Problem 99 - Prealgebra

Question

Write 2 as a fraction with denominator 8 .

Step-by-Step Solution

Answer

2 as a fraction with denominator 8 is \( \frac{16}{8} \).

Step 1: Understanding the Problem

We need to express the number 2 as a fraction that has 8 as its denominator. This means we are looking for a fraction \( \frac{n}{8} \) that is equivalent to 2.

Step 2: Express 2 as a Fraction with Denominator 1

Initially, consider the number 2 as a fraction \( \frac{2}{1} \). Every whole number can be expressed as a fraction by placing it over 1.

Step 3: Adjust to the Desired Denominator

To convert \( \frac{2}{1} \) into a fraction with a denominator of 8, we need to multiply both the numerator and the denominator by the same number to keep the value of the fraction unchanged. Calculate how 1 becomes 8 by figuring out what to multiply it by. In this case, we need to multiply by 8.

Step 4: Perform the Multiplication

Multiply both the numerator and denominator of \( \frac{2}{1} \) by 8:\[\frac{2 \times 8}{1 \times 8} = \frac{16}{8}\] This changes the denominator to 8, while keeping the value of the fraction the same as the original number 2.

Step 5: Verify the Result

Check that \( \frac{16}{8} \) is equal to 2 by performing the division 16 ÷ 8, which equals 2. This confirms that \( \frac{16}{8} \) is indeed equivalent to 2.

Key Concepts

Fractions with Different Denominators

When you're working with fractions, one of the things you'll often need to do is convert them into a format where they share the same denominator. This is especially necessary when you're trying to add or subtract fractions. A common denominator is simply a shared multiple of the denominators of each fraction involved in the operation. For example, if you have the fractions \( \frac{1}{4} \) and \( \frac{1}{3} \), they have different denominators of 4 and 3, respectively. In order to add or compare these fractions, you would first find a common denominator, which in this case would be 12. By converting each fraction to have the same denominator, they become \( \frac{3}{12} \) and \( \frac{4}{12} \). Now, you can easily add or compare them.Understanding how to manipulate fractions so that they share the same denominator is crucial for many types of mathematical problems, especially those involving fractions with different denominators.

Equivalent Fractions

Equivalent fractions may look different but they are actually the same in value. This means that you can change the appearance of a fraction without changing its value by multiplying or dividing both the numerator and the denominator by the same number. For instance, \( \frac{1}{2} \) is equivalent to \( \frac{2}{4} \), \( \frac{3}{6} \), or \( \frac{4}{8} \). They are all equal because if you simplify any of these by dividing the numerator and the denominator by their greatest common factor, you'll get \( \frac{1}{2} \). The concept of equivalent fractions can also help us convert fractions. For example, if we need a fraction that is equivalent to \( 2 \) but with a denominator of 8, we express \( 2 \) as \( \frac{2}{1} \) and then multiply both the numerator and the denominator by 8, turning it into \( \frac{16}{8} \). Equivalent fractions are a fundamental concept that allows us to work flexibly with different kinds of fraction problems.

Multiplying Fractions

Multiplying fractions is a straightforward process and essential for solving more complex fractional problems. When you multiply fractions, you simply multiply the numerators together and the denominators together. Let's take the fractions \( \frac{2}{3} \) and \( \frac{4}{5} \) as an example. To multiply these, you'll multiply the numerators: \( 2 \times 4 = 8 \) and the denominators: \( 3 \times 5 = 15 \), resulting in \( \frac{8}{15} \). It’s important to note that sometimes you might have to simplify the resulting fraction if the numerator and the denominator have a common factor. But in this case, \( 8 \) and \( 15 \) have no common factors other than 1, so \( \frac{8}{15} \) is already in its simplest form.Multiplying fractions also comes in handy when altering fractions to have a different denominator, as demonstrated when converting \( \frac{2}{1} \) to \( \frac{16}{8} \). Multiplying both parts of a fraction by the same number is an application of this basic multiplication principle.