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Chapter 4: Problem 159

Divide, and write the answer in simplified form. \(\frac{12 r}{25} \div \frac{18 s}{35}\)

Short Answer

Expert verified
\frac{14r}{15s}

Step by step solution

01

- Write the Division of Fractions as Multiplication

To divide by a fraction, multiply by its reciprocal. So, \[\frac{12r}{25} \div\ \frac{18s}{35}\] can be written as: \[ \frac{12r}{25} \times\ \frac{35}{18s} \].
02

- Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together: \[ \frac{12r \times\ 35}{25 \times\ 18s} \].
03

- Simplify the Fraction

Now simplify the fraction. First, calculate the product of the numerators and the denominators: \[ \frac{12r \times\ 35}{25 \times\ 18s} = \frac{420r}{450s} \]. Next, find the greatest common divisor (GCD) of 420 and 450, which is 30. Divide both the numerator and the denominator by 30: \[ \frac{420r}{450s} \div\ 30 = \frac{14r}{15s} \].

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Key Concepts

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

Reciprocal
When dealing with fraction division, understanding the concept of the reciprocal is key. The reciprocal of a fraction is created by flipping the numerator and the denominator.
For example, the reciprocal of \(\frac{18s}{35}\) is \(\frac{35}{18s}\).
When you divide fractions, you multiply by the reciprocal of the divisor. This is because division by a fraction is the same as multiplying by its reciprocal.
So, \(\frac{12r}{25} \div \frac{18s}{35}\) becomes \( \frac{12r}{25}\ \times \frac{35}{18s}\).
Always remember to switch the division problem into a multiplication problem using the reciprocal.
Numerator
A fraction is made up of two parts: the numerator and the denominator. The numerator is the top number and represents how many parts we have.
In our case, \(\frac{12r}{25} \div \frac{18s}{35}\), the numerator of the first fraction is 12r and the numerator of the second fraction is 18s.
When multiplying fractions, you multiply the numerators together. Hence, \(12r \times 35 = 420r\).
The product of the numerators will be the new numerator of the resulting fraction.
Denominator
Like the numerator, the denominator is very important when working with fractions. The denominator is the bottom number and tells us how many parts the whole is divided into.
In our case, \(\frac{12r}{25} \div \frac{18s}{35}\), the denominators are 25 and 18s.
When you multiply fractions, you need to multiply the denominators together.
Therefore, \(25 \times 18s = 450s\).
The product of the denominators will be the new denominator of the resulting fraction.
Simplification
The final step in fraction division is simplification. Simplification makes your fraction easier to understand and use.
After multiplying the numerators and the denominators, you can end up with large numbers. Here, \(\frac{420r}{450s}\) is the result before simplification.
You then find the greatest common divisor (GCD) of the numerator and denominator to reduce the fraction. The GCD of 420 and 450 is 30.
Divide both the numerator and the denominator by their GCD, so \(\frac{420r}{450s} \div 30 = \frac{14r}{15s}\).
Simplification helps present the fraction in its simplest form for ease of understanding.

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Most popular questions from this chapter

In the following exercises, perform the indicated operation. $$ 2 \frac{11}{12}-1 \frac{7}{12} $$

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In the following exercises, perform the indicated operations and simplify. $$\frac{y}{10}-\frac{1}{3}$$

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