Problem 75 - Prealgebra

Question

Multiply or divide as indicated. \(\frac{11}{8} \cdot \frac{29}{8}\)

Step-by-Step Solution

Answer

The product is \(\frac{319}{64}\).

Step 1: Multiply the Numerators

To multiply two fractions, we start by multiplying the numerators. For the fractions \(\frac{11}{8}\) and \(\frac{29}{8}\), multiply the numerators: \(11 \times 29 = 319\). This will be the numerator of our result.

Step 2: Multiply the Denominators

Next, multiply the denominators of the fractions. For \(\frac{11}{8}\) and \(\frac{29}{8}\), multiply the denominators: \(8 \times 8 = 64\). This will be the denominator of our result.

Step 3: Write the Result

After multiplying the numerators and denominators, we write the result as a single fraction. The product of \(\frac{11}{8} \cdot \frac{29}{8}\) is \(\frac{319}{64}\).

Step 4: Verify Simplification

Check if the fraction \(\frac{319}{64}\) can be simplified. Since 319 and 64 have no common divisors other than 1, the fraction is already in its simplest form.

Key Concepts

Understanding Numerators

The numerator of a fraction is the top number in a fraction. It tells you how many parts of a whole are being considered. In the fraction \(\frac{11}{8}\), 11 is the numerator. When multiplying fractions, you focus on the numerators first.To multiply the numerators, simply multiply them together as you would with whole numbers. For example, with \(\frac{11}{8} \cdot \frac{29}{8}\), you multiply 11 by 29. This gives you 319, which becomes the new numerator for the resulting fraction. Remember, the numerator acts like a counter indicating how much of the whole you have. In complex fractions, the same rule applies irrespective of the size of the numbers involved. Always deal with numerators first before moving to the bottom numbers, which are known as denominators.

Exploring Denominators

A denominator is the bottom part of a fraction. It shows how many equal parts the whole is divided into. For instance, in the fraction \(\frac{11}{8}\), the number 8 represents the denominator.When multiplying fractions, once you've dealt with the numerators, you should move to multiplying the denominators. This is similar to handling numerators: just multiply them as you would ordinary numbers. For our example, \(8 \times 8\) gives 64.The denominator defines the size of each part in the whole. When dealing with fractions, larger denominators mean each part is smaller, while smaller denominators indicate larger parts. Therefore, when forming a new fraction after multiplying, always make sure you calculate this part accurately.

The Art of Simplifying Fractions

Simplifying fractions means reducing them to their simplest form so they're easier to understand at a glance. This involves ensuring both the numerator and denominator share no common factors, except the number 1. To simplify, divide both parts by their greatest common divisor (GCD). In our multiplication result \(\frac{319}{64}\), check for common divisors. Here, 319 and 64 have no shared divisors other than 1, meaning the fraction is already simplified.Simplification is important as it helps in easily comparing and understanding fractions. Always check after performing operations like multiplication if further reduction is possible. However, if no common factors exist, as with our example, feel confident it’s already as simple as can be.