Problem 59 - Prealgebra

Question

Convert the given decimal to an improper fraction. Do not simplify your answer. 3.9

Step-by-Step Solution

Answer

3.9 as an improper fraction is \( \frac{39}{10} \).

Step 1: Identify the Whole Number and the Decimal

The given number is 3.9. Here, 3 is the whole number and 0.9 is the decimal part.

Step 2: Convert the Decimal Part to a Fraction

The decimal 0.9 can be expressed as the fraction \( \frac{9}{10} \) because there is one digit after the decimal point, meaning the denominator should be 10.

Step 3: Combine Whole and Fraction Parts

To combine the whole number 3 and the fraction \( \frac{9}{10} \), you first convert the whole number 3 into a fraction with the same denominator, which is \( \frac{30}{10} \).

Step 4: Add Fractions

Now, add the fractions \( \frac{30}{10} \) and \( \frac{9}{10} \). This gives \( \frac{30}{10} + \frac{9}{10} = \frac{39}{10} \). Thus, 3.9 as an improper fraction is \( \frac{39}{10} \).

Key Concepts

Decimal to Fraction Conversion

Converting decimals to fractions is an essential skill in mathematics. Decimals represent parts of whole numbers. To write them as fractions, you follow a few straightforward steps.

Let's take the decimal 0.9 as an example. Decimals can be read with place values, where "0.9" is in the tenths place. This means 0.9 is equivalent to nine tenths, or \( \frac{9}{10} \).

Steps for Conversion:

• Write down the decimal number as the numerator (top part) of the fraction.
• The denominator (bottom part) is the place value of the last digit. For 0.9, it's 10 because the last digit is in the tenths place.
• Combine these to make a fraction: \( \frac{9}{10} \).

Remember, this method applies to more complex decimals as well. For example, in the case of 0.75, the fraction would be \( \frac{75}{100} \), which further simplifies to \( \frac{3}{4} \). However, simplification is not always required.

Adding Fractions

Adding fractions is a process of combining numbers that have the same or different denominators. It’s essential to ensure the fractions have the same denominator before adding them. This is because you can only add fractions directly when their denominators match.

Steps to Add Fractions:

• If the denominators are the same, simply add the numerators (top numbers).
• Keep the denominator the same when adding the numerators.
• If denominators differ, find the smallest common multiple to rewrite them with a common denominator.

For example, if you need to add \( \frac{30}{10} \) and \( \frac{9}{10} \), since both have a common denominator of 10, you directly add:\( 30 + 9 = 39 \). Thus, \( \frac{30}{10} + \frac{9}{10} = \frac{39}{10} \). Ensure both fractions have the same denominator to make the process smooth and accurate.

Combining Whole Numbers and Fractions

Bringing together whole numbers and fractions is about converting the whole number into fraction form. This allows the operation (such as addition or subtraction) to proceed easily.

Procedure:

• Convert the whole number into a fraction by assigning it a denominator of 1.
• If a common denominator is already present in another fraction, multiply the converted whole number so that its denominator matches.
• Once matched, add or subtract the fractions as needed.

Using our example, convert a whole number like 3 into a fraction with the same denominator as the decimal-converted fraction \( \frac{9}{10} \). Therefore, 3 becomes \( \frac{30}{10} \). By doing this, you ensure that both parts - the whole and the fraction - can be added cleanly without any fuss, resulting in \( \frac{39}{10} \). This combined fraction represents both the original whole number and the fractional part.