Problem 64 - Elementary Algebra

Question

A cork board is made by gluing together wine corks. Each cork is \(\frac{3}{4}\) inch wide. Find the number of corks needed to create a board that is 24 inches wide.

Step-by-Step Solution

Answer

32 corks are needed.

Step 1: Identify the width of one cork

The width of each wine cork is given as \(\frac{3}{4}\) inch.

Step 2: Identify the width of the board

The desired width of the cork board is given as 24 inches.

Step 3: Set up the division problem

To find the number of corks needed, divide the total width of the board by the width of one cork: \(\frac{24}{\frac{3}{4}} \).

Step 4: Perform the division

Perform the division \(\frac{24}{\frac{3}{4}} \). This is equivalent to multiplying 24 by the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\): \(\frac{24}{\frac{3}{4}} = 24 \times \frac{4}{3} = 32\).

Step 5: Interpret the result

The result of the division is 32. Therefore, 32 corks are needed to create a 24-inch wide board.

Key Concepts

fraction multiplication

In this exercise, we need to multiply fractions to find our solution. Fraction multiplication is quite straightforward. To multiply two fractions, you simply multiply the numerators (top numbers) together and the denominators (bottom numbers) together. For example, if we have \(\frac{a}{b}\) and \(\frac{c}{d}\), the result of their multiplication will be \(\frac{a \times c}{b \times d}\).
This concept comes into play in our exercise where we multiply the whole number by the reciprocal of a fraction to perform division.

reciprocal of a fraction

Understanding the reciprocal of a fraction is key to solving division problems involving fractions. The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). Basically, you flip the numerator and the denominator.
In our exercise, we found the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\). This reciprocal is used to convert the division problem into a multiplication problem, making it easier to solve. Remember, multiplying by a reciprocal achieves the same result as dividing by the original fraction.

basic division

Basic division is a fundamental arithmetic operation where a number is divided into equal parts. In the context of fractions, division can be made simpler by using the reciprocal.
For example, in our exercise, we divided 24 inches by \(\frac{3}{4}\) inches, turning it into a multiplication problem: \(\frac{24}{\frac{3}{4}}\) becomes \24 \times \frac{4}{3}\. This helped us find that the number of corks needed was 32.
Remember, division of fractions relies heavily on understanding multiplication of fractions and the concept of reciprocals.