Problem 28 - Algebra 1

Question

Find the difference. $$ \frac{4}{3}-\frac{7}{3} $$

Step-by-Step Solution

Answer

The difference between \( \frac{4}{3} \) and \( \frac{7}{3} \) is -1.

Step 1: Identifying denominators

The first step in subtracting fractions is to check if the denominators, the bottom numbers of the fractions, are the same. In this case, both denominators are 3, so, the fractions can be subtracted directly

Step 2: Subtracting the fractions

The numerators, the top numbers of the fractions, can be subtracted while keeping the denominator constant, because our denominators are identical in this case. So, \( \frac{4}{3}-\frac{7}{3} = \frac{4-7}{3} = \frac{-3}{3} \)

Step 3: Simplifying the result

Dividing -3 by 3 gives -1. Therefore, the simplification of the fraction \( \frac{-3}{3} \) results in -1

Key Concepts

Fraction Simplification

Fraction simplification is all about making a fraction as straightforward as possible. In our original example, \( \frac{-3}{3} \), simplification means reducing this fraction to its simplest form. Since the numerator (-3) and the denominator (3) are divisible by the same number, we divide both by this common factor, which is 3.
Performing the division:

• -3 divided by 3 results in -1 for the numerator.
• 3 divided by 3 results in 1 for the denominator.

Thus, \( \frac{-3}{3} \) simplifies to \(-1\). Simplifying fractions helps in recognizing equivalent fractions and efficiently handling them in mathematical calculations.

Numerators and Denominators

Understanding what numerators and denominators are is vital in dealing with fractions. They are the building blocks:

• Numerator: This is the top number in a fraction. It signifies how many parts of the whole are being considered. In the example, 4 and 7 are the numerators.
• Denominator: This is the bottom number in a fraction. It tells how many equal parts the whole is divided into. Here, the denominator is 3 for both fractions.

When subtracting fractions like \( \frac{4}{3} - \frac{7}{3} \), if denominators are the same, you keep the denominator constant and subtract the numerators directly. Understanding these components ensures clarity in fractions' operations. This foundational knowledge is crucial for arithmetic involving different fractions.

Fraction Arithmetic

Fraction arithmetic refers to performing basic math operations such as addition, subtraction, multiplication, and division with fractions. Here's a focus on subtraction:
When subtracting fractions with the same denominators, like in \( \frac{4}{3} - \frac{7}{3} \), you directly subtract the numerators:

• Keep the denominator the same, which ensures the fractions remain on a common scale.
• Subtract the first fraction's numerator from the second fraction's numerator, giving \(-3\) in this case.
• Combine the results to form a new fraction \( \frac{-3}{3} \).

The key steps involve aligning the fractions to the same base, if they are not, and then concentrating on the numerators. Mastery of these operations offers confidence in handling more advanced math problems involving fractions.