Problem 49 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{5}{4} \cdot \frac{6}{7}$$

Step-by-Step Solution

Answer

The result of \(\frac{5}{4} \cdot \frac{6}{7}\) simplified to its lowest terms is \(\frac{15}{14}\).

Step 1: Multiply Numerators

First, multiply the numerators of the fractions together. In other words, multiply 5 and 6 together. This gives \(5 \cdot 6 = 30\). This wins to be the numerator of the answer.

Step 2: Multiply Denominators

Then multiply the denominators of the fractions, which are 4 and 7. This gives \(4 \cdot 7 = 28\). This will be the denominator of the answer.

Step 3: Write out Multiplication Result

Next, write out the multiplication result of the two fractions which is \(\frac{30}{28}\).

Step 4: Simplify to Lowest Terms

Finally, simplify the fraction to its lowest terms. The numbers 30 and 28 both have a common factor of 2. So dividing numerator and denominator by 2 gives \(\frac{15}{14}\) which cannot be simplified further.

Key Concepts

Simplifying Fractions

When working with fractions, simplifying is an essential step to present your answer in the most straightforward form. Simplifying a fraction means to make the numerator (top number) and the denominator (bottom number) as small as possible while keeping the same value. It involves finding the greatest common divisor (GCD) for both numbers and dividing them by it.

For example, if you have the fraction \(\frac{20}{25}\), you would notice that both numbers can be evenly divided by 5. So, to simplify, you divide the numerator and denominator by 5 to get the fraction \(\frac{4}{5}\), which is the simplified form.

Multiplying Numerators

The multiplication of fractions starts with the numerators: the top numbers representing parts of a whole. To perform this operation, simply multiply the numerators of both fractions together. This product becomes the numerator of the new fraction.

For instance, when multiplying \(\frac{3}{5}\) and \(\frac{2}{7}\), you would multiply 3 (the numerator of the first fraction) by 2 (the numerator of the second fraction), which equals 6. The new fraction starts off as \(\frac{6}{x}\), with x representing the product of the denominators which will be calculated separately.

Multiplying Denominators

Following the multiplication of numerators, the next step is to address the denominators. Multiply the denominator of the first fraction by the denominator of the second fraction. The result will serve as the denominator of your new, multiplied fraction.

Continuing our example with \(\frac{3}{5}\) and \(\frac{2}{7}\), you would multiply 5 (the denominator of the first fraction) by 7 (the denominator of the second fraction) to get 35. This gives us a new fraction \(\frac{6}{35}\), which now has both a new numerator and a new denominator.

Reducing Fractions to Lowest Terms

After multiplying the numerators and denominators, it is possible that the result can be simplified or reduced to its lowest terms. Reducing a fraction to its lowest terms involves dividing both the numerator and the denominator by their greatest common divisor (GCD). If they have no common factors aside from 1, the fraction is already in its simplest form.

Take \(\frac{30}{28}\) as an example. The GCD of 30 and 28 is 2. Dividing both by 2, the fraction reduces to \(\frac{15}{14}\), which has no common factors other than 1 and therefore is in simplest form. If the GCD were greater than 2, you would continue simplifying until no more common factors exist.