Problem 60 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{3}{7} \div \frac{1}{7}$$

Step-by-Step Solution

Answer

The reduced form of \(\frac{3}{7} \div \frac{1}{7}\) is 3.

Step 1: Identify the Dividend and the Divisor

In this exercise, \(\frac{3}{7}\) is the dividend (the number being divided) and \(\frac{1}{7}\) is the divisor (the number by which the dividend is being divided).

Step 2: Reciprocate the Divisor

The division of fractions is accomplished by multiplying the dividend by the reciprocal of the divisor. Thus, you'd need to find the reciprocal of the divisor. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. So, the reciprocal of \(\frac{1}{7}\) is \(\frac{7}{1}\), or simply 7.

Step 3: Perform the Multiplication

Now, multiply the dividend \(\frac{3}{7}\) by the reciprocal of the divisor \(7\). The multiplication of fractions is done by multiplying the numerators to get the new numerator, and multiplying the denominators to get the new denominator. Thus, \(\frac{3}{7} * 7 = \frac{3*7}{7*1} = \frac{21}{7}\).

Step 4: Simplify the Fraction

The fraction \(\frac{21}{7}\) simplifies to 3, because 21 divided by 7 equals to 3. Therefore, the fraction has been reduced to its lowest terms.

Key Concepts

Reciprocal of a Fraction

Understanding the reciprocal of a fraction is a key step when dividing fractions. A reciprocal flips a fraction, swapping its top and bottom numbers, known as the numerator and the denominator. For example, if you start with the fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \). This is the fundamental rule:

• The numerator and denominator exchange places.
• If you have a whole number, its reciprocal is simply \( \frac{1}{ \text{the whole number} } \).

In the given exercise, the divisor was \( \frac{1}{7} \). Flipping this fraction, the reciprocal becomes 7 or \( \frac{7}{1} \). This concept allows us to transform a division problem into a multiplication one, by multiplying with the reciprocal.

Simplifying Fractions

After calculating the product of fractions, you often need to simplify the resulting fraction. Simplifying makes fractions easier to understand and use. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1.

• Start by identifying any common factors of the numerator and denominator.
• Divide both by their greatest common factor (GCF) to simplify.

In our scenario, the fraction resulted in \( \frac{21}{7} \). Here, both 21 and 7 can be divided by their GCF, which is 7. Performing this division yields 3, leaving us with a whole number which is the simplest form.

Fraction Multiplication

Fraction multiplication comes into play when dividing fractions by multiplying by the reciprocal. To multiply fractions, follow these straightforward steps:

• Multiply the numerators together to find the new numerator.
• Multiply the denominators together to find the new denominator.
• The result is a new fraction: \( \frac{ \text{new numerator} }{ \text{new denominator} } \).

In the example provided, multiplying \( \frac{3}{7} \) by 7 (which we represent as \( \frac{7}{1} \)) we get:\[\frac{3 \times 7}{7 \times 1} = \frac{21}{7}\]This result then leads to simplification. Fraction multiplication is crucial because it transforms a division task into an easier multiplication problem, keeping operations simple.