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Chapter 7: Problem 67

Perform each indicated operation. See Section 1.3. $$ \frac{13}{20} \div \frac{2}{9} $$

Short Answer

Expert verified
\( \frac{117}{40} \) or as a mixed number, \( 2\frac{37}{40} \).

Step by step solution

01

Understand Division of Fractions

When you divide by a fraction, it is equivalent to multiplying by its reciprocal. The reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). Therefore, dividing by a fraction is the same as multiplying by the inverse of that fraction.
02

Write the Multiplication Expression

Convert the division operation to a multiplication operation by writing the reciprocal of \( \frac{2}{9} \). The expression becomes: \[ \frac{13}{20} \times \frac{9}{2} \]
03

Multiply the Fractions

Multiply the numerators and the denominators: \[ \frac{13 \times 9}{20 \times 2} \] This results in: \[ \frac{117}{40} \]
04

Simplify the Fraction

Check if the fraction can be simplified. Since 117 and 40 have no common factors other than 1, \( \frac{117}{40} \) is already in its simplest form. You can also express it as a mixed number: \[ 2\frac{37}{40} \] by performing the division 117 梅 40.

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Key Concepts

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

Reciprocal of a Fraction
The reciprocal of a fraction is essentially flipping the fraction. It involves swapping the numerator and the denominator with each other. For any fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). Reciprocals are particularly handy when it comes to dividing fractions because division by a fraction is the same as multiplication by its reciprocal.
  • Example: The reciprocal of \( \frac{2}{9} \) is \( \frac{9}{2} \).
Remember, to find a reciprocal, simply switch the top and bottom numbers, but ensure the original fraction is non-zero to avoid division by zero issues.
Multiplying Fractions
Multiplying fractions might seem tricky, but it's straightforward. Just multiply the numerators together and the denominators together. In simplest terms, align the top numbers and the bottom numbers, then multiply straight across.
  • Example: \( \frac{13}{20} \times \frac{9}{2} = \frac{13 \times 9}{20 \times 2} = \frac{117}{40} \).
There's no need to find a common denominator when multiplying, which is different from adding or subtracting fractions.
Simplifying Fractions
Simplifying fractions involves reducing a fraction to its simplest form, where the numerator and denominator have no common factors other than 1. To simplify, find the greatest common factor (GCF) of both numbers and divide them by it.
  • Example: With \( \frac{117}{40} \), the GCF of 117 and 40 is 1, so the fraction is already in simplest form.
If simplifying manually is tricky, prime factorization or using a calculator can help confirm that a fraction is already simplified.
Mixed Numbers
A mixed number combines a whole number and a fraction and is often used to make improper fractions easier to understand. You can convert an improper fraction to a mixed number by dividing the numerator by the denominator.
  • Example: For \( \frac{117}{40} \), dividing 117 by 40 gives 2 with a remainder of 37, so it becomes \( 2\frac{37}{40} \).
Mixed numbers provide a clearer picture of how much more there is beyond the complete wholes.

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Most popular questions from this chapter

Solve. See the Concept Check in the Section. Which of the following are equivalent to \(\frac{\frac{a}{7}}{\frac{b}{13}} ?\) a. \(\frac{a}{7} \cdot \frac{b}{13}\) b. \(\frac{a}{7} \div \frac{b}{13}\) c. \(\frac{a}{7} \div \frac{13}{b}\) d. \(\frac{a}{7} \cdot \frac{13}{b}\)

Perform each indicated operation. The length of a rectangle is \(\frac{3}{y-5}\) feet, while its width is \(\frac{2}{y}\) feet. Find its perimeter and then find its area. IMAGE CANNOT COPY!

For a certain model fax machine, the manufacturing cost \(C(x)\) per machine is given by the function $$ C(x)=\frac{250 x+10,000}{x} $$ where \(x\) is the number of fax machines manufactured and \(\operatorname{cost} C(x)\) is in dollars per machine. a. Find the cost per fax machine when manufacturing 100 fax machines. b. Find the cost per fax machine when manufacturing 1000 fax machines. c. Does the cost per machine decrease or increase when more machines are manufactured? Explain why this is so.

Perform each indicated operation. A board of length \(\frac{3}{x+4}\) inches was cut into two pieces. If one piece is \(\frac{1}{x-4}\) inches, express the length of the other piece as a rational expression. IMAGE CANNOT COPY!

Perform the indicated operations. Addition, subtraction, multiplication, and division of rational expressions are included here. $$ \frac{8 x+7}{3 x+5}-\frac{2 x-3}{3 x+5} $$
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