Problem 24 - Algebra and Trigonometry

Question

\(21-26=\) Perform the indicated operations. \(\begin{array}{ll}{\text { (a) }\left(3+\frac{1}{4}\right)\left(1-\frac{4}{5}\right)} & {\text { (b) }\left(\frac{1}{2}-\frac{1}{3}\right)\left(\frac{1}{2}+\frac{1}{3}\right)}\end{array}\)

Step-by-Step Solution

Answer

(a) \(\frac{13}{20}\); (b) \(\frac{5}{36}\).

Step 1: Calculate Fraction Addition in Part (a)

Add the whole number 3 to the fraction \( \frac{1}{4} \). The fraction form of 3 is \( \frac{12}{4} \), so the addition becomes \( \frac{12}{4} + \frac{1}{4} = \frac{13}{4} \).

Step 2: Calculate Fraction Subtraction in Part (a)

Subtract \( \frac{4}{5} \) from 1 in the expression \( 1 - \frac{4}{5} \). This can be rewritten as \( \frac{5}{5} - \frac{4}{5} = \frac{1}{5} \).

Step 3: Multiply Results from Part (a)

Multiply the results obtained in Steps 1 and 2: \( \frac{13}{4} \times \frac{1}{5} = \frac{13}{20} \).

Step 4: Calculate Fraction Subtraction in Part (b)

Subtract \( \frac{1}{3} \) from \( \frac{1}{2} \). To do this, make the denominators the same: \( \frac{1}{2} = \frac{3}{6} \) and \( \frac{1}{3} = \frac{2}{6} \). Thus, \( \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \).

Step 5: Calculate Fraction Addition in Part (b)

Add \( \frac{1}{2} \) to \( \frac{1}{3} \). Rewrite with a common denominator: \( \frac{1}{2} = \frac{3}{6} \) and \( \frac{1}{3} = \frac{2}{6} \). So, \( \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \).

Step 6: Multiply Results from Part (b)

Multiply the results obtained in Steps 4 and 5: \( \frac{1}{6} \times \frac{5}{6} = \frac{5}{36} \).

Key Concepts

Fraction Addition

Fraction addition might seem a bit daunting at first, but it's quite straightforward once you get the hang of finding a common denominator. Let's dive deeper into how this works using our example.
First, if you are adding two fractions, you need to make sure they have the same denominator. Denominators are the bottom numbers in fractions. For example, when adding \( \frac{1}{2} \) and \( \frac{1}{3} \), convert them to equivalent fractions with a common denominator.
Here, the least common denominator of 2 and 3 is 6. So, we convert:

• \( \frac{1}{2} = \frac{3}{6} \)
• \( \frac{1}{3} = \frac{2}{6} \)

After converting, simply add the numerators and keep the same denominator: \[ \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \]
This is how you perform fraction addition—always remember to find a common denominator first.

Fraction Subtraction

Just like with addition, subtracting fractions requires a common denominator. It's essential to ensure the denominators are the same so you can subtract the numerators straightforwardly. Let's simplify this with an example from our exercise.
In subtracting \( \frac{1}{3} \) from \( \frac{1}{2} \), find the least common denominator, which is 6 in this case. Convert the fractions:

• \( \frac{1}{2} = \frac{3}{6} \)
• \( \frac{1}{3} = \frac{2}{6} \)

Once converted, subtract the numerators: \[ \frac{3}{6} - \frac{2}{6} = \frac{1}{6} \]
Remember, the denominator remains unchanged, just like in addition. This method ensures that subtraction is performed smoothly and accurately.

Fraction Multiplication

Fraction multiplication is a bit different from addition or subtraction. It does not require a common denominator, making it simpler in this way. Let's look at how you would approach multiplying fractions using our example.
When you multiply fractions, multiply the numerators together and then the denominators. Here's how it works: if you have \( \frac{13}{4} \) and \( \frac{1}{5} \), multiply them as follows:

• Numerators: \( 13 \times 1 = 13 \)
• Denominators: \( 4 \times 5 = 20 \)

This results in \( \frac{13}{20} \).
If numbers allow, always check if you can simplify the fraction further. For example, \( \frac{5}{10} \) can be simplified to \( \frac{1}{2} \). However, in our case, \( \frac{13}{20} \) is already in its simplest form.
With multiplication, simplicity is key—just multiply across numerators and denominators, and simplify if possible.