Problem 45 - Prealgebra

Question

Simplify each expression. $$\left(\frac{7}{15}-\frac{11}{30}\right)^{2}$$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{1}{100} \).

Step 1: Find a Common Denominator

The fractions \( \frac{7}{15} \) and \( \frac{11}{30} \) need a common denominator to be subtracted. The least common multiple of 15 and 30 is 30. So, the common denominator is 30.

Step 2: Rewrite the First Fraction

Convert \( \frac{7}{15} \) to an equivalent fraction with a denominator of 30. Multiply both the numerator and the denominator by 2 to get \( \frac{14}{30} \).

Step 3: Subtract the Fractions

Now, subtract \( \frac{11}{30} \) from \( \frac{14}{30} \):\[\frac{14}{30} - \frac{11}{30} = \frac{3}{30}\]After that, simplify \( \frac{3}{30} \) to \( \frac{1}{10} \) by dividing both the numerator and the denominator by 3.

Step 4: Square the Result

Now square the simplified result \( \frac{1}{10} \).\[\left(\frac{1}{10}\right)^2 = \frac{1^2}{10^2} = \frac{1}{100}\]The expression is simplified to \( \frac{1}{100} \).

Key Concepts

Common Denominator

When dealing with fractions, a common denominator is essential for operations like subtraction or addition. This means finding a shared bottom number (denominator) for both fractions involved in the operation.
To achieve this, you search for the least common multiple (LCM) of the denominators of the given fractions. In our exercise, we needed a common denominator to subtract \( \frac{7}{15} \) from \( \frac{11}{30} \). Here, the denominators are 15 and 30.

• Calculate the least common multiple (LCM) of 15 and 30, which is 30.
• This will serve as the common denominator.

Now that both fractions have the same denominator, you can carry out subtraction directly. Rewriting the first fraction, \( \frac{7}{15} \), becomes \( \frac{14}{30} \), and now subtraction using a common denominator is straightforward.

Squaring Fractions

Squaring fractions can sound complex, but it’s pretty simple once you break it down. When you square a fraction, you are essentially multiplying the fraction by itself.
In our problem, the fraction \( \frac{1}{10} \) needs to be squared. Here's how you do it:

• Square the numerator: \( 1^2 = 1 \).
• Square the denominator: \( 10^2 = 100 \).

After squaring both the numerator and the denominator, you get the fraction \( \frac{1}{100} \). This means \( \left( \frac{1}{10} \right)^2 \) simplifies neatly to \( \frac{1}{100} \). Squaring fractions involves straightforward multiplication but demands attention to each part of the fraction.

Equivalent Fractions

Equivalent fractions are fractions that may look different but represent the same value or proportion. To transform a fraction into an equivalent one, you multiply or divide both the numerator and the denominator by the same number.
For instance, in our step-by-step solution, the fraction \( \frac{7}{15} \) transformed into \( \frac{14}{30} \) to match the common denominator. Here's how it works:

• Multiply both the numerator and denominator of \( \frac{7}{15} \) by 2 to get \( \frac{14}{30} \).

This multiplication doesn’t change the value of the fraction, just how it appears. Understanding equivalent fractions lets you easily compare, add, or subtract fractions without changing their actual value. This is especially useful when dealing with operations involving multiple fractions with different denominators.