Problem 45 - Prealgebra

Question

Simplify each of the following as much as possible, and write all answers as decimals. $$\frac{5}{8}+0.35\left(\frac{1}{2}\right)$$

Step-by-Step Solution

Answer

The simplified form of the expression is 0.8.

Step 1: Simplify the Fraction Multiplication

Start by simplifying the multiplication part of the expression: \(0.35 \times \frac{1}{2}\). To multiply a decimal by a fraction, multiply the decimal by the numerator, and divide by the denominator. So, we have: \[0.35 \times \frac{1}{2} = \frac{0.35 \times 1}{2} = \frac{0.35}{2} = 0.175 \] Thus, the multiplication results in 0.175.

Step 2: Convert Fraction to Decimal and Add

We need to add the result from Step 1 to \(\frac{5}{8}\). First, we convert \(\frac{5}{8}\) into a decimal. Divide 5 by 8 to get: \[\frac{5}{8} = 0.625\]Now, add the decimal results: \[0.625 + 0.175 = 0.800\]

Step 3: Final Simplification and Answer

Check the addition to ensure the simplification is correct. Thus, the expression \(\frac{5}{8} + 0.35 \left(\frac{1}{2}\right)\) simplifies to 0.800. So, the final simplified answer is 0.8, as decimals are usually expressed to one significant figure.

Key Concepts

Fraction to Decimal

Converting a fraction to a decimal is essential when you want to perform operations with numbers expressed in different formats, like fractions and decimals.

The conversion process involves two main steps: dividing the numerator (the top part of the fraction) by the denominator (the bottom part). Let’s consider the fraction \( \frac{5}{8} \) from our exercise.

• Perform the division: 5 divided by 8. This operation gives us the decimal 0.625.
• Using a calculator can be very helpful here to ensure precision.

By converting fractions to decimals, you make calculations simpler, especially when you need to add, subtract, multiply, or divide them with decimals.

Decimal Addition

Adding decimals is a fundamental math operation, often necessary in exercises involving different number forms. In this exercise, once you've converted your fraction into a decimal form, you find yourself performing an operation with two decimals: 0.625 and 0.175.

To add these two decimals, you need to align them by their decimal points. This ensures that each digit is placed in the correct column whether it be units, tenths, hundredths, etc. Here’s a quick guide:

• Write the numbers one above the other with the decimal points lined up.
• Add each column starting from the right (hundredths place) to the left.
• If any column adds up to more than 9, carry over to the next left column.

In our example, 0.625 plus 0.175 equals 0.800.

Decimal addition is simple when following these steps, ensuring accuracy in calculations.

Multiplication with Fractions

When multiplying a decimal by a fraction, the process can initially seem complicated due to the mixed number format. However, breaking it down into straightforward steps makes it much easier. With our exercise, you're multiplying 0.35 by \( \frac{1}{2} \). Here’s how to handle this operation:

First, multiply the decimal and the fraction's numerator. In our exercise:

• Multiply 0.35 by 1 (the numerator of \( \frac{1}{2} \)), which equals 0.35.
• Then, divide the result (0.35) by the fraction's denominator, which is 2.

This division gives you the result 0.175.

This step-by-step method simplifies the multiplication and helps in finding accurate results when dealing with problems that have both fractions and decimals.