Problem 39 - Basic College Mathematics

Question

Multiply and simplify. $$ \frac{8}{10} \cdot \frac{45}{100} $$

Step-by-Step Solution

Answer

\( \frac{9}{25} \)

Step 1: Multiply the Numerators

Multiply the numerators of both fractions. \[8 \times 45 = 360\]

Step 2: Multiply the Denominators

Multiply the denominators of both fractions. \[10 \times 100 = 1000\]

Step 3: Write the Fraction

Combine the results from Steps 1 and 2 into a single fraction. \[\frac{360}{1000}\]

Step 4: Simplify the Fraction

Find the greatest common divisor (GCD) of 360 and 1000, which is 40, and divide both numerator and denominator by the GCD.\[\frac{360 \div 40}{1000 \div 40} = \frac{9}{25}\]

Key Concepts

simplifying fractions

Simplifying fractions makes them easier to work with and understand. It involves reducing the fraction to its simplest form where the numerator and the denominator have no common factors other than 1.
To do this, we need to find the greatest common divisor (GCD) of the numerator and the denominator and divide both by this number.
For instance, in the exercise, we simplify \(\frac{360}{1000}\) by finding the GCD of 360 and 1000, which is 40. Then, we divide both the numerator and the denominator by 40, resulting in \(\frac{9}{25}\). Now the fraction is simplified.

greatest common divisor

The greatest common divisor (GCD), also known as the greatest common factor, is the largest number that can exactly divide both the numerator and the denominator.
To find the GCD of 360 and 1000, you can use either the Euclidean algorithm or the prime factorization method.

• Using the Euclidean algorithm, you start with the two numbers and repeatedly apply the division algorithm until the remainder is zero.
  The last non-zero remainder is the GCD.
• With prime factorization, you list the prime factors of each number. The GCD is the product of the highest powers of all common prime factors.

This is how we determined that the GCD of 360 and 1000 is 40, helping us simplify \(\frac{360}{1000}\) to \(\frac{9}{25}\).

numerator and denominator

In a fraction, the numerator is the number above the fraction line, while the denominator is the number below. The numerator represents how many parts of the whole we have, and the denominator represents the total number of equal parts that make up the whole.
For example, in the fraction \(\frac{8}{10}\), 8 is the numerator, and 10 is the denominator. When multiplying fractions like \(\frac{8}{10} \cdot \frac{45}{100}\), you multiply the two numerators together and the two denominators together.
This results in a new fraction, which might need to be simplified. So, understanding the roles of numerator and denominator is essential for performing operations such as multiplication and simplification of fractions effectively.