Problem 75 - Elementary Algebra

Question

The thickness of one sheet of corrugated cardboard is \(\frac{3}{16}\) inch. The distance between shelves in a storage rack is 6 inches. Find the number of sheets of cardboard that can be stacked in this space.

Step-by-Step Solution

Answer

32 sheets

Step 1 - Understand the problem

We need to find out how many sheets of corrugated cardboard with a thickness of \(\frac{3}{16}\) inch can fit within a 6-inch distance between shelves.

Step 2 - Identify the required calculation

To solve this problem, we need to determine how many times the thickness of one sheet (\(\frac{3}{16}\) inch) can be contained within 6 inches.

Step 3 - Set up the division problem

We divide the total space (6 inches) by the thickness of one sheet (\(\frac{3}{16}\) inch): \[ \frac{6}{\frac{3}{16}} \]

Step 4 - Perform the division

When we divide by a fraction, we multiply by its reciprocal:\[ \frac{6}{\frac{3}{16}} = 6 \times \frac{16}{3} \]

Step 5 - Simplify the expression

Calculate the multiplication: \[ 6 \times \frac{16}{3} = \frac{96}{3} \]

Step 6 - Final calculation

Simplify \(\frac{96}{3}\) to get the final number of sheets: \[ \frac{96}{3} = 32 \]

Key Concepts

Fraction Arithmetic

Fraction arithmetic involves performing operations like addition, subtraction, multiplication, and division on fractions. In our exercise, we primarily deal with multiplication and division of fractions. The key point to remember is that fractions represent parts of a whole. For example, \( \frac{3}{16} \) means 3 parts out of 16 equal parts. To work with such numbers, ensure you understand how to manipulate both the numerator and the denominator. Additionally, always try to simplify fractions by finding the greatest common divisor before performing calculations.

Division by Fractions

When dividing by a fraction, many students get confused because it’s not a straightforward operation. To divide by a fraction, you multiply by its reciprocal. The reciprocal of a fraction is created by swapping its numerator and denominator. So, for example, the reciprocal of \( \frac{3}{16} \) is \( \frac{16}{3} \). This step-by-step approach helps to change the problem from a division into a multiplication, which is easier to solve. In our exercise, we transformed \( \frac{6}{\frac{3}{16}} \) into \( 6 \times \frac{16}{3} \). This change is essential as it simplifies the entire process of working with fractions.

Measurements in Algebra

Measurements often come into play in algebra problems, especially when dealing with real-life applications. In our exercise, we use measurements of inches to solve for the number of sheets of cardboard. When working with measurements, ensure consistency in units and understand the relationship between them. For our problem, we had the thickness of one sheet of cardboard as \( \frac{3}{16} \) inch and the total space available as 6 inches. The objective was to find how many such sheets fit into the available space. Converting the real-world problem into an algebraic equation makes it easier to solve using fraction arithmetic.

Simplifying Fractions

Simplifying fractions makes calculations more manageable and helps achieve the final result promptly. To simplify a fraction, divide the numerator and the denominator by their greatest common divisor (GCD). In our exercise, after finding \( \frac{96}{3} \) from \( 6 \times \frac{16}{3} \), we simplify it. Since both 96 and 3 can be divided by 3, our simplified fraction becomes 32. Simplifying fractions as you go along can save time and help avoid mistakes in larger calculations.