Problem 8 - Algebra A Combined Function

Question

Write each decimal as a fraction. $$ 892.7 $$

Step-by-Step Solution

Answer

892.7 as a fraction is \( \frac{8927}{10} \).

Step 1: Identify the Decimal Number

The given decimal number is 892.7. We need to express it as a fraction.

Step 2: Break Down the Decimal Number

The decimal 892.7 can be split into the whole number part 892 and the decimal part 0.7.

Step 3: Convert Decimal Part to Fraction

The decimal 0.7 can be expressed as a fraction. Since 0.7 is equivalent to 7 tenths, it can be written as \( \frac{7}{10} \).

Step 4: Combine Both Parts

Combine the whole number and the fractional part by adding them together: \( 892 + \frac{7}{10} = \frac{8920}{10} + \frac{7}{10} \).

Step 5: Find a Common Denominator & Add

Convert the whole number 892 into a fraction with the same denominator as the tenths: \( \frac{8920}{10} + \frac{7}{10} = \frac{8920 + 7}{10} = \frac{8927}{10} \).

Step 6: Final Fraction Form

The decimal 892.7 is represented as the fraction \( \frac{8927}{10} \).

Key Concepts

Decimal Representation

Decimals are a way to express numbers that are not whole. They use a decimal point to separate the whole part from the fractional part. For example, in the number 892.7, the part before the decimal point (892) is the whole number, and the part after the decimal point (0.7) represents a fraction of a whole.
Understanding this is key to converting decimals into fractions. When a number is written in decimal form, each digit after the decimal point represents a fraction with a denominator that is a power of 10. In our example, 0.7 means 7 tenths. Breaking down the decimal helps identify the fractional component that needs to be converted into a fraction.

• The first digit after the decimal point is tenths.
• The second digit is hundredths.
• The third digit is thousandths, and so on.

Recognizing which place value each decimal digit has is crucial for accurate conversion into a fraction.

Fractional Conversion

Converting decimals to fractions allows us to express numbers in another mathematical form, which can be simpler to deal with in some mathematical operations. To convert the decimal part of a number like 892.7, you focus on the portion after the decimal point: 0.7.

The process of fractional conversion involves a few steps:

• Identify the decimal part: 0.7.
• Recognize its least place value. Here, 7 is in the tenths place.
• Convert it to a fraction using the place value: 0.7 becomes \( \frac{7}{10} \).

Once you convert the decimal to a fraction, it becomes simpler to work with as part of mathematical operations, such as addition or multiplication. Understanding this process helps one handle complex calculations more effectively, as fractions can sometimes be more convenient than decimals for exact calculations.

Common Denominator

When adding fractions, it's important they share a common denominator. This means the base (bottom) number of each fraction should be the same. It ensures the fractions are on equivalent terms, making addition straightforward.

For example, when converting a number like 892.7 into a fraction, we separate it into a whole number part (892) and a fractional part (0.7, which is \( \frac{7}{10} \)). To combine these, we convert the whole number into a fraction with the same denominator:

• 892 is transformed into \( \frac{8920}{10} \) because \( 892 \times 10 = 8920 \).
• Add the fractions \( \frac{8920}{10} + \frac{7}{10} \) keeping the common denominator of 10.
• The result is \( \frac{8927}{10} \).

Having a common denominator simplifies the process, making it easy to add and convert decimals into proper fractions without confusion or errors. This strategy ensures a smooth transition from decimals to fractions, maintaining accuracy in calculations.