Problem 57 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\frac{3}{8} \cdot \frac{8}{9}$$

Step-by-Step Solution

Answer

The product is \( \frac{1}{3} \).

Step 1: Multiply the Numerators

In order to find the product of two fractions, multiply the numerators (the numbers on top). Multiply 3 and 8 to get:\[3 imes 8 = 24\]

Step 2: Multiply the Denominators

Next, multiply the denominators (the numbers on the bottom). Multiply 8 and 9 to get:\[8 imes 9 = 72\]

Step 3: Write the Product as a Fraction

Combine the results from Steps 1 and 2 to write the product as a fraction:\[\frac{24}{72}\]

Step 4: Simplify the Fraction

To reduce the fraction \(\frac{24}{72}\), find the greatest common divisor (GCD) of 24 and 72, which is 24. Divide both the numerator and the denominator by their GCD:\[\frac{24 \div 24}{72 \div 24} = \frac{1}{3}\]

Key Concepts

Simplifying Fractions

Simplifying fractions is the process of making the fraction as simple as possible. This often involves making the numbers as small as they can be, while still keeping the same value.

When you simplify a fraction, you look for a number that divides both the numerator (top number) and the denominator (bottom number) without leaving a remainder. This number is known as the greatest common divisor (GCD). Once you find this common number, you divide both the numerator and denominator by it.

For example, in the fraction \(\frac{24}{72}\), both 24 and 72 can be divided by 24, which is their GCD. So, dividing both by 24 simplifies the fraction to \(\frac{1}{3}\). This process ensures you have the simplest form of the fraction.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator evenly, meaning without leaving any remainder. It is a crucial part of simplifying fractions.

When you want to simplify a fraction, finding the GCD helps reduce the numbers involved. To find the GCD, list out all the divisors of both the numerator and the denominator. The largest number that appears in both lists is the GCD.

For instance, for the numbers 24 and 72:

• The divisors of 24 are: 1, 2, 3, 4, 6, 8, 12, 24.
• The divisors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.

The number 24 is the largest that appears in both lists, making it the GCD.

Numerators and Denominators

Understanding numerators and denominators is fundamental when working with fractions.

The numerator is the top number of a fraction and indicates how many parts of a whole we have. In the fraction \(\frac{3}{8}\), 3 is the numerator, suggesting we have 3 parts out of 8 equal parts of a whole.

The denominator is the bottom number and states the total number of equal parts the whole is divided into. So, in \(\frac{3}{8}\), the denominator is 8, meaning the whole is split into 8 parts.

In multiplication of fractions, both numerators are multiplied together as are both denominators. For \(\frac{3}{8} \cdot \frac{8}{9}\), the numerators 3 and 8 multiply to become 24, while the denominators 8 and 9 make 72, which gives \(\frac{24}{72}\) as an intermediate step before simplification.