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Chapter 4: Problem 75

Divide and simplify the answer to lowest terms. Write the answer as a fraction or whole number. $$\frac{4 x y}{3} \div \frac{14 x}{9}$$

Short Answer

Expert verified
\(\frac{6y}{7}\)

Step by step solution

01

Write the division as a multiplication

To divide by a fraction, multiply by its reciprocal. Therefore, rewrite the expression \(\frac{4xy}{3} \div \frac{14x}{9}\) as \(\frac{4xy}{3} \cdot \frac{9}{14x}\).
02

Multiply the fractions

Multiply the numerators together and the denominators together: \(\frac{4xy \cdot 9}{3 \cdot 14x} = \frac{36xy}{42x}\).
03

Simplify the fraction

Cancel out common factors in the numerator and the denominator. Divide both numerator and denominator by \(6x\): \(\frac{36xy \div 6x}{42x \div 6x} = \frac{6y}{7}\).
04

Write the fraction in its simplest form

The simplified form of the fraction is \(\frac{6y}{7}\).

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Key Concepts

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

Fraction Division
When dividing fractions in algebra, remember that division by a fraction is the same as multiplying by its reciprocal.
This is a fundamental concept that helps simplify many problems. For example, in our exercise, we started with \(\frac{4xy}{3} \div \frac{14x}{9}\).
Instead of dividing directly, we multiply the first fraction by the reciprocal of the second.

The reciprocal of \(\frac{14x}{9}\) is \(\frac{9}{14x}\).
So, the equation becomes \(\frac{4xy}{3} \cdot \frac{9}{14x}\).
This method simplifies the division process into a multiplication problem, which is easier to handle.
Simplifying Fractions
Simplifying fractions is the process of reducing them to their lowest terms.
This is done by dividing the numerator and denominator by their greatest common factor (GCF).

In our example, after multiplying the fractions, we got \(\frac{36xy}{42x}\).
To simplify this, we look for common factors.
Both 36 and 42 are divisible by 6, and there's also an \x\ in both the numerator and the denominator.
Dividing both by \6x\, we get \(\frac{36xy \div \6x}{42x \div \6x} = \frac{6y}{7}\).

The fraction \(\frac{6y}{7}\) cannot be simplified further, as 6 and 7 have no common factors other than 1.
Reciprocals
The reciprocal of a number is what you multiply by that number to get 1.

For fractions, the reciprocal is created by flipping the numerator and the denominator.
For example, the reciprocal of \(\frac{a}{b}\) is \(\frac{b}{a}\).

In our original problem, the given fractions were \(\frac{4xy}{3}\) and \(\frac{14x}{9}\).
To divide by \(\frac{14x}{9}\), we used its reciprocal \(\frac{9}{14x}\).
This is because multiplication by the reciprocal is equivalent to division.
Understanding reciprocals is crucial to working efficiently with fraction division.
Multiplying Fractions
To multiply fractions, you multiply the numerators together and the denominators together.

Let's see how it applies:
\(\frac{4xy}{3} \cdot \frac{9}{14x}\) translates to multiplying 4xy by 9 in the numerator and 3 by 14x in the denominator.
So, \(\frac{4xy \cdot 9}{3 \cdot 14x} = \frac{36xy}{42x}\).

This step simplifies the problem since multiplication is straightforward.
When multiplying fractions, always handle the numerators and denominators step-by-step to avoid mistakes.
Afterwards, simplify the resulting fraction by canceling common factors if present.

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Most popular questions from this chapter

Find the LCM. 21 and 14

Solve the equations. $$\frac{1}{3} b=-4$$

How many eighths are in \(2 \frac{3}{8} ?\)

Simplify to lowest terms by first reducing the powers of 10. $$\frac{720}{800}$$

One method for finding prime numbers is the sieve of Eratosthenes. The natural numbers from 2 to 50 are shown in the table. Start at the number 2 (the smallest prime number). Leave the number 2 and cross out every sccond number after the number 2. This will climinate all numbers that are multiples of 2. Then go back to the beginning of the chart and leave the number 3 , but cross out every third number after the number 3 (thus eliminating the multiples of 3 ). Begin at the next open number and continue this process. The numbers that remain are prime numbers. Use this process to find the prime numbers less than 50 . $$\begin{array}{c|c|c|c|c|c|c|c|c|c} & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ \hline 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ \hline 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\ \hline 31 & 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 & 40 \\ \hline 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 & 49 & 50 \\ \hline \end{array}$$
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