Problem 33 - Algebra A Combined Function

Question

Multiply or divide as indicated. See Examples 11 through 14 and \(21 .\) $$ \frac{1}{2} \cdot \frac{3}{4} $$

Step-by-Step Solution

Answer

The product is \(\frac{3}{8}\).

Step 1: Multiply the numerators

Start by multiplying the numerators of the two fractions. In this case, multiply the numerator of the first fraction (1) by the numerator of the second fraction (3). This gives us: \(1 \times 3 = 3\).

Step 2: Multiply the denominators

Next, multiply the denominators of the two fractions. Multiply the denominator of the first fraction (2) by the denominator of the second fraction (4).This gives us: \(2 \times 4 = 8\).

Step 3: Form the new fraction

With the result of the products from the previous steps, form a new fraction with the product of the numerators over the product of the denominators: \(\frac{3}{8}\).

Step 4: Simplify, if necessary

Check if the fraction \(\frac{3}{8}\) can be simplified. In this case, 3 and 8 have no common factors other than 1, so the fraction is already in its simplest form.

Key Concepts

Understanding the Numerator

The numerator is the top part of a fraction. When multiplying fractions, you start with the numerators. The numerator of the first fraction in our example is 1, while that of the second fraction is 3.

• The operation involves multiplying these two numbers to get a new numerator.
• In this case, the calculation is quite simple: \(1 \times 3 = 3\).

Think of the numerator as part of each whole that you are counting. It's the number of pieces you have compared to the total possible as defined by the denominator. In fraction multiplication, these pieces are simply multiplied together without altering their identity.

Demystifying the Denominator

The denominator is the bottom part of a fraction. It tells us into how many equal parts the whole is divided. When multiplying fractions, you do the same with the denominators as you do with the numerators.

• In our example, you multiply the two denominators, 2 and 4.
• This gives you \(2 \times 4 = 8\).

With these two steps, you've formed a new fraction! The denominator reflects the total possible pieces when you combine the wholes. While both numerators and denominators must be multiplied together, they each have specific roles in the structure of the fraction.

Simplifying Fractions

Simplifying fractions means reducing them to their most basic form where the numerator and the denominator have no common factors other than 1. Simplification is the final step after you're done multiplying both parts of the fraction.

• Check if both numbers can be divided by the same integer to make them smaller.
• For the fraction \(\frac{3}{8}\), 3 and 8 have no common factors, which means it's already in its simplest form.

While simplification can make fractions easier to interpret and work with, always remember it's only possible when the numerator and the denominator share common divisors. After simplifying, you should arrive with the fraction that accurately represents the same portion of the whole, only with smaller numbers.