Problem 44 - Fundamentals of Mathematics

Question

What part of 24 is \(9 ?\)

Step-by-Step Solution

Answer

9 is \( \frac{3}{8} \) of 24.

Step 1: Understand the Problem

To find 'what part' of 24 is 9, we’re looking for the ratio or the fractional part of 9 compared to 24. This means we need to find how many parts of 24 are equal to 9.

Step 2: Set Up the Fraction

Express the ratio of 9 to 24 as a fraction. This is represented as \( \frac{9}{24} \).

Step 3: Simplify the Fraction

To simplify \( \frac{9}{24} \), divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 9 and 24 is 3. So, divide 9 by 3 to get 3 and 24 by 3 to get 8. Thus, \( \frac{9}{24} = \frac{3}{8} \).

Step 4: Conclusion

The simplified fraction \( \frac{3}{8} \) represents what part of 24 is 9. Thus, 9 is \( \frac{3}{8} \) of 24.

Key Concepts

Ratios

When talking about math, a **ratio** is a way to compare two quantities by showing the relative size of one quantity to another. In the exercise, you wanted to compare the size of 9 to the size of 24. This involves creating a fraction:

• Write the number you want to compare (in this case, 9) as the numerator.
• Write the whole you want to compare it to (in this case, 24) as the denominator.

This gives you the ratio, or fraction, of their sizes expressed as \( \frac{9}{24} \).
Ratios help you understand how much one quantity takes up in relation to another. Practical examples include cooking recipes, where ingredients need to be in certain ratios to flavor the dish correctly.
Understanding ratios is crucial for comparing everything from distances on a map to amounts of ingredients in a recipe.

Simplifying Fractions

**Simplifying fractions** is the process of making a fraction as simple as possible. This usually involves making the numerator (top number) and the denominator (bottom number) the smallest possible integers that would give you the same ratio.
For example, in the fraction \( \frac{9}{24} \), you need to find a simpler form. Simplifying a fraction helps in better understanding and involving lesser numbers makes computation easier.
Here are the steps to simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide the numerator by the GCD.
• Divide the denominator by the GCD.

Once simplified, \( \frac{9}{24} \) turns into \( \frac{3}{8} \). This means that instead of 9 being a part of 24, it is straightforwardly described as 3 parts out of 8.
Remember, always aim to simplify fractions for clarity and ease of use in calculations.

Greatest Common Divisor

The **greatest common divisor** (GCD) is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder. Finding the GCD helps in simplifying fractions effectively.

• Consider the numbers in the fraction: For \( \frac{9}{24} \), the numbers are 9 and 24.
• List the divisors of each number: Divisors of 9 are 1, 3, 9; divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
• Identify the largest common divisor: Here, it's 3.

Divide both the numerator and the denominator by the GCD (which is 3 for this example). This simplifies \( \frac{9}{24} \) to \( \frac{3}{8} \).
Using the GCD is key to breaking down fractions to their simplest forms, which makes them easier to work with and understand. It’s a small calculation step but hugely impactful in maths.