Problem 19 - Elementary Algebra

Question

For the following problems, use the order of operations to find each value. $$\frac{(1+16-3)}{7}+5(12)$$

Step-by-Step Solution

Answer

Answer: The value of the given expression is \(62\).

Step 1: Evaluate the expression inside the parentheses

First, let's address the expression inside the parentheses, \((1+16-3)\). We have one addition and one subtraction operation: $$1+16-3 = 17-3 = 14$$ Now our expression looks like this: $$\frac{14}{7} + 5(12)$$

Step 2: Perform the division

Next, we'll perform the division operation, \(\frac{14}{7}\): $$\frac{14}{7} = 2$$ Now our expression has been simplified to: $$2 + 5(12)$$

Step 3: Perform the multiplication

Next, we'll perform the multiplication operation, \(5(12)\): $$5(12) = 60$$ Now our expression is simplified to: $$2 + 60$$

Step 4: Perform the addition

Lastly, we'll perform the addition operation, \(2 + 60\): $$2 + 60 = 62$$ Thus, the value of the given expression is \(\bold{62}\).

Key Concepts

Parentheses

Understanding how to tackle parentheses is the first step in mastering the order of operations. Parentheses are used to indicate that the operations inside them should be performed before anything else in the equation. This prioritization helps in organizing complex mathematical problems.
Mathematically, you will often see grouped operations like \(1+16-3\). Here, both addition and subtraction occur within the parentheses. Simply follow the sequence from left to right and resolve the operations in order. For example, first add \(1+16 = 17\) and then subtract \(17-3 = 14\).
Once the expression inside the parentheses is simplified, substitute this result into the rest of the equation. This forms the foundation for the next steps in solving your equation.

Division

After resolving parts of an equation within parentheses, division often comes next. Division is represented by the forward slash \(\frac{a}{b}\) which means "a divided by b."
When it’s time to tackle division, like going from \(\frac{14}{7}\) to \(2\), you simply calculate how many times the divisor (7) goes into the dividend (14). In our case, 7 goes into 14 exactly 2 times. This means that \(\frac{14}{7} = 2\).
After performing the division, replace the fraction with the result in your entire expression. Re-evaluate the updated equation as you prepare to proceed with multiplications next.

Multiplication

Following parentheses and division in the order of operations, multiplication comes next. In our example, we have the multiplication \(5(12)\).
Multiply 5 by 12 to get 60. Multiplication is a straightforward process of adding a number (12) to itself a specified number of times (5 times here).
Once you compute the multiplication correctly, plug the result back into the expression to further simplify it. This will move us towards tackling the final steps of addition and subtraction, streamlining the expression even more.

Addition and Subtraction

The last steps in the order of operations involve addition and subtraction. These operations come after you've completed any calculations in parentheses, division, and multiplication.
For the given expression \(2 + 60\), simply add the numbers together. Adding 2 to 60 results in 62. This is performed just like normal counting or combining groups – a fundamental arithmetic operation.
Addition and subtraction may appear simple, but they’re essential for synthesizing a final, correct answer from all prior operations. Concluding with addition confirms the solution to the original problem, where we see that \(2 + 60 = 62\). The correct evaluation from the original complex problem.