Log out Go to App
Go to App

Learning Materials

Features

Discover

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: \(\frac{4(a+b)}{ab} = \frac{4 \times 6}{8} = \frac{24}{8} = 3 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

He was totally engrossed in his work when the telephone started to ring. A. when B. while C. hardly Answer D. because E. as

Which of these two is the greater, \(x\) or \(y\) ? (1) \(1 / x>1 / y\) (2) Both are negative A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

Is \(x\) negative? (1) \(x y=3\) (2) \(x^2=y\) A. 1 alone, not 2 alone B. 2 alone, not 1 alone C. 1 and 2 together (need both) D. 1 alone or 2 alone E. 1 and 2 together are not sufficient

If \(y=1 / x+x\) and \(x=4\) then \(y\) is: (A) \(17 / 4\) (B) \(4 / 5\) (C) \(5 / 4\) (D) \(17 / 8\) (E) None of the above

I have know her for so long because we have live next door to each other for 12 years. A. know her for so long because we have live B. know her for so long because we have lived C. known her for so long because we have living D. know her for so long because we have been living E. known her for so long because we have lived
See all solutions

Recommended explanations on English Textbooks

Multiple Choice Questions

Read Explanation

English Grammar Summary

Read Explanation

Sociolinguistics

Read Explanation

Synthesis Essay

Read Explanation

Phonetics

Read Explanation

Research and Composition

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free