Problem 47 - Prealgebra

Question

Simplify the given expression by first converting the fraction into a terminating decimal. \(\frac{7}{5}+5.31\)

Step-by-Step Solution

Answer

The simplified expression is 6.71.

Step 1: Understand the Problem

We need to simplify the expression by first converting the fraction \( \frac{7}{5} \) into a decimal and then adding it to the decimal 5.31.

Step 2: Convert Fraction to Decimal

To convert the fraction \( \frac{7}{5} \) into a decimal, we divide the numerator (7) by the denominator (5). Performing the division, we get \( 7 \div 5 = 1.4 \).

Step 3: Add Decimals Together

Now, add the decimal from the fraction, 1.4, to the given decimal, 5.31. Performing the addition: \( 1.4 + 5.31 = 6.71 \).

Step 4: Final Simplified Expression

After adding the two decimals, the final simplified expression is 6.71.

Key Concepts

Fractions to Decimals

Fractions represent parts of a whole and are expressed as a ratio of two integers, with the numerator (top number) and the denominator (bottom number). Converting fractions to decimals involves division where the numerator is divided by the denominator.

• For example, take the fraction \(\frac{7}{5}\).
• Divide 7 by 5 to convert this fraction into a decimal.
• The result is \(7 \div 5 = 1.4\).

This process depends on whether the decimal terminates (ends at a certain point) or repeats. In our case, \(1.4\) is a terminating decimal because the division ends with a finite number of digits. This straightforward conversion process is necessary when solving problems involving decimals and fractions together.

Addition of Decimals

Adding decimals is similar to adding whole numbers, but it's crucial to align the decimal points. If the decimal points are not aligned, the numbers will be mismatched, leading to incorrect results. Consider the following simplified steps:

• Write down the numbers vertically, aligning the decimal points.
• Add zeros to the shorter decimal if necessary, to make them have the same number of digits after the decimal point.
• Start adding from the rightmost digit (the smallest decimal place) and move to the left, carrying over if needed.

For our example, \(1.4\) (from the fraction \(\frac{7}{5}\)) aligns well with the given decimal \(5.31\). Performing the addition gives us \(1.4 + 5.31 = 6.71\). This simple yet important process allows us to handle real-world problems where sums of decimals are common.

Simplifying Expressions

Simplifying expressions is the process of altering a mathematical expression into its simplest form. In arithmetic, this often involves performing operations like addition, subtraction, multiplication, and division to reach a single numerical result.
First, ensure all components of the expression are in compatible forms. This involves converting fractions to decimals as needed, as we've done here. The next step is straightforward arithmetic operations on the decimals.
To simplify \(\frac{7}{5} + 5.31\), we first converted the fraction to a decimal (\(1.4\)) and then added it to the given decimal. Therefore, the entire expression simplifies logically to \(6.71\). The simplified expression is straightforward and easily interpreted or used in further calculations.