Problem 22 - Prealgebra

Question

Express each ratio as a fraction in simplest form. \(\$ 0.99\) for 10 pencils

Step-by-Step Solution

Answer

The ratio as a fraction is \(\frac{99}{10}\).

Step 1: Convert Dollars to Cents

First, let's convert the dollar amount into cents to make the arithmetic straightforward. Since 1 dollar equals 100 cents, \($0.99\) is equivalent to 99 cents.

Step 2: Set Up the Ratio

The ratio of cost to pencils is \(\frac{99}{10}\), where 99 is the cent amount, and 10 is the number of pencils.

Step 3: Find the Greatest Common Divisor (GCD)

Identify the greatest common divisor of 99 and 10. The number 99 is divisible by 1, 3, 9, 11, 33, and 99, while 10 is divisible by 1, 2, 5, and 10. The GCD is 1.

Step 4: Simplify the Fraction

Since the GCD of 99 and 10 is 1, the fraction \(\frac{99}{10}\) is already in its simplest form.

Key Concepts

Fractions

Fractions are a way to represent a part of a whole or a division of quantities. When we deal with quantities like a price per item, we can express the relationship as a fraction to better understand it. For example, if you have $0.99 for 10 pencils, converting this amount into cents and dividing by the number of pencils gives us the fraction \(\frac{99}{10}\). This fraction shows that for every 10 pencils, you are dealing with 99 cents.
Fractions are helpful when comparing parts to a whole, especially in real-world situations like this one. In this context, the fraction form makes it clear how much you pay per group of items, enabling easier calculations and comparisons.

Simplification

Simplification is the process of reducing fractions to their simplest form. A fraction is simplified when you cannot divide its numerator and denominator by any common number other than 1. This reduction makes fractions easier to understand and work with.
For instance, consider the fraction \(\frac{99}{10}\). To simplify, you'd find any common divisors for the numerator (99) and the denominator (10). In this case, the greatest number you can divide both by is 1. Hence, the fraction is already in its simplest form. Simplification helps in further reducing complexity in calculations and ensures clarity when fractions are compared to each other.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is the highest number that can evenly divide two numbers. It plays a crucial role in simplifying fractions as it helps determine how much a fraction can be reduced.
To find the GCD, you list all the divisors of each number and choose the largest one common to both. For example, the divisors of 99 are 1, 3, 9, 11, 33, and 99, while those of 10 are 1, 2, 5, and 10. Here, the only common divisor is 1, making the GCD equal to 1.

• Finding the GCD helps reduce fractions to their simplest form by dividing both the numerator and denominator by this common number.
• It's an essential step in fraction simplification to ensure efficiency and accuracy in calculations.

Understanding GCD allows you to handle mathematical operations involving fractions with ease.