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Chapter 5: Problem 44

Divide. Simplify, if possible. $$ \frac{50}{9} \div \frac{6}{5} $$

Short Answer

Expert verified
\( \frac{125}{27} \)

Step by step solution

01

Write down the problem

You need to divide the fraction \(\frac{50}{9}\) by \(\frac{6}{5}\).
02

Convert division to multiplication

To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{6}{5}\) is \(\frac{5}{6}\). Therefore, \(\frac{50}{9} \div \frac{6}{5}\) becomes \(\frac{50}{9} \times \frac{5}{6}\).
03

Multiply the fractions

Multiply the numerators and the denominators: \(\frac{50}{9} \times \frac{5}{6} = \frac{50 \times 5}{9 \times 6} = \frac{250}{54}\).
04

Simplify the fraction

Simplify \(\frac{250}{54}\). First, find the greatest common divisor (GCD) of 250 and 54, which is 2. Then, divide both the numerator and the denominator by 2: \(\frac{250 \div 2}{54 \div 2} = \frac{125}{27}\).

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Key Concepts

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

Reciprocals
Understanding reciprocals is key for fraction division.
A reciprocal is simply a fraction flipped upside down.
For example, the reciprocal of \(\frac{6}{5}\) is \(\frac{5}{6}\).
When you divide by a fraction, you essentially multiply by its reciprocal.
This handy trick simplifies the process, making it easier to handle complex fractions.
Always remember: The reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).
This concept is crucial because it transforms division problems into multiplication problems, which are generally more straightforward.
Fraction Multiplication
Once you've converted a division problem into a multiplication problem, the next step is to multiply the fractions.
To do this, follow these simple steps:
  • Multiply the numerators (top numbers) together.
  • Multiply the denominators (bottom numbers) together.

For example, in the problem \(\frac{50}{9} \times \frac{5}{6}\), you multiply the numerators: 50 \times 5 = 250, and the denominators: 9 \times 6 = 54.
This gives you the product \(\frac{250}{54}\).
Multiplying fractions is much simpler than it seems at first glance, as long as you take it step-by-step.
Remember to always multiply straight across, meaning you don't need to cross-multiply or find common denominators.
Simplifying Fractions
The final step in most fraction problems is to simplify your answer.
Simplifying a fraction means reducing it to its smallest form.
To do this, find the greatest common divisor (GCD) of the numerator and the denominator.
Divide both the top and bottom by this number.
  • For \(\frac{250}{54}\), the GCD is 2.
  • Divide both 250 and 54 by 2, resulting in \(\frac{125}{27}\).

Always check if the fraction can be simplified further.
Simplifying fractions ensures your final answer is in the cleanest, most understandable format.
This step is essential for accurate and clear mathematics work.

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Most popular questions from this chapter

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Hockey Goals. An official hockey goal is \(6 \mathrm{ft}\) wide. To make scoring more difficult, goalies often position themselves far in front of the goal to "cut down the angle." In Exercises 41 and 42 , suppose that a slapshot is attempted from point \(A\) and that the goalie is \(2.7 \mathrm{ft}\) wide. Determine how far from the goal the goalie should be located if point \(A\) is the given distance from the goal. (Hint: First find how far the goalie should be from point \(A\).) $$ 25 \mathrm{ft} $$

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