Problem 13 - Prealgebra

Question

Express each ratio as a fraction in simplest form. 6 ladybugs out of 27 insects

Step-by-Step Solution

Answer

The ratio is expressed as \( \frac{2}{9} \) in simplest form.

Step 1: Write Ratio as a Fraction

To express the ratio of ladybugs to insects in fraction form, write the number of ladybugs over the total number of insects as a fraction: \[ \frac{6}{27} \].

Step 2: Find the Greatest Common Divisor (GCD)

Determine the greatest common divisor (GCD) of the numerator and the denominator. For 6 and 27, the GCD is 3, as 3 is the largest number that evenly divides both 6 and 27.

Step 3: Simplify the Fraction

Divide both the numerator and the denominator by their GCD to simplify the fraction.\[ \frac{6 \div 3}{27 \div 3} = \frac{2}{9} \].

Step 4: Verify the Simplified Fraction

Check that the fraction \( \frac{2}{9} \) is in simplest form by ensuring that the numerator and denominator have no common divisors other than 1. Since 2 and 9 are coprime, \( \frac{2}{9} \) is indeed simplified.

Key Concepts

Fractions

Fractions are ways to represent parts of a whole. They are written with two numbers—one on top of the other—separated by a line. The number on top is called the numerator. It shows how many parts we have. The number below is the denominator. It indicates into how many equal parts the whole is divided.
For example, in the fraction \( \frac{6}{27} \), the numerator, 6, tells us there are 6 ladybugs, and the denominator, 27, tells us the total number of insects. Fractions can express ratios, where they depict the relative sizes of two quantities. A fraction like \( \frac{6}{27} \) can represent how one type of insect compares to the total number.

GCD (Greatest Common Divisor)

The Greatest Common Divisor, or GCD, is the largest number that divides two numbers without leaving a remainder. Finding the GCD is a crucial step in simplifying fractions, as it helps reduce the fraction to its simplest form.

• To determine the GCD of 6 and 27, list their divisors.
• Divisors of 6 are: 1, 2, 3, 6
• Divisors of 27 are: 1, 3, 9, 27

The numbers that are common divisors are 1 and 3. Among these, the largest number is 3. Thus, the GCD of 6 and 27 is 3. Using this GCD allows us to simplify fractions like \( \frac{6}{27} \) more easily.

Simplifying Fractions

Simplifying fractions means reducing them to their smallest possible forms. This is done by dividing both the numerator and denominator by their greatest common divisor (GCD). Simplifying makes fractions easier to understand and compare.
To simplify \( \frac{6}{27} \):

• First, find the GCD of 6 and 27, which is 3.
• Divide both the numerator (6) and denominator (27) by this GCD.
• \[ \frac{6 \div 3}{27 \div 3} = \frac{2}{9} \]

The simplified fraction is \( \frac{2}{9} \). In this form, the numerator and denominator have no other common divisors except 1, ensuring that this fraction cannot be simplified further. This simplification confirms that the fraction is in its simplest form.