Problem 16 - Elementary Algebra

Question

For the following problems, use the order of operations to find each value. $$6(4-1)+8(3+7)-20$$

Step-by-Step Solution

Answer

Answer: The value of the expression is 78.

Step 1: Parentheses

First, perform the operations inside the parentheses. $$6(4-1)+8(3+7)-20$$ Becomes: $$6(3)+8(10)-20$$

Step 2: Multiplication

Next, perform the multiplication operations. $$6\times 3+8\times 10-20$$ Becomes: $$18+80-20$$

Step 3: Addition and Subtraction

Finally, perform the addition and subtraction operations from left to right. $$18+80-20$$ First, add 18 and 80 to get 98: $$98-20$$ Then, subtract 20 from 98 to get the final value of 78. $$78$$ Hence, the value of the expression $$6(4-1)+8(3+7)-20$$ is 78.

Key Concepts

Parentheses

Parentheses are an essential part of mathematical expressions. They indicate which operations need to be performed first. In the order of operations, you should always evaluate the expressions within parentheses before handling any other part of the formula.
Consider the example from the original problem:

• The given expression is: \(6(4-1)+8(3+7)-20\).
• Start by solving what’s inside the parentheses: \(4-1\) becomes \(3\), and \(3+7\) becomes \(10\).
• As a result, the expression simplifies to: \(6(3)+8(10)-20\).

Parentheses can alter the way an expression is calculated, grouping terms to change the usual left-to-right sequence, so always address them first.

Multiplication

Once you've dealt with any parentheses, the next step is to handle multiplication (and division, although this isn't in our example). Multiplication is performed before addition and subtraction, which ensures the correct value is computed following the order of operations.
From the simplified expression:

• \(6(3)+8(10)-20\) changes to \(6 \times 3 + 8 \times 10 - 20\).
• Next, compute each multiplication: \(6 \times 3 = 18\) and \(8 \times 10 = 80\).
• This then converts the expression to: \(18 + 80 - 20\).

Always tackle multiplication early on to ensure you keep your calculations in order and accurate.

Addition and Subtraction

The final operations in our order of operations are addition and subtraction. These should be performed from left to right, completing one step at a time to ensure accuracy.
In our expression:

• We have \(18 + 80 - 20\) from the previous step.
• Begin from the left: add 18 and 80 to form \(98\).
• Then, take that result and subtract 20: \(98 - 20\), leading to the final result of \(78\).

This step-by-step approach avoids errors, as mixing up addition and subtraction can easily happen if the proper order is not followed. Just remember, left to right as the final rule!