Problem 44 - Basic College Mathematics

Question

A child's baseball shirt requires \(\frac{5}{6}\) yd of fabric. How many shirts can be made from 25 yd of fabric?

Step-by-Step Solution

Answer

30 shirts can be made from 25 yards of fabric.

Step 1: Understand the Problem

We need to find out how many baseball shirts can be made from 25 yards of fabric, given that each shirt needs \( \frac{5}{6} \) yards of fabric.

Step 2: Determine the Mathematical Operation

To find out how many shirts can be made, we need to divide the total amount of fabric by the amount of fabric needed for one shirt: \( \text{Total Fabric} \div \text{Fabric per Shirt} \).

Step 3: Set Up the Division

Using the given values: \( 25 \div \frac{5}{6} \).

Step 4: Simplify the Division

Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, \( 25 \div \frac{5}{6} = 25 \times \frac{6}{5} \).

Step 5: Perform the Multiplication

Now, multiply: \( 25 \times \frac{6}{5} = 25 \times 1.2 = 30 \).

Step 6: Interpret the Result

The result means that 30 shirts can be made from 25 yards of fabric.

Key Concepts

Fractional Operations

Understanding fractional operations is key to solving many problems involving fractions. Fractions represent parts of a whole and can be added, subtracted, multiplied, and divided just like whole numbers. To perform operations with fractions:

• Ensure the fractions have a common denominator for addition and subtraction.
• For multiplication, simply multiply the numerators and denominators together.
• For division, flip (or find the reciprocal of) the second fraction and then multiply.

In our task, we use division of fractions, which involves flipping the divisor and turning the operation into multiplication.

Multiplication of Fractions

Multiplying fractions can seem tricky, but it's straightforward. You just multiply the numerators and the denominators:

• For instance, to multiply \( \frac{a}{b} \) by \( \frac{c}{d} \), you compute \( \frac{a \times c}{b \times d} \).

In our example, we needed to perform the multiplication step after converting the division of fractions into multiplication. Thus, we multiplied 25 by \( \frac{6}{5} \). Multiplying these gives us \( 25 \times \frac{6}{5} = 30 \). This shows how many shirts can be made with the given fabric.

Reciprocal of Fractions

The reciprocal of a fraction is simply flipping the numerator and the denominator. For example, the reciprocal of \( \frac{3}{4} \) is \( \frac{4}{3} \). The reciprocal changes division into multiplication:

• To divide by a fraction, multiply by its reciprocal.

In our exercise, we needed to divide 25 by \( \frac{5}{6} \). By finding the reciprocal of \( \frac{5}{6} \) to get \( \frac{6}{5} \), we turned the problem into a multiplication: \( 25 \times \frac{6}{5} \). This simplifies the problem, making it easier to solve.