Problem 58 - Prealgebra

Question

Convert the given decimal to an improper fraction. Do not simplify your answer. 2.893

Step-by-Step Solution

Answer

The improper fraction is \( \frac{2893}{1000} \).

Step 1: Write the Decimal as a Fraction

First, express the decimal number as a fraction. The decimal 2.893 can be written as \( \frac{2893}{1000} \). This is because there are three decimal places, so we use 1000 as the denominator.

Step 2: Convert the Mixed Number to an Improper Fraction

Since the decimal portion 2.893 indicates a whole number, we can express this as an improper fraction. First, recognize that 2 can be represented as \( \frac{2 \times 1000}{1000} = \frac{2000}{1000} \). Then add this to the fraction \( \frac{893}{1000} \).

Step 3: Calculate the Sum of Fractions

Now, add \( \frac{2000}{1000} \) and \( \frac{893}{1000} \). The sum is \( \frac{2000 + 893}{1000} = \frac{2893}{1000} \), which is the improper fraction representation of the number 2.893.

Key Concepts

Decimals to Fractions

Converting decimals to fractions is a foundational mathematical skill. It involves expressing numbers with decimal points as fractions, which are simply divisions of whole numbers. To convert a decimal to a fraction, follow these steps:

• Identify the place value of the last digit in the decimal. For example, in 2.893, the last digit "3" is in the thousandths place.
• Write the decimal as a fraction using the place value as the denominator. Here, 2.893 becomes \( \frac{2893}{1000} \).

This approach makes it easier to work with numbers in different mathematical operations by utilizing the more familiar structure of fractions.
Fractions represent parts of a whole, which often makes them easier to manipulate when performing calculations. Don't worry about simplifying the fraction unless specifically instructed; for many exercises, an improper fraction in this form is an acceptable final answer.

Mixed Numbers

Mixed numbers combine whole numbers with fractions. They are particularly useful when dealing with values that are not whole but have both an integer and a fractional component. For example, the decimal 2.893 can be seen as the mixed number 2 and 893/1000.
When converting a mixed number to an improper fraction:

• Multiply the whole number by the fraction's denominator.
• Add the result to the numerator of the fraction.
• The denominator remains the same.

This results in an improper fraction, where the numerator is greater than the denominator.
For 2.893 in our context:

• First, express the whole number "2" as a fraction: \( \frac{2 \times 1000}{1000} = \frac{2000}{1000} \).
• Then represent 893/1000 and join these to form \( \frac{2893}{1000} \).

This helps in various operations, especially when performing fraction addition or subtraction, where mixed numbers might be cumbersome.

Fraction Addition

Adding fractions is a fundamental operation when dealing with numbers in fractional form. To successfully add fractions, it is crucial that the fractions have the same denominator, often referred to as like denominators. Here's how you can go about it:

• Ensure both fractions have the same denominator. This makes the addition process straightforward as you're dealing with parts of the same whole.
• Add the numerators together while keeping the same denominator.

In the case of adding \( \frac{2000}{1000} \) and \( \frac{893}{1000} \):

• The denominators are already the same, so simply add the numerators: 2000 + 893 = 2893.
• Thus, the resulting fraction is \( \frac{2893}{1000} \).

This process results in a single, simplified fraction when the denominators are like. It's a method your mathematical toolkit will use often, ensuring you can handle even complex calculations with fractions effectively.