Problem 48 - Fundamentals of Mathematics

Question

For the following problems, find each part without using a diagram. $$\frac{8}{9}\( of \)\frac{3}{4}\( of \)\frac{2}{3}$$

Step-by-Step Solution

Answer

The product is \( \frac{4}{9} \).

Step 1: Understand the problem

The problem asks us to find the product of three fractions: \( \frac{8}{9} \), \( \frac{3}{4} \), and \( \frac{2}{3} \). We need to multiply these fractions together.

Step 2: Multiply the numerators

Multiply the numerators of the fractions together. For \( \frac{8}{9} \cdot \frac{3}{4} \cdot \frac{2}{3} \), the numerators are 8, 3, and 2. So, compute: \( 8 \times 3 \times 2 = 48 \).

Step 3: Multiply the denominators

Multiply the denominators of the fractions together. For \( \frac{8}{9} \cdot \frac{3}{4} \cdot \frac{2}{3} \), the denominators are 9, 4, and 3. So, compute: \( 9 \times 4 \times 3 = 108 \).

Step 4: Form the new fraction

Combine the products of the numerators and denominators to form a new fraction: \( \frac{48}{108} \).

Step 5: Simplify the fraction

Simplify \( \frac{48}{108} \) by finding the greatest common divisor (GCD) of 48 and 108, which is 12. Divide both the numerator and denominator by 12: \( \frac{48 \div 12}{108 \div 12} = \frac{4}{9} \).

Key Concepts

Numerators

In fraction multiplication, the numerator plays an essential role. The numerator is the top number in a fraction and represents how many parts of a whole are being considered.
For example, in the fraction \( \frac{8}{9} \), "8" is the numerator. When multiplying fractions, you multiply all the numerators together to find the new numerator of the resulting fraction.
Let's consider an example from our original problem: \( \frac{8}{9} \cdot \frac{3}{4} \cdot \frac{2}{3} \). Here, the numerators are "8", "3", and "2".
You multiply these numbers like so: \( 8 \times 3 \times 2 = 48 \). This is how you calculate the numerator of your resulting fraction after multiplication.

Denominators

Denominators are the numbers at the bottom of a fraction representing the total number of parts that make up a whole. They tell us into how many equal parts the whole is divided.
In our fractions from the problem include: \( \frac{8}{9} \), \( \frac{3}{4} \), and \( \frac{2}{3} \). The denominators are "9", "4", and "3" respectively. When multiplying fractions, you apply the same multiplication rule to the denominators to find the new denominator.
Multiply them as follows: \( 9 \times 4 \times 3 = 108 \). This is the denominator of the resulting fraction after multiplying the original fractions.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. It involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
The purpose is to make the fraction easier to understand without changing its value.

For example, after multiplying to form \( \frac{48}{108} \), you need to simplify it.
First, find the GCD of 48 and 108. This is the largest number that divides both 48 and 108 evenly.
In this case, it's "12".
Divide both the numerator and denominator by "12": \( \frac{48 \div 12}{108 \div 12} = \frac{4}{9} \). This gives us the simplified fraction.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor, is crucial when simplifying fractions.
It is the largest positive number that divides two or more numbers without leaving a remainder.
To find the GCD of two numbers, you can list the positive factors of each and choose the largest factor they share.

For example, the factors of 48 are:

• 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

The factors of 108 are:

• 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108

The largest factor common to both is "12".
This GCD allows you to simplify the fraction \( \frac{48}{108} \) to its simplest form, \( \frac{4}{9} \).
Finding the GCD efficiently helps in reducing fractions to their simplest form easily.