Problem 41 - Prealgebra

Question

Express each ratio or rate as a fraction in simplest form. 15 dollars out of 123 dollars

Step-by-Step Solution

Answer

The simplest form of the ratio is \( \frac{5}{41} \).

Step 1: Understand the Ratio

The ratio given is '15 dollars out of 123 dollars'. This can be expressed as a fraction where the numerator represents the part, which is 15 dollars, and the denominator represents the whole, which is 123 dollars. Hence, the fraction is \( \frac{15}{123} \).

Step 2: Find the Greatest Common Divisor (GCD)

To simplify a fraction, we need the greatest common divisor (GCD) of the numerator and the denominator. List the factors of 15: 1, 3, 5, 15. List the factors of 123: 1, 3, 41, 123. The largest common factor is 3.

Step 3: Divide by the GCD

Divide the numerator and the denominator of the fraction by their GCD. This gives: \( \frac{15 \div 3}{123 \div 3} = \frac{5}{41} \).

Step 4: Confirm the Fraction is in Simplest Form

Check if 5 and 41 have any common divisors other than 1. Since they do not, \( \frac{5}{41} \) is indeed the simplest form of the fraction.

Key Concepts

Simplifying Fractions

Simplifying fractions is all about making the numbers in a fraction as small as possible while keeping the same value. Imagine having a pie divided into 123 small pieces and you've taken 15 of them. This can seem a bit overwhelming. However, when simplified, it becomes easier to grasp. To simplify, we need to find a common number that both the numerator and the denominator can be divided by. Once we do that division, we'll have a smaller, more manageable fraction. This doesn't change the proportion, just the representation. For our given example,

• The original fraction is \( \frac{15}{123} \).
• This means for every 123 parts of something, you have 15 of them.
• We simplified by dividing both parts by 3, making it \( \frac{5}{41} \). This is the simplest form.

By simplifying, numbers become more relatable and easier to work with in calculations.

Greatest Common Divisor (GCD)

The Greatest Common Divisor, often abbreviated as GCD, is the largest number that divides two numbers without leaving a remainder. It's like a special key that unlocks the door to simplifying fractions if you think of the numbers as locked and the GCD as your key. To find the GCD of two numbers:

• List out all factors of each number. Factors are numbers that divide into another number completely, without leaving leftovers.
• For example, the number 15 has these factors: 1, 3, 5, and 15.
• For 123, the factors are: 1, 3, 41, and 123.

The largest factor that appears in both lists is the GCD. In our exercise, 3 is the GCD because it is the biggest number that divides both 15 and 123. Finding the GCD is critical in making a fraction simpler since it tells you exactly what number you can safely divide both the numerator and the denominator by.

Fraction Representation

Fractions are a way of representing numbers that are not whole. They show how many parts of a certain size make up a whole. When we write numbers like \( \frac{15}{123} \), this symbols tells us how 15 is related to 123. In a more practical way, it can describe ratios like a part of a quantity of money.Representing ratios as fractions is really handy because fractions give us a clear, standardized way to represent how two quantities relate to each other. Even when these numbers may seem unwieldy or complex, such as 15 and 123, turning them into fractions allows easy manipulation and comparison.When given a situation where comparing part of a whole makes sense, using a fraction helps.

• They allow us to compare the part against the total.
• The simplest form of a fraction gives the most efficient comparison.
• In our case, expressing \( \frac{15}{123} \) as \( \frac{5}{41} \) offers a direct snapshot of the relationship.

Understanding fraction representation is key to making sense of everyday relationships, be it financial budgets, recipe ingredients, or statistical data.