Problem 40 - Prealgebra

Question

Simplify each of the following as much as possible, and write all answers as decimals. $$6.7-\frac{1}{5}(0.45)$$

Step-by-Step Solution

Answer

6.61

Step 1: Evaluate the Fraction Multiplication

First, evaluate the product of the fraction \( \frac{1}{5} \) and the decimal 0.45. To do this, multiply the fraction and the decimal: \[ \frac{1}{5} \times 0.45 = 0.09 \]. This is because \( \frac{1}{5} = 0.2 \) and \( 0.2 \times 0.45 \) is calculated as follows: \( 0.2 \cdot 0.45 = 0.09 \).

Step 2: Subtract the Result from 6.7

Now that we have the result of the multiplication (0.09), subtract it from 6.7: \[ 6.7 - 0.09 = 6.61 \]. Perform the subtraction by aligning the decimal points, 6.70 - 0.09, which results in 6.61.

Key Concepts

Simplification

In mathematics, simplification refers to the process of reducing an expression or equation to its simplest form. It often involves combining like terms or performing operations to make expressions easier to work with and understand.

Simplification helps in reducing the complexity of a problem, making it more manageable. In the given exercise, the simplification involves evaluating the expression step by step. For example, we initially have two operations in the expression: multiplication and subtraction.

The priority is to follow the correct order of operations, which starts with multiplication before moving on to subtraction. This means we first focus on multiplying the fraction by the decimal before subtracting from 6.7. The final simplified step yields a clear, single decimal as the solution.

Fraction Multiplication

Multiplying fractions with decimals might seem complex at first, but it follows a straightforward method. A fraction multiplication involves multiplying the numerator (the top number of the fraction) by the other number involved in the operation, which can be a fraction or a decimal.

In this exercise, we are required to multiply the fraction \( \frac{1}{5} \) by the decimal 0.45. Since \( \frac{1}{5} \) converts to the decimal 0.2, the multiplication is \( 0.2 \times 0.45 \). When multiplying, you use standard arithmetic rules to compute the result, which is 0.09 in this case.

The product of the fraction and the decimal addresses the part of the expression we need to resolve before moving on to any other operations.

Subtraction

Subtraction is an arithmetic operation that finds the difference between numbers. In this exercise, it follows the fraction multiplication. We subtract the result derived from multiplying the fraction \( \frac{1}{5} \) and the decimal 0.45 from the number 6.7.

To perform subtraction with decimals, it's important to align the decimal points. Here, we have \( 6.7 - 0.09 \). Since 6.7 is equivalent to 6.70, lining up the decimals allows the operation to be conducted correctly.

By subtracting 0.09 from 6.70, we get the final result of 6.61. This subtraction finalizes the simplification of the expression, giving us the clean, simplified result as a decimal.