    ), and division (
    \div
    ).
  2. Apply BODMAS: Solve brackets first, resolving multiplication before addition inside, then the exponent, then division.
  3. Solve multiplication inside brackets:
    3 \times 2 = 6
    .
  4. Solve addition inside brackets:
    5 + 6 = 11
    .
  5. Solve exponent:
    11^2 = 121
    .
  6. Perform division:
    121 \div 4 = 30.25
    .
Final Answer: ( 30.25 )
Detailed Explanation:
  • BODMAS Rule: Inside brackets, multiplication precedes addition. After resolving brackets, the exponent is applied to the entire bracket result, followed by division.
  • Common Mistake: Applying the exponent to individual terms (e.g.,
    5^2 + (3 \times 2)^2
    ) or dividing before the exponent is incorrect.
  • Verification: Rewrite as
    (5 + (3 \times 2))^2 \div 4 = (5 + 6)^2 \div 4 = 11^2 \div 4 = 121 \div 4 = 30.25
    .
Summary of Key BODMAS Principles:
  • Brackets: Always resolve expressions inside brackets first.
  • Orders: Exponents and roots are computed after brackets.
  • Division and Multiplication: Have equal precedence and are performed left to right.
  • Addition and Subtraction: Have equal precedence and are performed left to right after division and multiplication.
  • Parentheses for Clarity: When in doubt, use parentheses to explicitly show the order of operations.
These problems demonstrate the application of BODMAS across various complexities, helping to reinforce the importance of operation precedence in mathematical calculations. If you’d like more problems or variations (e.g., with roots or nested brackets), let me know!

 

 

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