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Chapter 3: Problem 65

$$ \frac{1}{3} \div\left(\frac{1}{2}-\frac{1}{5}\right) \times \frac{1}{4}+\frac{1}{6} $$

Short Answer

Expert verified
\( \frac{4}{9} \)

Step by step solution

01

Simplify Inside the Parentheses

Start by simplifying the expression inside the parentheses: \ \ \ \ \ \ \( \frac{1}{2} - \frac{1}{5} \). To subtract these fractions, find a common denominator: \ \( \frac{1}{2} = \frac{5}{10} \) and \( \frac{1}{5} = \frac{2}{10} \). \ So, \( \frac{1}{2} - \frac{1}{5} = \frac{5}{10} - \frac{2}{10} = \frac{3}{10} \).
02

Divide Fractions

Now, divide \( \frac{1}{3} \) by \( \frac{3}{10} \): \ Divide fractions by multiplying the first fraction by the reciprocal of the second fraction: \ \( \frac{1}{3} \times \frac{10}{3} = \frac{10}{9} \).
03

Multiply Fractions

Next, multiply \( \frac{10}{9} \) by \( \frac{1}{4} \): \ \( \frac{10}{9} \times \frac{1}{4} = \frac{10 \times 1}{9 \times 4} = \frac{10}{36} \). Simplify this to \( \frac{5}{18} \).
04

Add Fractions

Lastly, add \( \frac{5}{18} \) and \( \frac{1}{6} \). Convert \( \frac{1}{6} \) to have a common denominator with \( \frac{5}{18} \): \ \( \frac{1}{6} = \frac{3}{18} \). \ Now add: \( \frac{5}{18} + \frac{3}{18} = \frac{8}{18} = \frac{4}{9} \).

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Key Concepts

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

Subtracting Fractions
Subtracting fractions might seem tricky, but it's really all about finding a common denominator. Here's what you need to do:
  • Make sure the fractions you are working with have the same denominator. This means you may need to find equivalent fractions.
  • For example, to subtract \(\frac{1}{2}\) and \(\frac{1}{5}\), convert them to equivalent fractions with the same denominator. The least common multiple (LCM) of 2 and 5 is 10.
  • So, \(\frac{1}{2} = \frac{5}{10}\), and \(\frac{1}{5} = \frac{2}{10}\).
  • Now, simply subtract the numerators: \(\frac{5}{10} - \frac{2}{10} = \frac{3}{10}\).
This method lets you subtract fractions without any hassle. Remember, getting the same denominator is key to simplifying the operation.
Dividing Fractions
When you need to divide fractions, think about multiplying by the reciprocal. Here's how you can do it:
  • Take the first fraction and multiply by the reciprocal of the second fraction.
  • For example, to divide \(\frac{1}{3}\) by \(\frac{3}{10}\), first find the reciprocal of \(\frac{3}{10}\), which is \(\frac{10}{3}\).
  • Now, multiply the first fraction by this reciprocal: \(\frac{1}{3} \times \frac{10}{3} = \frac{10}{9}\).
This method of turning division into multiplication makes things much simpler.
Multiplying Fractions
Multiplying fractions is straightforward: multiply the numerators together and the denominators together. Here's an example:
  • To multiply \(\frac{10}{9}\) by \(\frac{1}{4}\), just multiply the tops (numerators) and the bottoms (denominators).
  • You get \(\frac{10 \times 1}{9 \times 4} = \frac{10}{36}\).
  • Always simplify the fraction if you can. Here, \(\frac{10}{36}\) can be simplified to \(\frac{5}{18}\).
Simplifying fractions at the end ensures your answer is in its simplest form.
Adding Fractions
Adding fractions, similar to subtracting, requires a common denominator. Follow these steps:
  • Convert the fractions to have the same denominator.
  • For example, to add \(\frac{5}{18}\) and \(\frac{1}{6}\), convert \(\frac{1}{6}\) to \(\frac{3}{18}\).
  • Now, add the numerators: \(\frac{5}{18} + \frac{3}{18} = \frac{8}{18}\).
  • Simplify the fraction if possible. Here, \(\frac{8}{18} = \frac{4}{9}\).
Adding fractions is easy once the denominators are aligned.

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Most popular questions from this chapter

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

A recipe for strawberry punch calls for \(\frac{1}{5}\) qt of ginger ale and \(\frac{3}{5}\) qt of strawberry soda. How much liquid is needed? If the recipe is doubled, how much liquid is needed? If the recipe is halved, how much liquid is needed?

Simplify. $$ 10 \frac{3}{5}-4 \frac{1}{10}-1 \frac{1}{2} $$

Simplify. $$ \left(\frac{2}{3}+\frac{3}{4}\right) \div\left(\frac{5}{6}-\frac{1}{3}\right) $$

Simplify. Find the prime factorization of \(150 .[2.1 \mathrm{~d}]\)
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