Problem 41 - Prealgebra

Question

Add and subtract as indicated. $$(7.8-4.3)+2.5$$

Step-by-Step Solution

Answer

The result is 6.0.

Step 1: Understand the Expression

The given expression is \((7.8 - 4.3) + 2.5\). It involves both subtraction and addition operations. We will perform each operation in the correct order.

Step 2: Perform the Subtraction

First, subtract 4.3 from 7.8. This means computing \(7.8 - 4.3 = 3.5\).

Step 3: Perform the Addition

Next, add 2.5 to the result obtained from the subtraction. So, compute \(3.5 + 2.5 = 6.0\).

Step 4: Verify the Result

Check the arithmetic by calculating each step to ensure there are no calculation errors: \(7.8 - 4.3\) gives \(3.5\) and \(3.5 + 2.5\) indeed results in \(6.0\).

Key Concepts

Addition

Addition is one of the most fundamental operations in arithmetic. It involves combining two or more numbers to get a total, known as the sum.
When you add numbers, you are essentially bringing amounts together to form a larger value. In our exercise, we have an addition operation:

• The result of the subtraction, which is 3.5,
• is added to 2.5.

To perform this addition step, you would simply align the numbers, ensuring they share the same decimal places, and sum them directly. For example, lining up vertically: 7.8 + 2.5 = 6.0
The important part in addition, especially with decimals, is to align the decimal points, making sure each digit is in the correct place value, whether it's units, tenths, hundredths, etc.

Subtraction

Subtraction is the process of taking one number away from another. It helps us find out the difference between two numbers. In terms of arithmetic operations, subtraction is essentially the "inverse" of addition.
In our example, we see subtraction in action:

• We subtract 4.3 from 7.8.

To subtract correctly, especially with decimals, align the numbers by their decimal points and subtract each corresponding digit. For subtraction, it can be helpful to use borrowing when numbers in a particular column don’t add up easily. In this calculation: 7.8 - 4.3 = 3.5
Notice that subtraction can also tell us how much more or less one quantity is compared to another, which is valuable in many practical scenarios.

Order of Operations

The order of operations dictates the sequence in which different mathematical operations are carried out. It is crucial to follow the correct order to arrive at the right solution, especially when multiple operations are involved, as in our exercise.
The standard rules for the order of operations can be remembered by the acronym PEMDAS:

• P - Parentheses first
• E - Exponents (i.e. powers and square roots, etc.)
• MD - Multiplication and Division (left to right)
• AS - Addition and Subtraction (left to right)

In the exercise \( (7.8 - 4.3) + 2.5 \), the subtraction within the parentheses is performed first. Only after completing that operation do we proceed with the addition. Following this precise order ensures that we do not jumble up operations, which might lead to incorrect answers.