Problem 43 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$\frac{7}{10} \div \frac{10}{7}$$

Step-by-Step Solution

Answer

\( \frac{49}{100} \)

Step 1: Understand Division of Fractions

When dividing fractions, we use the rule: \( \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} \). This means we multiply the first fraction by the reciprocal (or inverse) of the second fraction.

Step 2: Find the Reciprocal of the Second Fraction

The reciprocal of \( \frac{10}{7} \) is \( \frac{7}{10} \). To find the reciprocal, you simply swap the numerator and the denominator.

Step 3: Rewrite the Division as a Multiplication

Rewriting the problem using the rule from Step 1, the problem \( \frac{7}{10} \div \frac{10}{7} \) becomes \( \frac{7}{10} \times \frac{7}{10} \).

Step 4: Multiply the Fractions

To multiply fractions, multiply the numerators together and the denominators together: \( 7 \times 7 = 49 \) and \( 10 \times 10 = 100 \).

Step 5: Write the Final Fraction

The product of the multiplication is \( \frac{49}{100} \). This fraction is already in its simplest form.

Key Concepts

Reciprocal

The concept of a reciprocal is fundamental when dealing with division of fractions. A reciprocal is essentially a fraction flipped upside down. So, if you have a fraction like \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). This is because multiplication by a reciprocal is equivalent to dividing by the original fraction.

For example, to find the reciprocal of \( \frac{10}{7} \), we swap the numerator and the denominator to get \( \frac{7}{10} \). This technique is essential for fraction division because it allows us to convert a division problem into a multiplication one, which is simpler to solve.

Multiplication of Fractions

Multiplying fractions is a key process that simplifies the division of fractions. After finding the reciprocal, you replace the division with multiplication. When you multiply fractions, you simply multiply their numerators and their denominators.

• The numerators are multiplied together: \( 7 \times 7 = 49 \).
• The denominators are multiplied together: \( 10 \times 10 = 100 \).

Thus, the multiplication of \( \frac{7}{10} \times \frac{7}{10} \) results in \( \frac{49}{100} \). This method is straightforward and eliminates the need to perform the more complex division operation.

Simplifying Fractions

Simplifying fractions means making a fraction as simple as possible. This involves ensuring that the numerator and denominator have no common factors other than 1. In our example, \( \frac{49}{100} \), the numbers 49 and 100 do not share any common factors, meaning this fraction is already in its simplest form.

However, in cases where fractions aren't simplified, you can find the greatest common divisor (GCD) and divide both the numerator and the denominator by this number to simplify the fraction.

Simplifying is crucial because it makes fractions easier to work with and helps in understanding ratios in a clearer manner.