Question

Explain to a friend why \(2+3 \cdot 4=14\) whereas \((2+3) \cdot 4=20\)

Short answer

Order of operations determines that multiplication comes before addition in \(2 + 3 \times 4 = 14\), while parentheses change the calculation by making addition come first in \((2 + 3) \times 4 = 20\).

Step-by-step solution

Understanding Order of Operations (PEMDAS/BODMAS)

To evaluate expressions correctly, follow the order of operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) or BODMAS (Brackets, Orders, Division, Multiplication, Addition, Subtraction).

Evaluate the First Expression without Parentheses

For the expression \(2 + 3 \times 4\), according to the order of operations, perform the multiplication before the addition. First calculate \( 3 \times 4 = 12 \). Then, add the result to 2: \( 2 + 12 = 14 \).

Evaluate the Second Expression with Parentheses

For the expression \((2 + 3) \times 4\), solve the parentheses first. Calculate \( 2 + 3 = 5 \). Then multiply the result by 4: \( 5 \times 4 = 20 \).

Compare the Results

Summarize the results of both expressions: \(2 + 3 \times 4\) results in 14 because multiplication is performed before addition. \((2 + 3) \times 4\) results in 20 because the parentheses indicate the addition should be performed first, followed by multiplication.

Key concepts

PEMDAS

PEMDAS is an acronym that helps us remember the correct order of operations in arithmetic. PEMDAS stands for:

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

This means that when you are solving an arithmetic expression, you should first do any calculations inside parentheses, then any exponents, followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right). For example, in the expression \(2 + 3 \times 4\), according to PEMDAS, you would first perform the multiplication (\(3 \times 4 = 12\)) before doing the addition, resulting in \(2 + 12 = 14\).

BODMAS

Another way to remember the order of operations is by using the acronym BODMAS, which stands for:

• Brackets
• Orders (another term for exponents)
• Division
• Multiplication
• Addition
• Subtraction

BODMAS is essentially the same as PEMDAS, but uses slightly different terminology. It is particularly common in countries that use British English. Just like with PEMDAS, operations enclosed in brackets should be performed first, followed by orders (or exponents), and then division and multiplication (from left to right), and finally addition and subtraction (from left to right). For example, in the expression \((2 + 3) \times 4\), the brackets tell us we need to add 2 and 3 first to get 5, and then multiply that result by 4 to get 20.

Parentheses in Arithmetic

Parentheses are used in arithmetic to indicate that the operations inside them should be performed first, before any other operations outside the parentheses. This can change the result of an expression significantly. For example, in the expression \(2 + 3 \times 4\), the multiplication is performed first, giving us 12, and then we add 2 to get 14. However, if we add parentheses and write \((2 + 3) \times 4\), the operation inside the parentheses (2 + 3) is performed first, giving us 5, and then we multiply by 4 to get 20. Using parentheses correctly is crucial for getting the intended result in arithmetic problems.

Multiplication before Addition

A common rule in both PEMDAS and BODMAS is that multiplication must be performed before addition (unless parentheses dictate otherwise). This is because multiplication is considered to have a higher priority than addition. For example, in the expression \(2 + 3 \times 4\), multiplication is performed first, resulting in \(3 \times 4 = 12\), and then the result is added to 2, giving \(2 + 12 = 14\). Ignoring this rule can lead to incorrect results. For instance, if you mistakenly add 2 and 3 first to get 5, and then multiply by 4, you would incorrectly get 20 instead of the correct answer, 14.