Problem 49 - Prealgebra

Question

Find the product of \(\frac{8}{15}\) and \(\frac{8}{35}\).

Step-by-Step Solution

Answer

The product is \(\frac{64}{525}\).

Step 1: Multiply the Numerators

To find the product of two fractions, begin by multiplying the numerators (the top numbers) together. For our fractions \(\frac{8}{15}\) and \(\frac{8}{35}\), the numerators are both 8. \(8 \times 8 = 64\).

Step 2: Multiply the Denominators

Next, multiply the denominators (the bottom numbers) of the fractions. The denominators we have are 15 and 35. \(15 \times 35 = 525\).

Step 3: Write the Fraction

Combine the results from Step 1 and Step 2 to write the new fraction as the product of the numerators over the product of the denominators: \(\frac{64}{525}\).

Step 4: Simplify the Fraction

Check if the fraction \(\frac{64}{525}\) can be simplified. Find the greatest common divisor (GCD) of 64 and 525. Since they have no common factors other than 1, \(\frac{64}{525}\) is already in its simplest form.

Key Concepts

Understanding Numerators

When we talk about numerators in fractions, we're focusing on the top part of the fraction. In the fraction \(\frac{a}{b}\), \(a\) is the numerator. It represents how many parts of the whole we're dealing with. For instance, in \(\frac{3}{4}\), the numerator 3 indicates that we have 3 parts out of a total of 4.Multiplying numerators is straightforward:

• Place the numerators next to each other in a multiplication expression.
• For example, with \(\frac{8}{15}\) and \(\frac{8}{35}\), we multiply 8 by 8 to get 64.
• This product forms the numerator of the resulting fraction.

Keeping numerators clear helps manage fraction multiplication and ensures accuracy.

Exploring Denominators

Denominators are the numbers under the line in a fraction, representing the total number of equal parts the whole is divided into. They are key in helping to determine the size of each part. In the fraction \(\frac{a}{b}\), \(b\) is the denominator.Here's how to handle them in multiplication:

• Line up the denominators for both fractions in a multiplication statement.
• Take our example of \(\frac{8}{15}\) and \(\frac{8}{35}\), where 15 and 35 are the denominators.
• Multiply 15 by 35, which equals 525, forming the new denominator.

This process ensures that you account for both fractions' total parts and manage the new fraction effectively.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator have no common divisors other than 1. This makes fractions easier to interpret and compare.To simplify \(\frac{64}{525}\):

• Identify if there's a greatest common divisor (GCD) between 64 and 525.
• In this case, the only common factor is 1, making \(\frac{64}{525}\) already in its simplest form.
• If there were a GCD greater than 1, divide both the numerator and denominator by this common factor.

Simplifying ensures fractions are concise and clear, aiding in better understanding and usage.