Problem 56 - Prealgebra

Question

Manufacturing. A clothing manufacturer is making scarves that require \(\frac{3}{8}\) yard of material each. How many can be made from 27 yards of material?

Step-by-Step Solution

Answer

The manufacturer can make 72 scarves.

Step 1: Understand the problem

We need to find out how many scarves can be created from 27 yards of material given that each scarf requires \( \frac{3}{8} \) yard of material.

Step 2: Set up the division problem

To solve this problem, divide the total amount of material by the amount of material needed for one scarf. This can be expressed as \( 27 \div \frac{3}{8} \).

Step 3: Dividing fractions by multiplying with the reciprocal

Dividing by a fraction \( \frac{3}{8} \) is the same as multiplying by its reciprocal \( \frac{8}{3} \). So, compute \( 27 \times \frac{8}{3} \).

Step 4: Carry out the multiplication

Multiply 27 by \( \frac{8}{3} \): \[ 27 \times \frac{8}{3} = \frac{27 \times 8}{3} = \frac{216}{3}. \]

Step 5: Simplify the result

Now, simplify the fraction \( \frac{216}{3} \). Divide 216 by 3 to get 72.

Step 6: Conclusion

The manufacturer can make 72 scarves from 27 yards of material.

Key Concepts

Multiplying Reciprocals

When dividing fractions, the process of multiplying by the reciprocal of the divisor is used. The reciprocal of a fraction, such as \( \frac{3}{8} \), is achieved by flipping the numerator and the denominator, transforming it into \( \frac{8}{3} \).
This technique essentially converts division into multiplication, making the arithmetic straightforward.

• To divide by a fraction, you multiply by its reciprocal.
• The operation simplifies because multiplying is often easier to visualize and perform than division.

For example, in the scarf manufacturing problem, the division \( 27 \div \frac{3}{8} \) is converted into a multiplication problem as \( 27 \times \frac{8}{3} \). This allows us to work with whole numbers and fractions in a more manageable way, making practical calculations less prone to error.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form. This involves dividing the numerator and the denominator by their greatest common divisor (GCD). In the case of the scarf problem, after multiplying \( 27 \times \frac{8}{3} \), we arrive at the fraction \( \frac{216}{3} \).
To simplify:

• Find the GCD of 216 and 3, which is 3.
• Divide both the numerator and the denominator by the GCD.

So, \( \frac{216}{3} \) simplifies to 72, because:

• 216 divided by 3 equals 72.
• The denominator becomes 1, resulting in a whole number.

Simplifying fractions is crucial in practical math as it helps to avoid complex numbers and makes interpretation of data easier.

Practical Math Problems

Practical math problems are those we encounter in everyday situations and often require basic operations like addition, subtraction, multiplication, and division. The key is to understand what operations are required and why.
Take the scarf example, where we need to calculate how many products can be made from a given amount of material.

• Recognize that this real-life scenario translates to a division problem.
• Use reciprocal multiplication to simplify the division process.
• Simplify results to achieve clear and meaningful answers.

These skills are fundamental because they allow us to convert real-world problems into mathematical expressions we can solve. In this case, understanding fraction division and simplification ensures the manufacturing process is efficient and helps in resource management.