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Chapter 3: Problem 52

If \(a+b=6\) and \(a b=8\) what is \(4 / a+4 / b\) ? (A) 6 (B) 8 (C) 4 (D) 2 (E) 3

Short Answer

Expert verified
The value is 3.

Step by step solution

01

Express the target expression in terms of a and b

We need to find \(\frac{4}{a} + \frac{4}{b}\). Notice that \(\frac{4}{a} + \frac{4}{b} \) can be written as \(\frac{4b + 4a}{ab} \).
02

Substitute values

From the problem statement, we have \( a + b = 6 \) and \( ab = 8 \). Substitute these into the expression: \(\frac{4a + 4b}{ab} = \frac{4(a+b)}{ab} \).
03

Simplify the expression

Substitute \( a + b = 6 \) and \( ab = 8 \) into the expression: \(\frac{4(a+b)}{ab} = \frac{4 \times 6}{8} = \frac{24}{8} = 3 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

algebra
Algebra helps us solve equations by finding unknown values, such as variables like 'a' and 'b'. In this problem, we have two algebraic expressions: 1) \(a + b = 6\) and 2) \(ab = 8\). Knowing these, we can express another algebraic form to solve the problem by finding \(\frac{4}{a} + \frac{4}{b}\). Understanding algebra allows us to manipulate and solve equations effectively. Focus on the given relationships and substitute the known values to simplify the expression.
problem-solving
Problem-solving in math involves breaking down the question and tackling it step by step. When approaching this exercise:
  • Identify the given information: \( a + b = 6 \) and \( ab = 8 \)
  • Translate the math problem into a more workable form: Convert \( \frac{4}{a} + \frac{4}{b} \) into \( \frac{4(a+b)}{ab} \).
  • Substitute known values: Replace \( a + b \) and \( ab \) with their given values.
  • Simplify: Perform arithmetic to find the final solution.
By systematically addressing each step, we ensure a clear path to the correct answer.
fractions
Understanding fractions is crucial in solving this problem. A fraction represents a part of a whole, shown by a numerator (top number) and a denominator (bottom number). In this problem, recognizing that \( \frac{4}{a} + \frac{4}{b} \) can be combined into a single fraction \( \frac{4(a+b)}{ab} \) makes the math simpler. This lets us collectively manipulate terms and reduce fractions easily. Remember, simplifying fractions often involves combining numerators and denominators with common factors, as we did by substituting and calculating \( \frac{24}{8} \) to get 3.
math shortcuts
Math shortcuts can save time and simplify complex problems. In this case, a useful shortcut involves quickly recognizing how to combine fractions: converting \( \frac{4}{a} + \frac{4}{b} \) into \( \frac{4(a+b)}{ab} \). This avoids unnecessary steps and makes use of given equations efficiently. Another shortcut used is substituting values early on: knowing \( a + b = 6 \) and \( ab = 8 \) right into the simplified form helps reach the answer faster. Always look for opportunities to replace complicated expressions with simpler equivalents.

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