Problem 20 - Elementary and Intermediate Algebra

Question

Translate each ratio into a fraction in simplest form. 11 cans to 121 cans

Step-by-Step Solution

Answer

\( \frac{1}{11} \)

Step 1: Write the Ratio as a Fraction

The ratio given is "11 cans to 121 cans". To convert it to a fraction, we write it as \( \frac{11}{121} \).

Step 2: Simplify the Fraction

To simplify \( \frac{11}{121} \), we need to find the greatest common divisor (GCD) of the numerator and denominator. Since both 11 and 121 are divisible by 11, we divide both by 11. This results in \( \frac{11 \div 11}{121 \div 11} = \frac{1}{11} \).

Key Concepts

Fraction Conversion

When you have a ratio, it's essentially a way of comparing two quantities. To understand it better, we often convert it into a fraction. This means transforming the ratio of two numbers into a division format. For example, the ratio "11 cans to 121 cans" can be rewritten as the fraction \( \frac{11}{121} \). By placing the first number over the second number, you create a fraction. This conversion is crucial for further calculations, like simplifying the fraction, which we will discuss later.

• This step is important; it sets the right foundation for further simplification.
• Always ensure that the terms of your ratio are in the correct order as you set up the fraction.

Converting ratios to fractions is a straightforward process that can help in many mathematical scenarios.

Greatest Common Divisor

To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that can divide both numbers without leaving a remainder. For our fraction \( \frac{11}{121} \), we need to identify the GCD of 11 and 121.
You can find the GCD by listing the factors of both numbers and choosing the greatest one that appears in both lists.

• The factors of 11 are: 1, 11.
• The factors of 121 are: 1, 11, 121.

In this case, the GCD is 11 since it is the largest number that divides evenly into both 11 and 121. Knowing how to find the GCD is a powerful tool that simplifies many math problems.

Simplifying Fractions

Once you have the fraction and the greatest common divisor, it's time to simplify. Simplifying means to make a fraction as simple as possible. This is done by dividing both the numerator and the denominator by the GCD. In our example, the fraction \( \frac{11}{121} \) has a GCD of 11.

• Divide 11 by 11 to simplify the numerator.
• Divide 121 by 11 to simplify the denominator.

This gives us \( \frac{1}{11} \) as the simplest form of the fraction. This is valuable because a simplified fraction is often easier to work with and understand. By learning to simplify fractions, you make your mathematical work more efficient and clear. Remember, the ultimate goal is to make the fraction as simple as possible while keeping it mathematically equivalent to the original ratio.