Problem 119 - Elementary Algebra

Question

\(3\left(\frac{1}{5}-\frac{2}{3}\right)-4^{2}\)

Step-by-Step Solution

Answer

The final answer is \( \frac{-87}{5} \).

Step 1: Distribute inside the parenthesis

Inside the parenthesis, we have fractions that need to be subtracted: \( \frac{1}{5} - \frac{2}{3} \). To subtract these fractions, a common denominator is needed. The least common multiple of 5 and 3 is 15.

Step 2: Convert the fractions to have a common denominator

Convert \( \frac{1}{5} \) and \( \frac{2}{3} \) to fractions with a denominator of 15: \(\frac{1}{5} = \frac{3}{15}\)and\(\frac{2}{3} = \frac{10}{15} \)

Step 3: Subtract the fractions

Subtract the fractions: \(\frac{3}{15} - \frac{10}{15} = \frac{3-10}{15} = \frac{-7}{15} \)

Step 4: Multiply by 3

Multiply the result by 3: \(3 \left( \frac{-7}{15}\right) = \frac{-21}{15} \). Simplifying \( \frac{-21}{15} \), divides both numerator and denominator by 3, resulting in: \(\frac{-7}{5} \).

Step 5: Calculate the exponent

Calculate \(4^{2}\): \4^{2} = 16

Step 6: Final subtraction

Now subtract 16 from the result: \(\frac{-7}{5} - 16 = \frac{-7}{5} - \frac{80}{5} = \frac{-87}{5} \). Thus, the final answer is \( \frac{-87}{5} \).

Key Concepts

Fractions

Fractions represent parts of a whole. A fraction is written as \( \frac{a}{b} \) where \(a\) is the numerator (the part) and \(b\) is the denominator (the whole). To work with fractions, you might need to add, subtract, multiply, or divide them. Each operation follows specific rules.
In our problem, we have fractions \( \frac{1}{5} \) and \( \frac{2}{3} \). To subtract these fractions, we had to first find a common denominator. This is essential so the fractions can be compared or combined.
Understanding how to handle different denominators simplifies the process of working with fractions.

Common Denominator

To compare or subtract fractions, they must have the same denominator. A common denominator allows us to rewrite the fractions so they share the same base value (denominator), making operations possible.
In our case, we needed to subtract \( \frac{1}{5} - \frac{2}{3} \). The least common multiple (LCM) of 5 and 3 is 15. Thus, we converted \( \frac{1}{5} = \frac{3}{15} \) and \( \frac{2}{3} = \frac{10}{15} \). Now, they have a common denominator, which simplifies subtraction to: \( \frac{3}{15} - \frac{10}{15} = \frac{-7}{15} \).
This reinforces the necessity of finding a common denominator to facilitate the operations.

Exponents

An exponent indicates how many times a number (the base) is multiplied by itself. It's written as \( a^b \), where 'a' is the base and 'b' is the exponent.
For example, \( 4^2 \) means 4 multiplied by itself once, resulting in 16 (i.e., \( 4 \times 4 \)).
In our problem, we needed to calculate \( 4^2 = 16 \). Exponents simplify repeated multiplication and are vital in algebra for handling expressions concisely.

Step-By-Step Solution

Breaking down problems into clear, manageable steps simplifies complex algebraic operations.
Our original problem was \( 3 \left( \frac{1}{5} - \frac{2}{3} \right) - 4^2 \). Here’s a quick review of the steps involved:

• First, find a common denominator to subtract the fractions: \( \frac{3}{15} - \frac{10}{15} = \frac{-7}{15} \).
• Then, multiply the result by 3: \( 3 \left( \frac{-7}{15} \right) = \frac{-21}{15} \), which simplifies to \( \frac{-7}{5} \).
• Calculate the exponent: \( 4^2 = 16 \).
• Finally, subtract 16 from the fractional result: \( \frac{-7}{5} - 16 \) becomes \( \frac{-87}{5} \).

This step-by-step approach ensures clarity and accuracy, helping students better understand and solve algebraic expressions.