Problem 18 - Fundamentals of Mathematics

Question

Determine each of the following quotients and then write a rule for this type of division. $$1 \div \frac{3}{4}$$

Step-by-Step Solution

Answer

\( 1 \div \frac{3}{4} = \frac{4}{3} \). Rule: Divide by multiplying with the reciprocal.

Step 1: Understand the Division of Fractions

The problem requires dividing 1 by the fraction \( \frac{3}{4} \). In fraction division, dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of a fraction is obtained by exchanging its numerator and denominator.

Step 2: Find the Reciprocal of the Fraction

The reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \).

Step 3: Multiply by the Reciprocal

Multiply 1 by the reciprocal of \( \frac{3}{4} \). This means calculating \( 1 \times \frac{4}{3} \).

Step 4: Perform the Multiplication

Since any number multiplied by 1 is the number itself, \( 1 \times \frac{4}{3} = \frac{4}{3} \).

Step 5: Write a Rule for Dividing by a Fraction

To divide by a fraction, multiply by its reciprocal.

Key Concepts

Reciprocal

To understand the concept of dividing fractions, it's essential to first know what a reciprocal is. Simply put, the reciprocal of a number or a fraction is what you multiply that number by to get 1. For a fraction, you find its reciprocal by swapping its numerator and denominator. This means if you have a fraction like \( \frac{a}{b} \), its reciprocal will be \( \frac{b}{a} \). It's always a good idea to quickly check if you've found the right reciprocal by multiplying the original fraction with its reciprocal. If you end up with 1, you know it's correct! Reciprocals are key in processes such as division of fractions, as they transform division into a much simpler operation—multiplication.

Fraction Multiplication

Multiplying fractions is straightforward once you grasp the process. Here's what to remember: multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. For example, when multiplying \( \frac{a}{b} \) by \( \frac{c}{d} \), the product is \( \frac{a \times c}{b \times d} \).

• Multiply the top numbers (numerators).
• Multiply the bottom numbers (denominators).
• Simplify the resulting fraction if possible.

Fraction multiplication is particularly useful in fraction division problems because once you find the reciprocal of the fraction you wish to divide by, you switch to multiplication, making the calculation smoother. Always keep an eye out for opportunities to simplify the fractions before multiplying, as this can save effort and time.

Fraction Division Rule

Dividing fractions might seem complex at first, but it becomes much easier with a simple rule: "To divide by a fraction, multiply by its reciprocal." This rule is the foundation of fraction division because it transforms the division problem into a multiplication one, which is typically more straightforward to compute. To apply this rule, follow these steps:

• Identify the fraction you are dividing by and find its reciprocal by switching its numerator and denominator.
• Change the division operation to multiplication.
• Multiply the first fraction by the reciprocal of the second fraction.
• Simplify the resulting fraction if necessary.

Using this method ensures you correctly and efficiently solve fraction division problems each time. Always verify your final answer by considering whether it could still be reduced to its simplest form to check for minor calculation errors. This rule streamlines the once-complicated division of fractions by turning it into an easier task of fraction multiplication.