Problem 22 - Prealgebra

Question

Write each decimal as a fraction or mixed number in simplest form. $$3.625$$

Step-by-Step Solution

Answer

3.625 as a fraction is \(3 \frac{5}{8}\).

Step 1: Understand the Decimal

First, note that the decimal 3.625 has a whole number part and a fractional part. The whole number part is 3.

Step 2: Convert the Decimal to a Fraction

The decimal part 0.625 can be written as a fraction by considering 625 over 1,000 since it is three decimal places to the right: \( \frac{625}{1000} \).

Step 3: Simplify the Fraction

Simplify \( \frac{625}{1000} \). The greatest common divisor (GCD) of 625 and 1000 is 125. Dividing both the numerator and the denominator by 125 gives: \( \frac{625 \div 125}{1000 \div 125} = \frac{5}{8} \).

Step 4: Write as a Mixed Number

Combine the whole number with the simplified fraction: \( 3 \frac{5}{8} \). This is the mixed number representation of 3.625.

Key Concepts

Decimal to Fraction Conversion

When converting a decimal to a fraction, you'll want to first split the decimal into its whole number and fractional components. Let's take the example of 3.625. Here, **3** is the whole number, while **0.625** is a fractional decimal part.

To convert the fractional part, consider how many digits lie after the decimal point. In 0.625, there are three digits. This means the decimal can be represented as 625 over 1,000 (because there are three decimal places, align with thousandths):

• The decimal point decides the power of 10 that acts as your denominator: 10, 100, 1,000, etc.
• For decimals like 0.625, which has three decimal places, use 1,000 as the denominator: \[ 0.625 = \frac{625}{1000} \]

This step helps translate the decimal into a fraction, leading us to the next crucial part: simplifying.

Simplifying Fractions

After converting a decimal to a fraction, often it can be simplified to make calculations easier and more understandable.

In the fraction \[ \frac{625}{1000} \], both the numerator and the denominator can be divided by their greatest common divisor (GCD). To find the GCD, list the factors of each number and find the largest one they share. In this case, the GCD is 125.

• Divide both 625 and 1000 by 125: \[ \frac{625 \div 125}{1000 \div 125} = \frac{5}{8} \]
• This gives you a simplified fraction and helps make further math tasks simpler.

Always simplify fractions when possible to make them easier to work with.

Mixed Numbers

A mixed number combines a whole number with a proper fraction, making it an effective way to represent a number that isn't whole, like 3.625. Once the decimal is converted to a simpler fraction, combine it with the whole number part. From the fractional step, we have "3 and \[ \frac{5}{8} \]".

These steps create your final mixed number:

• The whole number remains the same: **3**
• The simplified fraction from earlier is added: \[ 3 \frac{5}{8} \]

This mixed number represents the original decimal in an easily understandable and often more interpretable format for further mathematical operations.