Problem 11 - Fundamentals of Mathematics

Question

Perform the following multiplications. $$ \frac{25}{12} \cdot \frac{10}{45} $$

Step-by-Step Solution

Answer

\( \frac{25}{54} \)

Step 1: Multiply the Numerators

To perform the multiplication of two fractions, start by multiplying the numerators. Here, multiply 25 and 10: \[ 25 \times 10 = 250 \]

Step 2: Multiply the Denominators

Next, multiply the denominators of the fractions. Multiply 12 and 45: \[ 12 \times 45 = 540 \]

Step 3: Write the Fraction

Now, write the result of the multiplication of the numerators over the result of the multiplication of the denominators: \[ \frac{250}{540} \]

Step 4: Simplify the Fraction

To simplify \( \frac{250}{540} \), find the greatest common divisor (GCD) of 250 and 540, which is 10. Divide both the numerator and the denominator by 10:\[ \frac{250 \div 10}{540 \div 10} = \frac{25}{54} \]

Key Concepts

Numerators

The numerator is the number on the top of a fraction. It signifies how many parts of a whole we're dealing with. For instance, in the fraction \( \frac{25}{12} \), 25 is the numerator. In fraction multiplication, we begin by multiplying the numerators of the given fractions.
Think of the numerator as telling how many pieces you have. So, when you multiply the numerators (like 25 and 10 in this case), you are essentially finding out how many parts you will have after multiplication.

• Numerators convey the number of pieces.
• Multiply numerators to get the total "pieces" for the product.
• Example: \( 25 \times 10 = 250 \)

Remember, in every multiplication problem involving fractions, your new numerator is a product of the original numerators.

Denominators

The denominator is the bottom part of a fraction and tells us into how many equal parts the whole is divided. Using \( \frac{25}{12} \) as our example, the denominator is 12. When multiplying fractions, the second step after the numerators is to tackle the denominators.
Multiplying the denominators helps us find the total number of equal pieces into which our entire product is divided. During this step, you multiply the denominators of the fractions involved, much like you multiply the numerators.

• Denominators show how every piece fits into the whole.
• Multiplied together to determine total segments.
• Example: \( 12 \times 45 = 540 \)

It's a process of recalibrating the size of each piece, as you explore the fractional world of multiplication. Once the denominators are multiplied, you know into how many parts your product is divided.

Simplifying Fractions

Simplifying a fraction means making it as simple as possible, reducing both the numerator and the denominator to their smallest possible integers while keeping the fraction's value unchanged.
To simplify, you will need to find the greatest common divisor (GCD) of the numerator and the denominator. This is the largest number that evenly divides both numbers, allowing you to shrink them down while maintaining equivalence.
Once you determine the GCD—in our case, 10 for \( \frac{250}{540} \)—you can simplify:

• Divide both the numerator and the denominator by the GCD to get the simplest form.
• The new fraction maintains its size compared to the original but is visually cleaner.
• Example: \( \frac{250}{540} \div \frac{10}{10} = \frac{25}{54} \)

When simplifying, the goal is to reduce clutter and see the simplest fraction that still represents the same value or ratio. This makes work with fractions easier and tidier.