Problem 43 - Basic College Mathematics

Question

Simplify. $$ \left(\frac{3}{5}\right)^{2}-\frac{1}{10} \cdot 2 \frac{1}{2} $$

Step-by-Step Solution

Answer

\( \frac{11}{100} \)

Step 1 - Simplify the Power

Evaluate \(\bigg(\frac{3}{5}\bigg)^2\). To do this, square both the numerator and the denominator: \[ \bigg(\frac{3}{5}\bigg)^2 = \frac{3^2}{5^2} = \frac{9}{25} \]

Step 2 - Convert Mixed Number to Improper Fraction

Convert the mixed number \(2 \frac{1}{2}\) to an improper fraction. First, multiply the whole number by the denominator and add the numerator: \[ 2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{5}{2} \]

Step 3 - Multiply the Fractions

Multiply \(\frac{1}{10}\) by \(\frac{5}{2}\). To do this, multiply the numerators and the denominators: \[ \frac{1}{10} \times \frac{5}{2} = \frac{1 \times 5}{10 \times 2} = \frac{5}{20} = \frac{1}{4} \]

Step 4 - Subtract the Fractions

Subtract \( \frac{1}{4} \) from \( \frac{9}{25} \). To do this, find a common denominator: \[ \frac{9}{25} - \frac{1}{4} = \frac{9 \times 4}{25 \times 4} - \frac{1 \times 25}{4 \times 25} = \frac{36}{100} - \frac{25}{100} = \frac{11}{100} \]

Key Concepts

Exponentiation

Exponentiation is a fundamental mathematical operation that involves raising a number to a power. In the given exercise, we start by evaluating the expression \( \bigg(\frac{3}{5}\bigg)^2 \). This means we need to square both the numerator and the denominator. To do this, we calculate \( 3^2 = 9 \) and \( 5^2 = 25 \). Therefore, \( \bigg(\frac{3}{5}\bigg)^2 = \frac{9}{25} \). This simplified form helps make subsequent operations, like subtraction, easier to manage.

Mixed Numbers

Mixed numbers consist of a whole number and a fraction. They are more intuitive but can be complex for arithmetic operations. In our exercise, we encounter the mixed number \( 2 \frac{1}{2} \). To simplify its use, we convert it to an improper fraction. We do this by multiplying the whole number part (2) by the denominator (2) and adding the numerator (1), resulting in \( \frac{5}{2} \). This conversion is crucial for multiplication or division involving mixed numbers.

Improper Fractions

Improper fractions have numerators larger than or equal to their denominators. They are useful for computation since they eliminate the mixed number's complexity. For instance, in our problem, the mixed number \( 2 \frac{1}{2} \) becomes the improper fraction \( \frac{5}{2} \). This step ensures a seamless multiplication process with other fractions to maintain accuracy.

Fraction Multiplication

Fraction multiplication is straightforward. To multiply fractions, multiply the numerators directly and the denominators directly. For example, \( \frac{1}{10} \cdot \frac{5}{2} = \frac{1 \times 5}{10 \times 2} = \frac{5}{20} \). However, we always need to simplify our final answer. Here, \( \frac{5}{20} = \frac{1}{4} \), as both the numerator and denominator can be divided by 5. This simplification step is essential in keeping our results tidy and easier to understand.

Fraction Subtraction

Subtracting fractions involves a common denominator. In our exercise, we need to subtract \( \frac{1}{4} \) from \( \frac{9}{25} \). The trick is to find a common denominator. In this case, the least common multiple (LCM) of 25 and 4 is 100. This converts our fractions to \( \frac{36}{100} \) and \( \frac{25}{100} \). Subtracting these, we have \( \frac{36}{100} - \frac{25}{100} = \frac{11}{100} \). This method ensures accurate and straightforward subtraction.