Problem 69 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$\frac{18}{14} \cdot \frac{21}{35} \cdot \frac{36}{7}$$

Step-by-Step Solution

Answer

The product is \( \frac{139}{35} \).

Step 1: Identify the Fractions

First, observe that the problem requires multiplying three fractions: \( \frac{18}{14} \), \( \frac{21}{35} \), and \( \frac{36}{7} \).

Step 2: Multiply the Numerators

Multiply the numerators: \( 18 \times 21 \times 36 \). Calculate one pair at a time: \( 18 \times 21 = 378 \), and then \( 378 \times 36 = 13608 \).

Step 3: Multiply the Denominators

Multiply the denominators: \( 14 \times 35 \times 7 \). Calculate one pair at a time: \( 14 \times 35 = 490 \), and then \( 490 \times 7 = 3430 \).

Step 4: Formulate the Combined Fraction

Combine the results from Steps 2 and 3 to give the fraction: \( \frac{13608}{3430} \).

Step 5: Simplify the Fraction

To simplify the fraction, find the greatest common divisor (GCD) of 13608 and 3430. The GCD is 98. Divide both the numerator and the denominator by 98: \( \frac{13608 \div 98}{3430 \div 98} = \frac{139}{35} \).

Step 6: Verify the Simplified Form

Verify the fraction \( \frac{139}{35} \) is in its simplest form by checking that 139 and 35 have no common divisors other than 1.

Key Concepts

Reducing Fractions

When working with fractions, reducing them to their simplest form is an essential step. This process simplifies the fraction by ensuring that the numerator and the denominator have no common factors other than 1.
Reducing fractions makes them easier to understand and work with. It involves dividing both the numerator and the denominator by their greatest common divisor (GCD).
For example, if you have a fraction like \( \frac{20}{28} \), find the GCD of 20 and 28, which is 4. Divide both by 4 to reduce the fraction to \( \frac{5}{7} \). Through these steps, fractions are simplified while keeping their value the same.

Greatest Common Divisor

The greatest common divisor, or GCD, is a key concept in simplifying fractions. It's the largest number that can evenly divide both the numerator and the denominator.
To find the GCD, you can use several methods, such as listing out the factors, using the Euclidean algorithm, or prime factorization of the numbers involved.
Let's take 36 and 24 as an example. The factors of 36 are:

• 1, 2, 3, 4, 6, 9, 12, 18, 36

The factors of 24 are:

• 1, 2, 3, 4, 6, 8, 12, 24

The common factors here are 1, 2, 3, 4, 6, and 12. Among these, the largest is 12.
By dividing both 36 and 24 by 12, you can reduce the fraction \( \frac{36}{24} \) to \( \frac{3}{2} \). Finding the GCD helps in achieving the simplest form effectively.

Multiplying Numerators and Denominators

Multiplying fractions involves a straightforward process of multiplying both the numerators and the denominators.
Consider multiplying two fractions: \( \frac{3}{4} \) and \( \frac{5}{6} \). Start by multiplying the numerators: 3 and 5, which gives 15. Next, multiply the denominators: 4 and 6, resulting in 24.
This gives you the new fraction \( \frac{15}{24} \). It's essential to always check if the resulting fraction can be simplified from this point. Multiply the numerators and denominators:

• Multiply the numerators together.
• Multiply the denominators together.
• Simplify where possible.

This method ensures accurate results for fraction multiplication, while simplifying fractions after multiplication keeps them manageable.