Problem 2 - Prealgebra

Question

Find each of the following products. (Multiply.) $$\frac{5}{6} \cdot \frac{7}{4}$$

Step-by-Step Solution

Answer

The product is \( \frac{35}{24} \) or 1 \( \frac{11}{24} \).

Step 1: Understand the operation

We need to multiply two fractions, \( \frac{5}{6} \) and \( \frac{7}{4} \). Multiplication of fractions involves multiplying the numerators and the denominators.

Step 2: Multiply the numerators

Multiply the numerators of the fractions: \( 5 \times 7 = 35 \).

Step 3: Multiply the denominators

Multiply the denominators of the fractions: \( 6 \times 4 = 24 \).

Step 4: Write the product as a fraction

Combine the products found in Steps 2 and 3 to form the fraction \( \frac{35}{24} \).

Step 5: Simplify the fraction

Determine if \( \frac{35}{24} \) can be simplified further by checking for any common factors. Since 35 and 24 have no common factors other than 1, \( \frac{35}{24} \) is already in its simplest form.

Step 6: Convert improper fraction to mixed number (if needed)

Since the result \( \frac{35}{24} \) is an improper fraction, convert it to a mixed number: 1 (whole number part) and \( \frac{11}{24} \) (the remainder).

Key Concepts

Improper Fractions

An improper fraction is a fraction where the numerator is larger than the denominator. This means the fraction represents a value greater than one whole unit.
For example, in the fraction \( \frac{35}{24} \), the numerator 35 is greater than the denominator 24.

Improper fractions can be useful, especially when multiplying or dividing fractions because they can sometimes simplify the arithmetic process.
To transform an improper fraction into a mixed number, divide the numerator by the denominator.

• The quotient becomes the whole number part.
• The remainder becomes the numerator of the fractional part.
• The denominator remains unchanged.

Understanding improper fractions and how to work with them lays the groundwork for performing operations like multiplication and division with fractions efficiently.

Mixed Numbers

A mixed number combines a whole number part and a proper fraction. Proper fractions are those where the numerator is less than the denominator.
Mixed numbers make it easier for us to understand and visualize values that are greater than a whole unit.

To convert an improper fraction like \( \frac{35}{24} \) into a mixed number:

• First, divide 35 by 24.
• In this case, 24 goes into 35 once (1 whole number), which leaves a remainder of 11.
• This remainder of 11 becomes the numerator of the fractional part, so the mixed number is \( 1\frac{11}{24} \).

Mixed numbers can be particularly helpful in everyday situations like cooking or measuring materials because they give a clear sense of how much more than a whole something is.
Working with mixed numbers can also simplify adding or subtracting fractions.

Simplifying Fractions

Simplifying a fraction means reducing it to its simplest form, where the numerator and the denominator have no common factors other than 1.
Simplifying makes fractions easier to compare and compute with, providing a cleaner and more intuitive result.

To simplify a fraction, find the greatest common divisor (GCD) of the numerator and the denominator.

• If they have the GCD other than 1, divide both the numerator and the denominator by this number.
• If no such common divisor exists—other than 1—the fraction is already in its simplest form. For instance, the fraction \( \frac{35}{24} \) cannot be simplified because 35 and 24 share no common factors other than 1.

Learning to simplify fractions ensures that all fractions are presented in the easiest form to interpret and use in further mathematical operations. Reducing fractions to their simplest form also helps maintain accuracy, especially in complex calculations involving multiple fractions.