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Chapter 4: Problem 44

Simplify the following. $$\frac{3-\frac{1}{2}}{4+\frac{1}{5}}$$

Short Answer

Expert verified
The simplified result is \(\frac{25}{42}\).

Step by step solution

01

Subtract Fractions in the Numerator

Start by simplifying the numerator, which is the expression \(3 - \frac{1}{2}\). To subtract these, convert \(3\) into an equivalent fraction with a denominator of 2, i.e., \(\frac{6}{2}\). Now, subtract: \(\frac{6}{2} - \frac{1}{2} = \frac{5}{2}\). So, the simplified numerator is \(\frac{5}{2}\).
02

Add Fractions in the Denominator

Next, simplify the denominator, \(4 + \frac{1}{5}\). Convert \(4\) into an equivalent fraction with a denominator of 5, i.e., \(\frac{20}{5}\). Now, add the fractions: \(\frac{20}{5} + \frac{1}{5} = \frac{21}{5}\). Therefore, the simplified denominator is \(\frac{21}{5}\).
03

Divide the Fractions

Now, we have \(\frac{\frac{5}{2}}{\frac{21}{5}}\). Dividing fractions involves multiplying by the reciprocal of the denominator. So, multiply \(\frac{5}{2}\) by the reciprocal of \(\frac{21}{5}\), which is \(\frac{5}{21}\). Thus, we compute: \(\frac{5}{2} \times \frac{5}{21} = \frac{25}{42}\).
04

Simplified Result

The fraction \(\frac{25}{42}\) cannot be simplified further as 25 and 42 have no common factors other than 1. Therefore, the simplified form is \(\frac{25}{42}\).

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Key Concepts

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

Subtracting Fractions
To subtract fractions, it's important to ensure they have a common denominator. This means the bottom number of each fraction should be the same. In our example, the operation is \(3 - \frac{1}{2}\). Since 3 can be rewritten as a fraction with a denominator of 2, we convert it to \(\frac{6}{2}\). Now both fractions have the same denominator, allowing us to subtract their numerators easily: \(\frac{6}{2} - \frac{1}{2} = \frac{5}{2}\).
  • Convert whole numbers to fractions with the same denominator as the other fraction involved.
  • Keep the denominator the same and subtract the numerators.
By following these steps, you can subtract fractions with ease.
Adding Fractions
Adding fractions is similar to subtracting them; you also need a common denominator. In our problem, the operation is \(4 + \frac{1}{5}\). Here, 4 is rewritten as \(\frac{20}{5}\), harmonizing the denominators. You then add the numerators: \(\frac{20}{5} + \frac{1}{5} = \frac{21}{5}\).
  • Ensure both fractions have the same denominator before proceeding.
  • Add the numerators while maintaining the common denominator.
Remember, fractions require uniformity in denominators to be combined correctly.
Dividing Fractions
Dividing fractions involves a neat trick: multiplying by the reciprocal. Given \(\frac{\frac{5}{2}}{\frac{21}{5}}\), dividing turns into using the reciprocal of \(\frac{21}{5}\), which is \(\frac{5}{21}\). Multiply \(\frac{5}{2}\) by this reciprocal: \(\frac{5}{2} \times \frac{5}{21} = \frac{25}{42}\).
  • Change the division into multiplication by using the reciprocal.
  • Multiply the numerators together and the denominators together.
This method gracefully turns division into a simpler multiplication task.
Reciprocal of a Fraction
The reciprocal of a fraction is simply flipping its numerator and denominator. For any fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). In the exercise solution, the reciprocal of \(\frac{21}{5}\) is used to divide fractions effectively.
  • Swap the top and bottom numbers of the fraction.
  • Use the reciprocal to transform division into multiplication.
Understanding reciprocals is key to handling division in fraction operations successfully.

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Most popular questions from this chapter

Solve. In your own words, describe how to divide mixed numbers.

The perimeter of the square below is \(23 \frac{1}{2}\) feet. Find the length of each side.

Solve. True or false? When the fraction \(\frac{7}{12}\) is rewritten with a denominator of \(48,\) the result is \(\frac{11}{48} .\) If false, give the correct fraction.

Recall that to find the average of two numbers, we find their sum and divide by \(2 .\) For example, the average of \(\frac{1}{2}\) and \(\frac{3}{4}\) is \(\frac{\frac{1}{2}+\frac{3}{4}}{2}\). Find the average of each pair of numbers. Two positive numbers, \(a\) and \(b\), are graphed below. Where should the graph of their average lie?

Perform the indicated operations. $$ -4 \frac{2}{5} \cdot 2 \frac{3}{10} $$
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