Problem 65 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\frac{3}{7} \cdot \frac{14}{18} \cdot \frac{6}{2}$$

Step-by-Step Solution

Answer

The product simplifies to 1.

Step 1: Convert the Problem into a Single Fraction

Write the product of the fractions as a single fraction by multiplying the numerators together and the denominators together. This will give us: \[\frac{3 \cdot 14 \cdot 6}{7 \cdot 18 \cdot 2}\]

Step 2: Multiply the Numerators and Denominators

Multiply all the numerators together and all the denominators together separately:\[Numerator: 3 \times 14 \times 6 = 252\]\[Denominator: 7 \times 18 \times 2 = 252\]

Step 3: Simplify the Fraction

Now we have the fraction:\[\frac{252}{252}\]This fraction simplifies to:\[\frac{252 \div 252}{252 \div 252} = 1\]

Step 4: Verify the Simplification

Since the numerator and the denominator are the same, the fraction simplifies to 1. We also verified by checking any common factors during the multiplication step, confirming the answer is correct.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential step when working with fractions, whether you're multiplying, adding, or subtracting them. Simplifying means making a fraction as simple as possible. This involves reducing the fraction to its lowest terms. In other words, you want both the numerator (the top number) and the denominator (the bottom number) to be as small as they can be while still having the same value.

Here's how you can simplify a fraction:

• Identify the greatest common factor (GCF) of the numerator and the denominator. The GCF is the largest number that divides both numbers without leaving a remainder.
• Divide both the numerator and the denominator by their GCF.
• Check if the resulting numbers can be simplified further.

In our exercise, we ended up with the fraction \(\frac{252}{252}\). Because the numbers on top and bottom are the same, they divide to 1, which is the simplest form.

Numerator and Denominator

Understanding numerators and denominators is crucial when dealing with fractions. A fraction consists of two numbers separated by a horizontal line. The number above the line is the numerator, and it tells you how many parts you have. The number below the line is the denominator, which tells you how many parts make up a whole.

For instance, in the fraction \(\frac{3}{7}\), 3 is the numerator and 7 is the denominator. You have 3 parts out of a total of 7.

Here's how they work together in multiplication:

• The numerators are multiplied together to form the new numerator.
• The denominators are multiplied together to form the new denominator.

This process turns the separate fractions into one single fraction that you can then simplify, as demonstrated in the exercise.

Fraction Multiplication Steps

Multiplying fractions might seem tricky at first, but it's quite straightforward once you break it down into steps. Here's a simple guide on how to multiply fractions:

• First, simply multiply all the numerators (the numbers on top of the fractions) together to get a new numerator.
• Next, multiply all the denominators (the numbers at the bottom) together to create a new denominator.
• After that, you have a single fraction that may need to be simplified.
• Lastly, simplify the fraction by finding any common factors between the new numerator and denominator, as we did in our exercise where it simplified to 1.

This method allows you to go from multiple fractions to one easy-to-manage fraction which reflects the total product. Each step builds on the previous one, making it simple to follow and ensuring you end with the correct result.