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Chapter 6: Problem 66

Simplify each complex fraction. Assume no division by 0. $$\frac{\frac{a}{b}}{\frac{a-1}{b}}$$

Short Answer

Expert verified
The simplified version of the complex fraction \( \frac{\frac{a}{b}}{\frac{a-1}{b}} \) is \( \frac{a}{a-1} \).

Step by step solution

01

Understand the Problem

Here, the goal is to simplify a complex fraction, which is a fraction where the numerator, the denominator, or both are fractions. Here, both the numerator and denominator themselves are fractions. The complex fraction is \( \frac{\frac{a}{b}}{\frac{a-1}{b}} \).
02

Reciprocal of the Denominator

To simplify a complex fraction, the main step is to multiply the top and bottom of the fraction by the reciprocal of the denominator. The reciprocal of a fraction is obtained by interchanging the numerator and denominator. Thus, the reciprocal of \( \frac{a-1}{b} \) is \( \frac{b}{a-1} \).
03

Multiply by Reciprocal

Now, multiply the complex fraction by the reciprocal of the denominator. This gives \( \frac{\frac{a}{b} * \frac{b}{a-1}}{1} = \frac{ab}{b(a-1)} \). Notice that the denominator became 1, which is a consequence of multiplying a fraction by its reciprocal.
04

Simplify the Fraction

In the fraction \( \frac{ab}{b(a-1)} \), 'b' in the numerator and the 'b' in the denominator cancel out each other and we get \( \frac{a}{a-1} \) as the simplified version of the complex fraction.

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Key Concepts

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

Simplifying Fractions
Simplifying fractions is a fundamental concept in mathematics, making complex expressions easier to work with. When you simplify a fraction, you reduce it to its simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

For example, if you come across a fraction like \( \frac{12}{16} \), you find that the GCD of 12 and 16 is 4. So, dividing both by 4 gives \( \frac{3}{4} \). This is the simplified form.

  • Simplification makes calculations easier.
  • It reveals the fraction in its most basic form.
  • Always check if further simplification is possible.
By simplifying fractions, we can quickly see relationships between numbers, which is quite useful in various math applications.
Reciprocal of Fractions
The reciprocal of a fraction is a simple yet crucial concept. It involves flipping the numerator and the denominator. So, if you have a fraction \( \frac{p}{q} \), its reciprocal would be \( \frac{q}{p} \).

Finding the reciprocal is especially important when dividing fractions, as you can't easily divide by a fraction but you can multiply by its reciprocal.

Here鈥檚 a quick guide:
  • To find a reciprocal, swap the numerator and denominator.
  • Reciprocals are useful in fraction multiplication and division.
  • A fraction multiplied by its reciprocal equals 1: \( \frac{p}{q} \times \frac{q}{p} = 1 \).
Understanding reciprocals makes working with complex fractions simpler and aids in both simplification and solving equations.
Fraction Multiplication
Multiplying fractions might sound intricate, but it's actually quite straightforward. To multiply fractions, you multiply the numerators together and the denominators together.

Consider two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \). The product would be \( \frac{a \cdot c}{b \cdot d} \).

Multiplication rules:
  • First, multiply the numerators to get the new numerator.
  • Next, multiply the denominators to get the new denominator.
  • Simplify the resulting fraction if needed.
This method is extremely useful in simplifying complex fractions and solving fraction problems efficiently. When you multiply by the reciprocal of a fraction, you effectively perform division, which simplifies the process.

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Most popular questions from this chapter

What numbers are equal to their own reciprocals?

Set up and solve a proportion. A recipe for chocolate chip cookies calls for \(1 \frac{1}{4}\) cups of flour and 1 cup of sugar. The recipe will make \(3 \frac{1}{2}\) dozen cookies. How many cups of flour will be needed to make 12 dozen cookies?

Set up and solve a proportion. A road ascends 750 feet in every 2,500 feet of travel. By how much will the road rise in a trip of 10 miles?

Find two different-looking but correct answers for the following problem. $$\text { Simplify: } \frac{y^{2}+5(2 y+5)}{25-y^{2}}$$

Perform the operations. Assume that no denominators are \(0 .\) $$\frac{x}{3 y}+\frac{2 x}{3 y}-\frac{x}{3 y}$$
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