Problem 62 - Fundamentals of Mathematics

Question

For the following problems, find each value. $$\frac{3}{16} \cdot \frac{9}{8} \cdot \frac{6}{5}$$

Step-by-Step Solution

Answer

The value is \( \frac{81}{320} \).

Step 1: Identify the Given Problem

The problem asks us to find the value of the expression \( \frac{3}{16} \cdot \frac{9}{8} \cdot \frac{6}{5} \). This involves multiplying three fractions together.

Step 2: Multiply the Numerators

Start by multiplying the numerators (the top numbers) of each fraction. Compute: \( 3 \times 9 \times 6 = 162 \).

Step 3: Multiply the Denominators

Next, multiply the denominators (the bottom numbers) of each fraction: \( 16 \times 8 \times 5 = 640 \).

Step 4: Form the Fraction

Combine the results from steps 2 and 3 to express the product as a single fraction: \( \frac{162}{640} \).

Step 5: Simplify the Fraction

Simplify \( \frac{162}{640} \) by finding the greatest common divisor (GCD) of 162 and 640, which is 2. Divide both the numerator and denominator by 2: \( \frac{162 \div 2}{640 \div 2} = \frac{81}{320} \). As 81 and 320 have no common factors other than 1, this fraction is in its simplest form.

Key Concepts

Understanding Numerators

In fractions, the *numerator* is the top number that indicates how many parts of a whole we are looking at. It's crucial to handle numerators correctly when multiplying fractions.

• For example, in the fraction \( \frac{3}{16} \), "3" is the numerator, meaning you have 3 parts out of 16.
• To multiply numerators, simply take the numerators of all the fractions involved and multiply them together. So in our problem: \( 3 \times 9 \times 6 = 162 \).
• This step combines the count of parts from each fraction as one comprehensive numerator for the final product.

The result will be the numerator of the fraction you're forming with the denominator, allowing you to proceed with finding the product.

Identifying Denominators

The *denominator* is the bottom number of a fraction and shows the total number of equal parts in the whole or a set. Handling denominators is essential when multiplying fractions.

• In our example, the denominators are 16, 8, and 5. They tell us about the size of each part from each fraction.
• To find the denominator of the product, multiply each of these denominators: \( 16 \times 8 \times 5 = 640 \).
• This process combines the size of the whole across all fractions into a single denominator.

The new denominator is used with the multiplied numerator to form a new fraction representing the product of the original set of fractions.

Simplifying Fractions

After multiplying fractions, you often end up with a fraction that can be simplified. Simplifying fractions involves reducing them to their smallest equivalent form.

• To simplify \( \frac{162}{640} \), we look for any common factors between the numerator and the denominator so we can divide them by these factors until no more common factors exist.
• Begin with the largest possible number that divides both, but any small factor greater than 1 will work for a start.
• In our example, dividing by 2 simplifies the fraction to \( \frac{81}{320} \).

Once there are no other common factors than 1, the fraction is fully simplified, giving you a model in its simplest form.

The Greatest Common Divisor

The *greatest common divisor* (GCD) is the largest number that divides both the numerator and the denominator of a fraction without a remainder. Finding the GCD is key to simplifying fractions efficiently.

• For the fraction \( \frac{162}{640} \), we determine that the GCD is 2.
• When simplifying, divide both numerator and denominator by the GCD: \( 162 \div 2 = 81 \) and \( 640 \div 2 = 320 \).
• This provides a simplified form of the fraction: \( \frac{81}{320} \).

Using the GCD ensures that the fraction is reduced to its simplest and clearest form, making it easier to compare with other fractions or use in further calculations.