Problem 16 - Basic Engineering Mathematics

Question

Determine the value of $$ 13-2 \times 3+14 \div(2+5) $$

Step-by-Step Solution

Answer

The value is 9.

Step 1: Solve Inside the Parentheses

First, we resolve any expressions inside parentheses. The expression within the parentheses is \(2+5\). Calculate that first to get:\[2+5 = 7\]

Step 2: Division

Next, perform the division within the expression. Replace the expression inside the parentheses with the calculated result from Step 1:\[14 \div 7\]Calculate the division to get:\[14 \div 7 = 2\]

Step 3: Multiplication

Now handle multiplication. The expression becomes:\[13-2 \times 3 + 2\]Calculate the multiplication:\[2 \times 3 = 6\]

Step 4: Final Calculation

Replace the multiplication in the expression and solve the rest:\[13 - 6 + 2\] First, subtract:\[13 - 6 = 7\]Then, add the result to 2:\[7 + 2 = 9\]

Key Concepts

Parentheses

Parentheses signify that the enclosed expression must be addressed first in any mathematical equation. This is one of the most critical rules in the order of operations. It acts like a priority signal directing us where to start.
Here's how to handle them:

• Identify the part of the equation enclosed by parentheses.
• Solve the expression inside the parentheses first before moving on to other operations.

In our example, the parentheses contain the expression:- Inside the parentheses: \(2+5\)We start by calculating the value within these parentheses, which gives us the result of 7. Subsequently, we replace \(2+5\) in the original equation with 7. This step simplifies the equation and allows us to focus on the remaining operations.

Multiplication and Division

Once expressions inside parentheses are resolved, the next focus is on multiplication and division. They are of equal precedence, meaning you handle them from left to right, in the order they appear.
The sequence for solving these is straightforward:

• Look through the equation from left to right.
• Perform any multiplication or division as they appear.

In our specific example:- We first replace the part in parentheses, leading to the sub-expression \(14 \div 7\). Solving this gives us 2.- Then, move to the multiplication \(2 \times 3\), and compute it as 6.By performing these operations, we continue transforming the original equation into a simpler form without altering its value. This structured approach ensures clarity and accuracy in solving complex expressions.

Addition and Subtraction

Finally, we tackle addition and subtraction, the last steps in solving the equation. These operations are at the same level of precedence and are executed from left to right as they appear.
Here's a quick guide on what to do:

• Scan the equation from left to right.
• Perform addition or subtraction operations sequentially.

In the example equation, after tackling multiplication and division, we substitute the results back into the expression:- Start by subtracting \(13 - 6\) which equals 7.- Now, add 2 to 7, resulting in 9.These final steps ensure we have carefully transformed the expression to reveal its value. Remember, by carefully following the order of operations, you arrive at the correct solution efficiently.