Problem 75 - Fundamentals of Mathematics

Question

Perform each multiplication and division. $$ \frac{1}{10} \cdot \frac{5}{12} $$

Step-by-Step Solution

Answer

The result is \(\frac{1}{24}\).

Step 1: Understand the Question

The expression given is \( \frac{1}{10} \times \frac{5}{12} \). The task is to multiply the two fractions.

Step 2: Multiply the Numerators

To multiply fractions, multiply the numerators (top numbers) together. Here, that means multiplying 1 and 5. \[1 \times 5 = 5\]

Step 3: Multiply the Denominators

Next, multiply the denominators (bottom numbers) together. Here, that means multiplying 10 and 12.\[10 \times 12 = 120\]

Step 4: Write the New Fraction

Combine the solutions from Steps 2 and 3 to form a new fraction: \[ \frac{5}{120} \]

Step 5: Simplify the Fraction

To simplify \( \frac{5}{120} \), find the greatest common divisor (GCD) of 5 and 120, which is 5. Divide both the numerator and the denominator by their GCD:\[\frac{5 \div 5}{120 \div 5} = \frac{1}{24}\]

Step 6: Conclusion

The simplified form of the original multiplication is \( \frac{1}{24} \). Thus, the result of the multiplication \( \frac{1}{10} \times \frac{5}{12} \) is \( \frac{1}{24} \).

Key Concepts

Understanding Numerators

Numerators are the top parts of fractions, and they represent how many parts of a whole are being considered. When multiplying fractions, you multiply the numerators together to find the new numerator of the result.
For example, in the fraction multiplication \( \frac{1}{10} \times \frac{5}{12} \), the numerators are 1 and 5.

• Multiply 1 by 5 to get 5.

This step gives you the numerator for the fraction resulting from the multiplication, as seen in the original exercise: the numerator becomes 5. Understanding how to handle the numerators correctly is crucial as they dictate part of the amount in the final answer. Always ensure you multiply them directly to obtain the complete result.

Role of Denominators

Denominators are found at the bottom of the fraction, showing into how many equal parts the whole is divided. In fraction multiplication, you multiply the denominators to get the denominator of the new fraction.
Considering our example, \( \frac{1}{10} \times \frac{5}{12} \), the denominators are 10 and 12.

• Multiply 10 by 12 to get 120.

The new fraction formed will thus have 120 as its denominator, illustrating the addition of a scaled part size in the multiplication process. Notably, multiplying denominators ensures that the entire scaling and division concept is maintained correctly. This is a key step to maintain the integrity of the fraction's representation.

Simplifying Fractions

Simplifying fractions is often necessary to express the fraction in its simplest form, which makes it easier to understand and use in further calculations. You achieve simplification by dividing both the numerator and the denominator by their greatest common divisor.

• For the fraction \( \frac{5}{120} \), divide both by their GCD, which is 5.
• The fraction simplifies to \( \frac{1}{24} \).

Converting to the simplest form is beneficial for clarity and ensures that the fraction is as straightforward as possible. In practical terms, this simplification allows for easier comparison with other fractions and a clearer understanding of the fraction's real-world quantity.

Greatest Common Divisor (GCD)

The greatest common divisor comes into play when simplifying fractions. It is the largest number that can divide both the numerator and the denominator without leaving a remainder.
Using the example fraction \( \frac{5}{120} \), the GCD of 5 and 120 is 5.

• Divide 5 by 5 to get 1.
• Divide 120 by 5 to get 24.

This results in the simplified fraction \( \frac{1}{24} \). Calculating the GCD is a foundational skill for simplifying fractions, as it ensures that you reduce the fraction by the correct factor every time, providing the most reduced and straightforward form.