Problem 42 - Prealgebra

Question

Simplify each of the following as much as possible, and write all answers as decimals. $$\frac{1}{8}(0.7)+\frac{3}{8}(0.7)$$

Step-by-Step Solution

Answer

The simplified decimal is 0.35.

Step 1: Distribute the Fraction to the Decimal

Firstly, distribute \( \frac{1}{8} \) to \( 0.7 \). This gives us \( \frac{1}{8} \times 0.7 = 0.0875 \).

Step 2: Distribute the Other Fraction to the Decimal

Next, distribute \( \frac{3}{8} \) to \( 0.7 \). This gives us \( \frac{3}{8} \times 0.7 = 0.2625 \).

Step 3: Add the Results from Step 1 and Step 2

Add the results from Step 1 and Step 2: \( 0.0875 + 0.2625 = 0.35 \).

Key Concepts

Simplification

Simplification in mathematics means making an expression easier to understand or work with. By reducing complexity, we make it simpler to find an answer or gain insight into a problem.

In the given exercise, simplification involves combining like terms or evaluating expressions to their fullest. For example, reducing fractions, executing multiplications, and performing additions help achieve a simpler form.

• Start by executing operations within parentheses or dealing with fractions and decimals as needed.
• When working with decimals, write them as fractions if necessary to make calculations more direct.
• Simplified results are usually expressed in the smallest possible terms.

Remember that simplification might also involve applying arithmetic operations to reach a singular, consolidated answer, consistent with the context of your initial problem.

Fractions

Fractions represent parts of a whole number and are written in the form \( \frac{a}{b} \), where \( a \) is the numerator and \( b \) is the denominator. In mathematical expressions, fractions can often be converted to decimals for ease of use, but retaining their fractional form can sometimes simplify calculations and comparisons.

In the original exercise, we dealt with fractions \( \frac{1}{8} \) and \( \frac{3}{8} \). These fractions are multiplied by the decimal 0.7, showing how fractions can interact with other numbers.

• Multiply the numerators straight across to evaluate: \(\frac{1}{8} \times 0.7 = 0.0875\)
• Combining fractions is often easier if they share a common denominator, but multiplication does not require this.
• Fractions can become decimals during calculations, as seen in the result of the multiplication.

Recognizing when and how to convert between fractions and decimals is crucial for efficient problem-solving.

Distributive Property

The distributive property is a fundamental mathematical property used to expand expressions by multiplying a single term with two or more terms inside parentheses. This property can simplify calculations, especially when dealing with multiplication across large sums or when fractions and decimals are involved.

The distributive property states that for any numbers \(a\), \(b\), and \(c\), \(a(b + c) = ab + ac\).

In our exercise, the distributive property simplifies the expression by handling each fraction and the decimal separately:

• First, distribute \( \frac{1}{8} \times 0.7\), then \(\frac{3}{8} \times 0.7\).
• Add the results to consolidate the expression: \(0.0875 + 0.2625\).

This method ensures each part of the expression is evaluated thoroughly, leading to a simplified and complete solution. Using the distributive property, we break down complex equations into manageable parts for ease of calculation.