Problem 35 - Algebra and Trigonometry

Question

Perform the indicated operations. \(\left(3+\frac{1}{4}\right)\left(1-\frac{4}{5}\right)\)

Step-by-Step Solution

Answer

The result is \(\frac{13}{20}\).

Step 1: Convert Mixed Number to Improper Fraction

The mixed number \(3 + \frac{1}{4}\) can be changed into an improper fraction. Multiply the whole number 3 by 4 (the denominator of the fraction \(\frac{1}{4}\)), and then add the numerator of \(1\) to get \(\frac{13}{4}\).

Step 2: Simplify the Other Fraction

Simplify \(1 - \frac{4}{5}\) by expressing 1 as a fraction with the same denominator: \(\frac{5}{5}\) and subtracting: \(\frac{5}{5} - \frac{4}{5} = \frac{1}{5}\).

Step 3: Perform Multiplication of Fractions

Now multiply the fractions \(\frac{13}{4}\) and \(\frac{1}{5}\). Multiply the numerators: \(13 \cdot 1 = 13\), and the denominators: \(4 \cdot 5 = 20\). This gives us \(\frac{13}{20}\).

Step 4: Simplify the Result, if Possible

Check if \(\frac{13}{20}\) can be simplified. Since 13 and 20 have no common factors besides 1, \(\frac{13}{20}\) is already in its simplest form.

Key Concepts

Mixed Numbers Conversion

A mixed number is a combination of a whole number and a fraction. Sometimes in math, we need to convert these mixed numbers into improper fractions to make calculations easier. An improper fraction has a numerator larger than or equal to its denominator.

Let's see how the conversion works. Imagine a mixed number like \(3 + \frac{1}{4}\). To convert it:

• First, multiply the whole number (3) by the denominator of the fraction (4). So, \(3 \times 4 = 12\).
• Next, add the numerator of the fraction (1) to this result: \(12 + 1 = 13\).
• This gives us the improper fraction \(\frac{13}{4}\).

This process makes calculations, like multiplication, simpler because working with improper fractions avoids extra steps later on.

Subtracting Fractions

Subtracting fractions involves working with fractions that share the same denominator. If they don't, you'll need to adjust them first. By converting whole numbers to fractions, such as "1" into a fraction like \(\frac{5}{5}\), we easily facilitate subtraction.

Let's try an example: subtract \(\frac{4}{5}\) from a whole number 1:

• Represent 1 as a fraction with a denominator of 5: \(\frac{5}{5}\).
• Subtract the fraction \(\frac{4}{5}\): \(\frac{5}{5} - \frac{4}{5} = \frac{1}{5}\).

By having the same denominator, the subtraction becomes a straightforward process of taking one numerator from the other without changing the denominator.

Fraction Simplification

After performing operations like multiplication or addition on fractions, it is often necessary to simplify the result. Simplification means reducing the fraction to its simplest form, where the numerator and denominator have no common factors other than 1.

Consider the fraction \(\frac{13}{20}\) obtained after multiplying fractions. To check for simplification, follow these steps:

• Identify the greatest common divisor (GCD) of the numerator and denominator. For 13 and 20, the only common factor is 1.
• Since we cannot divide both by any number other than 1, \(\frac{13}{20}\) is already in its simplest form.

Simplifying fractions is vital because it ensures results are easier to interpret and compare with other numbers.