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Chapter 7: Problem 12

Simplify each complex fraction. Use either method. $$ \frac{\frac{q-5}{q}}{\frac{q+5}{3}} $$

Short Answer

Expert verified
\[ \frac{3(q-5)}{q(q+5)} \]

Step by step solution

01

- Rewrite the Complex Fraction

Rewrite the complex fraction as a division of two fractions: \[ \frac{\frac{q-5}{q}}{\frac{q+5}{3}} = \frac{q-5}{q} \times \frac{3}{q+5} \]
02

- Simplify the Multiplication

Multiply the numerators together and the denominators together: \[ \frac{q-5}{q} \times \frac{3}{q+5} = \frac{(q-5) \times 3}{q \times (q+5)} \]
03

- Simplify the Resulting Fraction

Simplify the resulting fraction, if possible. The fraction \[ \frac{3(q-5)}{q(q+5)} \] cannot be simplified further because there are no common factors between the numerator and the denominator.

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Key Concepts

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

complex fractions
A complex fraction is a fraction where the numerator, the denominator, or both contain fractions themselves. Essentially, it is a fraction with one or more fractions in it. Take, for example, the problem: \[ \frac{\frac{q-5}{q}}{\frac{q+5}{3}} \]. Here, both the numerator and the denominator are fractions. The aim is to simplify such complex fractions into a simpler form. This process often involves rewriting the complex fraction as a division problem, which makes it more manageable to work with. To start with complex fractions:
  • Identify the fractions in the numerator and the denominator.
  • Convert the problem into a division of two fractions.
  • Simplify the resulting division.
Simplifying complex fractions allows mathematicians and students to make equations more readable and easier to solve.
multiplication of fractions
Understanding how to multiply fractions is key to simplifying complex fractions. When you rewrite a complex fraction as a division problem, you'll often change it to a multiplication problem. For example, \[ \frac{\frac{q-5}{q}}{\frac{q+5}{3}} \] can be rewritten as \[ \frac{q-5}{q} \times \frac{3}{q+5} \]. Multiplying fractions is simple if you follow these steps:
  • Multiply the numerators (top numbers) together.
  • Multiply the denominators (bottom numbers) together.
For example, multiply \( (q-5) \times 3 \) for the numerator and \( q \times (q+5) \) for the denominator to get \[ \frac{(q-5) \times 3}{q \times (q+5)} \]. Remember, after you multiply the fractions, you'll need to simplify the resulting fraction if possible.
simplifying fractions
Simplifying fractions is the final step in handling both simple and complex fractions. To simplify a fraction, you divide the numerator and the denominator by their greatest common factor (GCF). In some cases, like our example \[ \frac{3(q-5)}{q(q+5)} \], no further simplification is possible because there are no common factors. However, when common factors exist:
  • Identify any common factors between the numerator and the denominator.
  • Divide both the numerator and the denominator by their GCF.
Simplifying makes the fraction easier to understand and use in further calculations. Always check to ensure that the final fraction is expressed in its simplest form, as this helps in achieving accurate results in mathematical problems.

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Most popular questions from this chapter

Heather Schaefer, a high school mathematics teacher, gave a test on perimeter, area, and volume to her geometry classes. Working alone, it would take her 4 hr to grade the tests. Her student teacher, Courtney Slade, would take 6 hr to grade the same tests. How long would it take them to grade these tests if they work together?

Solve each equation for \(k\) $$ 92=\frac{k}{2} $$

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{2 m^{2}-5 m-12}{m^{2}-10 m+24} \cdot \frac{m^{2}-9 m+18}{4 m^{2}-9}$$

Multiply or divide. Write each answer in lowest terms. See Examples \(3,6,\) and 7 . $$\frac{6 s^{2}+17 s+10}{s^{2}-4} \cdot \frac{s^{2}-2 s}{6 s^{2}+29 s+20}$$

Solve each variation problem. For a constant area, the length of a rec tangle varies inversely as the width. The length of a rectangle is \(27 \mathrm{ft}\) when the width is \(10 \mathrm{ft}\). Find the width of a rectangle with the same area if the length is \(18 \mathrm{ft}\).
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