Problem 81 - Elementary Algebra

Question

Solve the word problems. If it takes \(3 \frac{1}{4}\) yards of material to make one dress, how much material is needed for 20 dresses?

Step-by-Step Solution

Answer

65 yards of material is needed.

Step 1: Convert Mixed Number to Improper Fraction

The amount of material needed for one dress is given as a mixed number: \(3 \frac{1}{4}\). First, we need to convert this into an improper fraction. \(3 \frac{1}{4}\) can be expressed as \(\frac{4 \times 3 + 1}{4} = \frac{13}{4}\).

Step 2: Determine Total Material Needed

To find the total amount of material needed for 20 dresses, multiply the material for one dress by the number of dresses. This is calculated as: \[\frac{13}{4} \times 20\].

Step 3: Perform Multiplication

Multiply the numerators and the denominators separately: \[13 \times 20 = 260\] and \[4 \times 1 = 4\]. Thus, the fraction is \(\frac{260}{4}\).

Step 4: Simplify the Fraction

Divide the numerator by the denominator to simplify: \(\frac{260}{4} = 65\). Thus, 65 yards of material is needed.

Key Concepts

Mixed Numbers

A mixed number is a type of fraction that consists of two parts: a whole number and a proper fraction. For instance, the mixed number \(3 \frac{1}{4}\) is made up of the whole number 3 and the fraction \(\frac{1}{4}\). Mixed numbers are often used to represent quantities that are larger than a single whole but not enough to form the next whole unit. To understand mixed numbers better, imagine you have 3 entire pizzas and an extra quarter of another pizza. This can be represented as \(3 \frac{1}{4}\) pizzas. Mixed numbers can easily be converted into improper fractions, which makes arithmetic operations simpler. Here’s how you do it:

• Multiply the whole number by the denominator of the fraction.
• Add the result to the numerator of the fraction.
• Place the resulting sum over the original denominator.

For the number \(3 \frac{1}{4}\):

• Multiply 3 (whole number) by 4 (denominator) to get 12.
• Add 1 (the numerator) to 12, resulting in 13.
• The improper fraction is \(\frac{13}{4}\).

Understanding mixed numbers and how to convert them is important when performing operations like multiplication or division in word problems.

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. For example, \(\frac{13}{4}\) is an improper fraction because 13 is greater than 4. They represent values equal to or greater than one whole.Improper fractions can be easier to use in arithmetic calculations than mixed numbers, particularly in multiplication and division. To work effectively with improper fractions:

• Simplify: Sometimes, it helps to simplify the fraction by dividing both the numerator and denominator by their greatest common factor.
• Convert: If necessary, you can convert improper fractions back to mixed numbers, particularly if the context of the problem makes it easier to interpret the results.

To convert an improper fraction like \(\frac{13}{4}\) back to a mixed number:

• Divide 13 by 4, which gives a quotient of 3 and a remainder of 1.
• The mixed number is \(3 \frac{1}{4}\).

Understanding how to manage improper fractions is essential in simplifying complex operations and interpreting the result correctly in word problems.

Multiplication of Fractions

Multiplying fractions can seem complex, but it's quite straightforward once you understand the steps. Whether you're working with proper fractions, improper fractions, or mixed numbers (which should be converted to improper first), the process remains the same.Here’s how to perform the multiplication of fractions:

• Multiply the numerators: This will give you the numerator of the product.
• Multiply the denominators: This will be the denominator of the product.
• Simplify: After multiplying, simplify the result, if possible, by dividing both the numerator and the denominator by their greatest common factor.

For example, consider the multiplication \(\frac{13}{4} \times 20\):

• The numerator is calculated as \(13 \times 20 = 260\).
• The denominator is \(4 \times 1 = 4\), since 20 can be written as \(\frac{20}{1}\).
• This results in the fraction \(\frac{260}{4}\), which simplifies to 65 because 260 divided by 4 is 65.

By understanding the multiplication of fractions, you're equipped to tackle a wide range of algebra word problems effectively.