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Chapter 9: Problem 61

Perform the indicated operations and simplify. $$\frac{5}{x} \div 10$$

Short Answer

Expert verified
The simplified form is \(\frac{1}{2x}\).

Step by step solution

01

Express division as multiplication

To simplify the expression \(\frac{5}{x} \div 10\), convert the division operation into a multiplication by the reciprocal. The given expression becomes: \(\frac{5}{x} \div 10 = \frac{5}{x} \times \frac{1}{10}\)
02

Multiply the fractions

Now, multiply the two fractions: \(\frac{5}{x} \times \frac{1}{10}\). When multiplying fractions, multiply the numerators and the denominators respectively: \[\frac{5 \times 1}{x \times 10}\ = \frac{5}{10x}\]
03

Simplify the fraction

Finally, simplify the fraction \(\frac{5}{10x}\) by dividing the numerator and the denominator by the greatest common divisor. Here, 5 is the greatest common divisor: \(\frac{5}{10x} = \frac{1}{2x}\).

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Key Concepts

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

division in algebra
In algebra, division can be approached similarly to division with numbers, but it often involves variables. One handy trick is to convert division into multiplication by using the reciprocal. This helps simplify the problem and makes the work more straightforward.
For example, if you have \(\frac{5}{x} \div 10\), you can rewrite this as \(\frac{5}{x} \times \frac{1}{10}\). Using the reciprocal of 10 (which is \(\frac{1}{10}\)) turns the division into multiplication, which is simpler to handle.
multiplication of fractions
When multiplying fractions, it's as easy as multiplying their numerators and denominators. You don't need to worry about common denominators like you would in addition or subtraction.
For instance, with \(\frac{5}{x} \times \frac{1}{10}\), multiply the numerators (5 and 1) and the denominators (x and 10).
This yields \(\frac{5 \times 1}{x \times 10} = \frac{5}{10x}\). Multiplying fractions is quite direct if handled step-by-step.
simplifying fractions
Simplifying fractions involves reducing them to their simplest form. This means dividing the numerator and the denominator by their greatest common divisor (GCD).
In our example, \(\frac{5}{10x}\), the greatest common divisor of 5 and 10 is 5.
So, by dividing the numerator and the denominator by 5, we get \(\frac{5}{10x} = \frac{1}{2x}\). This is our simplified fraction.
Remember to always look for the GCD to simplify your fractions to their most basic form.

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Most popular questions from this chapter

If the exercise is an equation, solve it and check. Otherwise, perform the indicated operations and simplify. $$\frac{1}{x+3}=\frac{x+1}{x^{2}-9}$$

Ricardo can run at the rate of 8 mph and walk at the rate of 4 mph. If he wants to run and walk a total of 13 miles in 2 hours, how far should he run?

If the exercise is an equation, solve it and check. Otherwise, perform the indicated operations and simplify. $$\frac{c-5}{10}-\frac{c-2}{4}=c$$

If the exercise is an equation, solve it and check. Otherwise, perform the indicated operations and simplify. $$\frac{y}{3}-\frac{3}{y}=\frac{y-3}{3}$$

Make up a verbal problem involving a ratio that would give rise to the following equation (be sure to indicate what the variable would represent): $$ \frac{x}{600}=\frac{9}{10} $$
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