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Magazine Chapter 1: Problem 1
Evaluate the following arithmetic expressions using the rules for priority of operations: (a) \(12 \div 2 \times 6\) (b) \(12 \times 6 \div 2\) (c) \(12 \div(2 \times 6)\) (d) \((12 \div 2) \times 6\)(e) \(12+6 \div 2\) (f) \(12 \div 6+2\) (g) \(\frac{8-2}{2+1}\) (h) \(\frac{-10 \div 2}{-6+1}\)
Short Answer
Expert verified
(a) 36, (b) 36, (c) 1, (d) 36, (e) 15, (f) 4, (g) 2, (h) 1.
Step by step solution
01
Solve Expression (a)
For expression (a) \(12 \div 2 \times 6\), we follow the order of operations. First, perform the division: \(12 \div 2 = 6\). Then, multiply the result by 6: \(6 \times 6 = 36\). The answer is 36.
02
Solve Expression (b)
For expression (b) \(12 \times 6 \div 2\), the operations are multiplication and division, which we perform from left to right. First, \(12 \times 6 = 72\). Next, divide 72 by 2: \(72 \div 2 = 36\). The answer is 36.
03
Solve Expression (c)
For expression (c) \(12 \div (2 \times 6)\), handle the parentheses first by performing the multiplication: \(2 \times 6 = 12\). Then, perform the division: \(12 \div 12 = 1\). The answer is 1.
04
Solve Expression (d)
For expression (d) \((12 \div 2) \times 6\), solve the operation inside the parentheses first: \(12 \div 2 = 6\). Then, multiply by 6: \(6 \times 6 = 36\). The answer is 36.
05
Solve Expression (e)
For expression (e) \(12+6 \div 2\), follow the order of operations by performing division first: \(6 \div 2 = 3\). Then add the result to 12: \(12 + 3 = 15\). The answer is 15.
06
Solve Expression (f)
For expression (f) \(12 \div 6 + 2\), perform division first: \(12 \div 6 = 2\). Then add 2: \(2 + 2 = 4\). The answer is 4.
07
Solve Expression (g)
For expression (g) \(\frac{8-2}{2+1}\), calculate the numerator and denominator separately. The numerator is \(8 - 2 = 6\) and the denominator is \(2 + 1 = 3\). Therefore, divide: \(\frac{6}{3} = 2\). The answer is 2.
08
Solve Expression (h)
For expression (h) \(\frac{-10 \div 2}{-6 + 1}\), begin with the division in the numerator: \(-10 \div 2 = -5\). Then, calculate the denominator: \(-6 + 1 = -5\). The fraction becomes \(\frac{-5}{-5} = 1\). The answer is 1.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Arithmetic Expressions
Arithmetic expressions are combinations of numbers and operators that need to be evaluated according to specific rules. These operators include:
When solving arithmetic expressions, it's crucial to understand the order of operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This order ensures that expressions are solved consistently and accurately.
For example, in expressions like \(12 + 6 \div 2\) or \(12 \times 6 \div 2\), division and multiplication are performed before addition and subtraction, following the accepted rules.
- Addition (+)
- Subtraction (-)
- Multiplication (脳)
- Division (梅)
When solving arithmetic expressions, it's crucial to understand the order of operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This order ensures that expressions are solved consistently and accurately.
For example, in expressions like \(12 + 6 \div 2\) or \(12 \times 6 \div 2\), division and multiplication are performed before addition and subtraction, following the accepted rules.
Division and Multiplication
Division and multiplication in arithmetic expressions must be handled with care. They are equally prioritized operations that should always be executed sequentially from left to right in an expression.
Consider the expression \(12 \div 2 \times 6\). According to the rules, start with the division: \(12 \div 2 = 6\). Then multiply: \(6 \times 6 = 36\). If an expression includes both these operations, performing them in left-to-right order is key to arriving at the correct result.
Similarly, with \(12 \times 6 \div 2\), the multiplication comes first: \(12 \times 6 = 72\). Then, proceed with the division: \(72 \div 2 = 36\). This uniform method of resolving operations helps reduce errors and maintains clarity.
Consider the expression \(12 \div 2 \times 6\). According to the rules, start with the division: \(12 \div 2 = 6\). Then multiply: \(6 \times 6 = 36\). If an expression includes both these operations, performing them in left-to-right order is key to arriving at the correct result.
Similarly, with \(12 \times 6 \div 2\), the multiplication comes first: \(12 \times 6 = 72\). Then, proceed with the division: \(72 \div 2 = 36\). This uniform method of resolving operations helps reduce errors and maintains clarity.
Parentheses in Calculations
Parentheses in arithmetic expressions serve to alter the default order of operations. They mark off sub-expressions that should be evaluated first, overriding the standard PEMDAS rules.
For instance, in the expression \(12 \div (2 \times 6)\), calculations inside the parentheses need to be completed first. Therefore, \(2 \times 6 = 12\); only after this is done do we proceed with the division: \(12 \div 12 = 1\).
Using parentheses appropriately is crucial for ensuring that arithmetic expressions represent the intended calculations. They help group parts of an expression to reflect different priorities and ensure that calculations are performed correctly.
For instance, in the expression \(12 \div (2 \times 6)\), calculations inside the parentheses need to be completed first. Therefore, \(2 \times 6 = 12\); only after this is done do we proceed with the division: \(12 \div 12 = 1\).
Using parentheses appropriately is crucial for ensuring that arithmetic expressions represent the intended calculations. They help group parts of an expression to reflect different priorities and ensure that calculations are performed correctly.
Numerator and Denominator Evaluations
In expressions involving fractions, it is necessary to evaluate both the numerator and the denominator separately before performing the division.
Take the expression \(\frac{8-2}{2+1}\) as an example. First, calculate the numerator: \(8 - 2 = 6\). Then, solve the denominator: \(2 + 1 = 3\). Finally, divide the two results: \(\frac{6}{3} = 2\).
This method ensures that each part of the fraction is handled accurately, preventing mistakes and producing the correct result every time. It is especially useful when fractions involve more complex arithmetic operations, maintaining balance and correctness in calculations.
Take the expression \(\frac{8-2}{2+1}\) as an example. First, calculate the numerator: \(8 - 2 = 6\). Then, solve the denominator: \(2 + 1 = 3\). Finally, divide the two results: \(\frac{6}{3} = 2\).
This method ensures that each part of the fraction is handled accurately, preventing mistakes and producing the correct result every time. It is especially useful when fractions involve more complex arithmetic operations, maintaining balance and correctness in calculations.
