Problem 73 - Elementary and Intermediate Algebra

Question

Perform the indicated operation and, if possible, simplify. If there are no variables, check using a calculator. $$ \frac{7}{6} \div \frac{3}{5} $$

Step-by-Step Solution

Answer

The answer is \(\frac{35}{18}\).

Step 1 - Rewrite the Division as Multiplication

To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{3}{5}\) is \(\frac{5}{3}\). Thus, rewrite the expression: \(\frac{7}{6} \times \frac{5}{3} \).

Step 2 - Multiply the Fractions

Multiply the numerators together and the denominators together: \(\frac{7 \times 5}{6 \times 3} \), which simplifies to \(\frac{35}{18} \).

Step 3 - Simplify the Fraction Further, if Possible

In this case, \(\frac{35}{18} \) is already in its simplest form, as the numerator and the denominator have no common factors.

Step 4 - Check Using a Calculator

Divide \(\frac{7}{6}\) by \(\frac{3}{5}\) on a calculator to verify that it equals about 1.9444, which matches the decimal form of \(\frac{35}{18}\).

Key Concepts

reciprocal

When working with fractions, understanding the concept of the reciprocal is essential, especially for fraction division. The reciprocal of a fraction is just that fraction flipped upside down. For example, the reciprocal of \(\frac{3}{5}\) is \(\frac{5}{3}\). To get the reciprocal of a fraction, we simply swap the numerator (top number) and the denominator (bottom number). This is because multiplying a fraction by its reciprocal always equals 1, which is the identity element of multiplication. Whenever we need to divide by a fraction, we multiply by its reciprocal instead. This approach transforms a division problem into a simpler multiplication one.

multiplication of fractions

Multiplying fractions is straightforward once we have the reciprocal. To multiply two fractions, we follow these steps:

• Multiply the numerators (top numbers) together to get the new numerator.
• Multiply the denominators (bottom numbers) together to get the new denominator.

For example, to multiply \(\frac{7}{6}\) by \(\frac{5}{3}\), we multiply the numerators: \ 7 \times 5 = 35 \ and the denominators: \ 6 \times 3 = 18. \ This results in the fraction \(\frac{35}{18}\). Multiplication of fractions often results in fractions that need further simplification, which we will cover next.

simplifying fractions

Once we have our multiplied fraction, we should check if it can be simplified. Simplifying a fraction involves reducing it to its lowest terms. This means dividing both the numerator and the denominator by their greatest common divisor (GCD). However, our fraction \(\frac{35}{18}\) is already in the simplest form since 35 and 18 have no common factors other than 1. Simplifying helps in better understanding and using fractions in further calculations or comparisons.

using a calculator to verify

Finally, it's always good practice to verify our manual calculations using a calculator. This ensures accuracy, especially in complex problems. To verify the division of fractions using a calculator:

• First, divide 7 by 6 to get the decimal form of \(\frac{7}{6}\), which is about 1.1667.
• Then, divide this result by the decimal form of \(\frac{3}{5}\) (which is 0.6).
• The calculator should give you approximately 1.9444.

Compare this with the decimal form of our simplified fraction \(\frac{35}{18}\), which is also about 1.9444. Matching these values confirms the correctness of our solution!