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Chapter 6: Problem 14

Solve. If no solution exists, state this. $$ \frac{1}{6}+\frac{1}{8}=\frac{1}{t} $$

Short Answer

Expert verified
The solution is \( t = \frac{24}{7} \).

Step by step solution

01

Find a common denominator

To add \( \frac{1}{6} \) and \( \frac{1}{8} \), find the least common multiple (LCM) of 6 and 8, which is 24.
02

Rewrite fractions with the common denominator

Convert \( \frac{1}{6} \) and \( \frac{1}{8} \) so they have the common denominator 24: \( \frac{1}{6} = \frac{4}{24} \) and \( \frac{1}{8} = \frac{3}{24} \).
03

Add the fractions

Now add the fractions: \( \frac{4}{24} + \frac{3}{24} = \frac{7}{24} \).
04

Set the sum equal to the given equation

Set \( \frac{7}{24} \) equal to \( \frac{1}{t} \): \( \frac{7}{24} = \frac{1}{t} \).
05

Solve for t

To solve for \( t \), take the reciprocal of both sides: \( t = \frac{24}{7} \).

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Key Concepts

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

Common Denominator
When you need to add or subtract fractions, finding a common denominator is crucial. The common denominator is a shared multiple of the denominators of the fractions you're working with.

For example, in the problem involving \ \( \frac{1}{6} \) and \ \( \frac{1}{8} \), we need to find a number that both 6 and 8 can divide evenly into.

This number helps us rewrite the fractions so they have a common base, making addition or subtraction possible without changing their values.
Least Common Multiple
To find a common denominator efficiently, we use the least common multiple (LCM). LCM is the smallest number that is a multiple of two or more numbers.

For 6 and 8, the LCM can be found through several methods, such as prime factorization or listing multiples. The multiples of 6 are 6, 12, 18, 24, etc. The multiples of 8 are 8, 16, 24, etc. The smallest number they both share is 24.

Therefore, the LCM of 6 and 8 is 24, which we use as our common denominator to add the fractions \ \( \frac{1}{6} \) and \ \( \frac{1}{8} \).
Reciprocal
A reciprocal of a number is simply flipping its numerator and denominator. For example, the reciprocal of \ \( \frac{2}{3} \) is \ \( \frac{3}{2} \).

In solving the equation \ \( \frac{7}{24} = \frac{1}{t} \), finding the value of \ t \ involves taking the reciprocal of both sides. Hence, \ t \ becomes \ \( \frac{24}{7} \).

This step is often used in algebra to isolate variables and simplify equations, making it a key concept in solving rational equations.

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