Question

Evaluate each expression. $$ \frac{5}{9} \cdot \frac{3}{10} $$

Short answer

\( \frac{1}{6} \)

Step-by-step solution

Identify the Numerators and Denominators

In the given expression, the numerators are 5 and 3, and the denominators are 9 and 10.

Multiply the Numerators

Multiply the numerators of the fractions: \( 5 \times 3 = 15 \)

Multiply the Denominators

Multiply the denominators of the fractions: \( 9 \times 10 = 90 \)

Form the New Fraction

Construct the new fraction with the multiplied numerator and denominator:\( \frac{15}{90} \)

Simplify the Fraction

Simplify \( \frac{15}{90} \) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 15.\( \frac{15 \div 15}{90 \div 15} = \frac{1}{6} \)

Key concepts

Numerators and Denominators

In any fraction, the numerator is the number above the fraction line, while the denominator is the number below the fraction line.
To understand this using the given exercise, consider the fraction\( \frac{5}{9} \). Here, 5 is the numerator, and 9 is the denominator.
Similarly, in the fraction \( \frac{3}{10} \), 3 is the numerator, and 10 is the denominator.
These two numbers define the value of the fraction. The numerator tells us how many parts we have, while the denominator tells us into how many parts the whole is divided.
Understanding this fundamental concept is crucial before performing fraction multiplication.

Simplifying Fractions

Whenever you multiply fractions, you often need to simplify the resulting fraction to its simplest form.
Simplifying a fraction means reducing it to the smallest possible numerator and denominator that have the same value.
For instance, in our exercise, we have the fraction \( \frac{15}{90} \).
To simplify \( \frac{15}{90} \), we need to divide both the numerator and the denominator by their greatest common divisor (GCD), which in this case, is 15.
After dividing: \( \frac{15 \bdiv 15}{90 \bdiv 15} = \frac{1}{6} \).
So, the simplified form of the fraction is \( \frac{1}{6} \).
Simplifying fractions helps in making calculations easier and results clearer.

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder.
Finding the GCD is very useful for simplifying fractions.
To find the GCD of 15 and 90:

• List the prime factors of each number. Prime factors are the prime numbers that multiply together to make the original number.
• For 15: the prime factors are 3 and 5.
• For 90: the prime factors are 2, 3, 3, and 5.
• The common prime factors between 15 and 90 are 3 and 5.
• Multiply these common factors together to get the GCD: 3 * 5 = 15.

Therefore, the GCD of 15 and 90 is 15, which we use to simplify our fraction.
Understanding the GCD concept will significantly help in solving a variety of fraction-related math problems.