Problem 26 - Prealgebra

Question

Multiply each of the following. Be sure all answers are written in lowest terms. $$\frac{3}{4} \cdot 20$$

Step-by-Step Solution

Answer

15

Step 1: Convert Whole Number to Fraction

Start by writing the whole number 20 as a fraction. You do this by placing it over 1, giving us \( \frac{20}{1} \).

Step 2: Multiply the Fractions

Multiply the numerators together and the denominators together. The fraction multiplication formula is \( \frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \). Applying this to our fractions, we get \( \frac{3}{4} \times \frac{20}{1} = \frac{3 \cdot 20}{4 \cdot 1} = \frac{60}{4} \).

Step 3: Simplify the Fraction

To simplify \( \frac{60}{4} \), divide both the numerator and the denominator by their greatest common divisor, which is 4. \( \frac{60 \div 4}{4 \div 4} = \frac{15}{1} \).

Step 4: Express the Answer

Since \( \frac{15}{1} \) simplifies further to the whole number 15, the final answer is 15.

Key Concepts

Simplifying Fractions

Simplifying a fraction means making the fraction as simple as possible. This usually involves reducing the fraction so that the numerator and the denominator are as small as possible. When a fraction is simplified, it becomes easier to understand and compare with other fractions. To simplify a fraction, first determine the greatest common divisor (GCD) of the numerator and the denominator. This is the largest number that can evenly divide both numbers. Once the GCD is known, divide both the numerator and the denominator by it. For example, in the fraction \( \frac{60}{4} \), the greatest common divisor is 4. Divide both 60 and 4 by their GCD, giving us:

• \( 60 \div 4 = 15 \)
• \( 4 \div 4 = 1 \)

So, \( \frac{60}{4} \) simplifies to \( \frac{15}{1} \), or just 15 when expressed as a whole number. Simplifying makes fractions easier to visualize and work with, especially in complex mathematical operations.

Converting Whole Numbers to Fractions

When working with fractions, sometimes you'll need to multiply by whole numbers. Since operations like multiplication are often performed on fractions, we convert whole numbers to fractions by placing them over 1. This doesn’t change the value of the number as anything divided by 1 is itself. For example, the whole number 20 can be expressed as a fraction: \( \frac{20}{1} \).

• Numerator: 20
• Denominator: 1

With this conversion, we can easily perform operations with other fractions. This method is particularly useful for maintaining the format necessary when multiplying by fractions, allowing you to apply consistent rules of arithmetic. By converting whole numbers to fractions, you simplify the process of multiplication, addition, subtraction, or division of mixed fractions and whole numbers.

Greatest Common Divisor

The greatest common divisor, or GCD, is an essential concept when working with fractions. It allows you to simplify fractions so they are easier to manage and understand. The GCD of two numbers is the largest number that divides both without leaving a remainder. Finding the GCD starts with identifying the common divisors of the numerator and denominator. From the common divisors, choose the largest one. For example, let’s find the GCD of 60 and 4:

• The divisors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
• The divisors of 4: 1, 2, 4
• The common divisors: 1, 2, 4

Here, the greatest common divisor is 4. Using the GCD is crucial in simplifying fractions, ensuring that you always present the fraction in its most reduced form, allowing for simpler calculations and clearer understanding.