Problem 21 - Prealgebra

Question

Express each ratio as a fraction in simplest form. \(\$ 5\) for 4 cans of tuna

Step-by-Step Solution

Answer

The ratio \( \$5 \) to 4 cans as a fraction is \( \frac{5}{4} \).

Step 1: Understanding the Problem

We are given a ratio of dollars to cans, specifically a ratio of $5 to 4 cans. Our task is to express this ratio as a fraction in simplest form.

Step 2: Write the Ratio as a Fraction

Convert the given ratio into a fraction. The ratio of $5 to 4 cans is written as the fraction \( \frac{5}{4} \).

Step 3: Simplification Check

Check if the fraction \( \frac{5}{4} \) can be simplified. Since 5 is a prime number and does not divide evenly into 4, this fraction is already in its simplest form.

Key Concepts

Understanding Fractions

Fractions are a way to represent numbers that aren't whole. They are made up of two parts: the numerator (the top number) and the denominator (the bottom number).
The numerator tells us how many parts we have, while the denominator indicates how many parts make up a whole. For example, in the fraction \( \frac{5}{4} \), 5 is the numerator, and 4 is the denominator.
Fractions are useful in many situations, such as when dividing objects into equal parts or expressing ratios as we see in ratio problems.

• Fraction: \( \frac{5}{4} \) means 5 parts of a whole divided into 4 equal parts.
• Numerator: Represents the number of parts we consider.
• Denominator: Represents the total number of equal parts in the whole.

With this understanding, you can express any ratio as a fraction by placing the first quantity as the numerator and the second as the denominator.

Simplifying Fractions

Simplifying fractions is the process of reducing fractions to their simplest form. This means that the numerator and the denominator have no common factors other than 1.
Simplifying makes fractions easier to work with and understand.
To simplify a fraction like \( \frac{5}{4} \), you check if both the numerator and the denominator can be divided by the same number evenly.
Here's how you do this:

• Check for common factors: Determine if there are any numbers that can divide both the numerator and the denominator without a remainder.
• Divide by the greatest common factor (GCF): If possible, find the largest factor common to both numbers and divide them. This results in a simplified fraction.

In the case of \( \frac{5}{4} \), as the solution mentions, since 5 is a prime number and doesn’t divide evenly into 4, the fraction is already in its simplest form.

Prime Numbers and Their Role

Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves.
They are important not only in number theory but also in simplifying fractions.
A prime number cannot be divided evenly by any other number, which means it doesn't form a simpler fraction unless the numerator and the denominator share no other factors.
For instance, if you have a fraction \( \frac{5}{4} \) and notice that 5 is a prime number, you know at once it can't be simplified by dividing both parts by a common number.

• Prime Number: A number greater than 1 with no divisors other than 1 and itself.
• Role in Fractions: Prime numerators often mean the fraction is already simplified.
• Realization: When simplifying fractions, recognizing a prime number instantly tells us if further simplification might be possible.

Recognizing prime numbers quickly can assist in determining the simplest form of any given fraction.