Problem 8 - Prealgebra

Question

Write each of the following ratios as a fraction in lowest terms. None of the answers should contain decimals. $$36 \quad to\quad 18$$

Step-by-Step Solution

Answer

The ratio 36 to 18 simplifies to 2.

Step 1: Identify the Given Ratio

The given ratio is 36 to 18. In mathematical terms, it can be expressed as a fraction: \( \frac{36}{18} \).

Step 2: Determine the Greatest Common Divisor (GCD)

To simplify the fraction, we need to find the greatest common divisor of the numerator (36) and the denominator (18). Using the Euclidean algorithm: 1. Divide 36 by 18 to get a remainder of 0, meaning 18 divides 36 perfectly. 2. Thus, the GCD of 18 and 36 is 18.

Step 3: Simplify the Fraction

We divide both the numerator and the denominator by the GCD obtained.\[ \frac{36}{18} = \frac{36 \div 18}{18 \div 18} = \frac{2}{1} \] Thus, the simplified fraction is \( \frac{2}{1} \).

Step 4: Express the Fraction in Simplest Terms

Since the denominator of the fraction is 1, it means the value of the fraction is simply 2. Therefore, \( \frac{2}{1} = 2 \).

Key Concepts

Ratios

A ratio is essentially a way to compare two quantities. When you see something like "36 to 18," it's just saying how many times one number contains another. In a ratio, the first number is called the antecedent and the second the consequent. In essence, ratios give us a relationship between two quantities, which can be expressed as a fraction.

• Compares two numbers or amounts.
• Expressed using the word "to." For example, 36 to 18.
• Can be written in fraction form, like \( \frac{36}{18} \).

This fraction form is beneficial because it allows us to simplify ratios easily by reducing the fraction to its lowest terms. When you simplify a ratio, you maintain the proportion between the numbers while making them easier to understand.

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept when it comes to simplifying fractions or ratios. It's the largest number that can divide both numbers in the ratio without leaving any remainder. Knowing the GCD helps us reduce fractions and ratios to their simplest form.

• The biggest number that divides two numbers without leaving a remainder.
• Helps simplify fractions by dividing both the numerator and denominator by the GCD.

For example, when you have the ratio 36 to 18, finding the GCD helps you know that both 36 and 18 can be divided perfectly by 18, which makes simplifying easy. It ensures accuracy and simplicity in mathematical calculations.

Euclidean Algorithm

The Euclidean algorithm is a method used to find the greatest common divisor of two numbers. It's an efficient process and works through a series of division steps. Here's a simple way to understand it:

• Start with two numbers, say 36 (numerator) and 18 (denominator).
• Divide the larger number by the smaller number.
• If you get a remainder of 0, you're done! The smaller number is the GCD.
• If not, repeat the process with the smaller number and the remainder you got.

Through this algorithm, you quickly find that the GCD of 36 and 18 is 18, since 36 divided by 18 leaves no remainder. This means 18 is the biggest number that can divide both 36 and 18 perfectly. Understanding this method helps make simplifying ratios and fractions much more straightforward.