Problem 164 - Intermediate Algebra

Question

In the following exercises, perform the indicated operation. $$ \frac{12 r}{25} \div \frac{18 s}{35} $$

Step-by-Step Solution

Answer

The result is \(\frac{14r}{15s}\).

Step 1: Understand the Division of Fractions

Dividing fractions involves multiplying by the reciprocal of the fraction you are dividing by. So, we need to invert the second fraction and change the division sign to multiplication.

Step 2: Write the Reciprocal

The reciprocal of \(\frac{18s}{35}\) is \(\frac{35}{18s}\).

Step 3: Set Up the Multiplication

Change the division into multiplication by writing \(\frac{12r}{25} \times \frac{35}{18s}\).

Step 4: Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together: \(\frac{12r \times 35}{25 \times 18s} = \frac{420r}{450s}\).

Step 5: Simplify the Fraction

Simplify \(\frac{420r}{450s}\) by finding the greatest common divisor (GCD) of 420 and 450. The GCD is 30. Divide both the numerator and the denominator by 30: \(\frac{420r \div 30}{450s \div 30} = \frac{14r}{15s}\).

Key Concepts

Reciprocal in Fractions

Fractions can be tricky at first, but understanding the concept of the reciprocal makes it much easier. The reciprocal of a fraction is simply switching its numerator and denominator. For example, the reciprocal of \(\frac{18s}{35}\) is \(\frac{35}{18s}\).
Reciprocals are essential when dividing fractions, as they allow you to convert the operation into a multiplication problem. This step simplifies the process and makes calculations more straightforward.
To find a reciprocal:

• Swap the numerator and the denominator
• Verify that the fraction is correctly inverted before proceeding with operations.
• In multiplication scenarios, using the reciprocal ensures accurate problem-solving.

Multiplication of Fractions

Once you have the reciprocal, division of fractions becomes multiplication. Instead of dividing by a fraction, you multiply by its reciprocal. Let's see how:
From the previous example, you flip the fraction \(\frac{18s}{35}\) to get \(\frac{35}{18s}\). Now, change the division sign into a multiplication sign: \(\frac{12r}{25} \times \frac{35}{18s}\).
Next, you multiply the numerators and denominators:

• Multiply the numerators: \12r \times 35 = 420r\
• Multiply the denominators: \25 \times 18s = 450s\

So, you get: \(\frac{420r}{450s}\)
This simplifies the problem and sets you up for the last step: simplification.

Simplifying Fractions

Simplifying fractions helps make them easier to understand and work with. You simplify by dividing both the numerator and the denominator by their greatest common divisor (GCD). In our example, \420\ and \450\ share a GCD of \30\.
Dividing both parts by \30\:

• Numerator: \420r \div 30 = 14r\
• Denominator: \450s \div 30 = 15s\

So, \(\frac{420r}{450s}\) simplifies to \(\frac{14r}{15s}\).
Simplified fractions are neat and easy to interpret. Remember, the goal is to reduce the fraction to its simplest form where the numerator and denominator are relatively prime.