Problem 35 - Fundamentals of Mathematics

Question

For the following problems, find each part without using a diagram. $$\frac{3}{5}\( of \)\frac{5}{12}$$

Step-by-Step Solution

Answer

The product is \( \frac{1}{4} \).

Step 1: Understand the Problem

The task is to find the product of two fractions: \( \frac{3}{5} \) and \( \frac{5}{12} \). This requires multiplying these two fractions.

Step 2: Multiply the Fractions

To multiply two fractions, multiply their numerators together and their denominators together. This means calculating: \[ \frac{3}{5} \times \frac{5}{12} = \frac{3 \times 5}{5 \times 12}. \]

Step 3: Simplify the Product

First, calculate the multiplication: \( 3 \times 5 = 15 \) and \( 5 \times 12 = 60 \). Hence, \( \frac{3 \times 5}{5 \times 12} = \frac{15}{60} \). Next, simplify this fraction by finding the greatest common divisor (GCD) of 15 and 60, which is 15. Divide both the numerator and the denominator by 15: \( \frac{15 \div 15}{60 \div 15} = \frac{1}{4} \).

Key Concepts

Fraction Simplification

When working with fractions, it's really important to be able to simplify them. Simplifying a fraction means making it as simple as possible without changing its value. You do this by dividing the numerator (the top number) and the denominator (the bottom number) of the fraction by the same number. Always try to find the largest number that can divide both the numerator and the denominator evenly.

This process makes the fraction smaller and easier to understand. It's like reducing it to its "bare bones". For example, if you end up with a fraction like \( \frac{15}{60} \), you simplify it by dividing both the top and bottom by their greatest common divisor, which is 15. After dividing, you get \( \frac{1}{4} \). Now, the fraction is fully simplified, showing you the simplest form of the division.

Greatest Common Divisor

The greatest common divisor (GCD) is a powerful tool used in simplifying fractions. It’s about finding the largest number that can perfectly divide two or more numbers. When simplifying fractions, the GCD helps find the biggest number by which you can divide both the numerator and the denominator without leaving a remainder.

In the fraction \( \frac{15}{60} \), you find that 15 is the greatest common divisor of both the numerator and the denominator. Knowing this, you can divide 15 by 15 and 60 by 15 to get the simpler form \( \frac{1}{4} \).

• The GCD makes it easy to simplify fractions quickly and efficiently.
• Helps avoid large and unwieldy numbers when working with fractional equations.
• Once fractions are simplified, their understanding becomes straightforward.

Mastering how to find the GCD not only makes fraction problems easier but also aids in solving more complex mathematical equations.

Numerator and Denominator Multiplication

To multiply fractions, the process involves straightforward multiplication of numerators and denominators. For example, if you're asked to find the product of \( \frac{3}{5} \) and \( \frac{5}{12} \), you multiply the numerators together and then the denominators.

This process would look like: \[ \frac{3}{5} \times \frac{5}{12} = \frac{3 \times 5}{5 \times 12}. \]
After calculating, it results in \( \frac{15}{60} \). Although the numbers have changed, the method remains simple:

• Multiply across the numerators to get a new numerator.
• Multiply across the denominators to get a new denominator.
• This gives the product of the fractions before simplification.

Understanding this basic method allows you to solve multiplication of fractions effectively. Remember, after multiplication, simplification helps arrive at the most straightforward answer.