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Chapter 8: Problem 300

Ironing Lenore can do the ironing for her family's business in \(h\) hours. Her daughter would take \(h+2\) hours to get the ironing done. If Lenore and her daughter work together, using 2 irons, the number of hours it would take them to do all the ironing is \(\frac{1}{\frac{1}{h}+\frac{1}{h+2}}\) (a) Simplify the complex fraction \(\frac{1}{\frac{1}{h}+\frac{1}{h+2}}\). (b) Find the number of hours it would take Lenore and her daughter, working together, to get the ironing done if \(h=4\).

Short Answer

Expert verified
(a) \( \frac{h(h+2)}{2(h+1)} \). (b) 2.4 hours.

Step by step solution

01

- Set Up the Given Fraction

Start with the complex fraction provided: \ \ \ \ \ \( \frac{1}{\frac{1}{h}+\frac{1}{h+2}} \).
02

- Find a Common Denominator

Combine the fractions in the denominator by finding a common denominator: \ \ \( \frac{1}{h} + \frac{1}{h+2} = \frac{(h+2) + h}{h(h+2)} = \frac{2h+2}{h(h+2)} \).
03

- Rewrite the Complex Fraction

Rewrite the original complex fraction by dividing 1 by the combined fraction: \ \ \( \frac{1}{\frac{2h+2}{h(h+2)}} = \frac{h(h+2)}{2h+2} \).
04

- Simplify the Fraction

Simplify the fraction by canceling common factors: \ \ \( \frac{h(h+2)}{2(h+1)} \).
05

- Substitute Values for Part (b)

For part (b), substitute \(h = 4\) into the simplified fraction: \ \ \( \frac{4(4+2)}{2(4+1)} = \frac{24}{10} = 2.4 \) hours.

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Key Concepts

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

complex fractions
Complex fractions appear when a fraction includes one or more fractions in the numerator or the denominator. These can seem confusing at first but can be simplified step-by-step.

For the given exercise, the complex fraction is \(\frac{1}{\frac{1}{h}+\frac{1}{h+2}}\). Here are some tips to manage complex fractions:
  • First, deal with the fractions in the denominator by finding a common denominator.
  • Next, simplify the fraction in the denominator into a single fraction.
  • Finally, perform the necessary divisions to simplify the complex fraction into a simpler form.
simplifying fractions
Simplifying fractions means reducing fractions to their simplest form. This process makes fractions easier to work with, especially in complex computations.

Let's look at step-by-step simplification for our exercise:

1. **Find the Common Denominator:** \(\frac{1}{h} + \frac{1}{h+2}\).
Combine these fractions by finding a common denominator: \(\frac{h+2}{h(h+2)} + \frac{h}{h(h+2)} = \frac{(h+2) + h}{h(h+2)} = \frac{2h + 2}{h(h+2)}\).

2. **Simplify the Complex Fraction:**
Rewrite it as \(\frac{1}{\frac{2h+2}{h(h+2)}} = \frac{h(h+2)}{2h+2}\).

3. **Reduce by Canceling Common Factors:**
\

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Most popular questions from this chapter

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