Problem 93 - Fundamentals of Mathematics

Question

Perform each multiplication and division. $$ 20 \cdot \frac{18}{4} $$

Step-by-Step Solution

Answer

The result is 90.

Step 1: Simplify the Fraction

First, simplify the fraction \( \frac{18}{4} \) by dividing both the numerator and the denominator by their greatest common divisor. The greatest common divisor of 18 and 4 is 2. So, \( \frac{18}{4} = \frac{18 \div 2}{4 \div 2} = \frac{9}{2} \).

Step 2: Multiply

Now, multiply 20 by \( \frac{9}{2} \). This can be written as: \[ 20 \cdot \frac{9}{2} = \frac{20 \cdot 9}{2} \]Calculate the multiplication of the numerator: \( 20 \cdot 9 = 180 \)

Step 3: Simplify the Result

Finally, divide the result by the denominator 2:\[ \frac{180}{2} = 90 \]

Key Concepts

Fraction Simplification

Fraction simplification is the process of reducing fractions to their simplest form. This is done by dividing the numerator and the denominator by their greatest common divisor (GCD).
If both numbers can be divided by the same number, you will get a simpler fraction equivalent to the original.

• Why Simplify Fractions? Simplifying makes calculations easier and fractions more understandable. It gives a cleaner, clearer representation.
• Example: Consider the fraction \( \frac{18}{4} \). Simplifying it involves dividing both the top and bottom by their GCD, which is 2: \( \frac{18 \div 2}{4 \div 2} = \frac{9}{2} \). This is now in its simplest form.

Greatest Common Divisor

The greatest common divisor is a number that divides two or more numbers without leaving a remainder, and is the highest such number. Finding the GCD is crucial for simplifying fractions.

• How to Find the GCD: List all the factors of each number, and find the largest number they share.
• Example: For 18 and 4:
  • Factors of 18: 1, 2, 3, 6, 9, 18
  • Factors of 4: 1, 2, 4
  • The greatest shared factor is 2, making 2 the GCD.

Numerator

A numerator is the top number in a fraction. It shows how many parts of a whole or a set we are considering. Understanding the numerator is important, especially when modifying or using fractions in operations.

• Role of the Numerator: When multiplying, add or subtract, the numerator directly affects the size or value of the fraction.
• Example in Multiplication: In the operation \( 20 \cdot \frac{9}{2} \), the numerator 9 is multiplied by 20, resulting in 180. This gives the number of parts represented in the product.

Denominator

The denominator is the bottom part of a fraction. It tells us how many equal parts make up a whole. A clear understanding of the denominator helps in defining the size or scope of each fractional part.

• Function of the Denominator: It helps standardize the value of the fraction and is critical in operations like division or finding equivalent fractions.
• Example During Simplification: In \( \frac{9}{2} \), the denominator 2 indicates that the whole is divided into two equal parts. When multiplying \( 20 \cdot \frac{9}{2} \), after finding \( 20 \cdot 9 = 180 \), we divide by the denominator 2, resulting in 90.