91Ó°ÊÓ

Division is one of the four basic arithmetic operations that involves finding out how many times one number is contained within another. In this exercise, we start by dividing 12 by 34.
To express this mathematically, we rewrite the division as a fraction, giving us \( \frac{12}{34} \).
This conversion is crucial because it allows us to handle the operation using properties of fractions.

• Write the division as a fraction.
• Identify the greatest common divisor (GCD) to simplify.

Once we have the fraction, we simplify it. Simplification involves finding the GCD of 12 and 34, which is 2, then dividing both the numerator and the denominator by this number.
Hence, \( \frac{12}{34} = \frac{12 \div 2}{34 \div 2} = \frac{6}{17} \).
By doing this, we obtain a simpler fraction that is easier to work with in subsequent steps.

Multiplication of fractions might seem daunting, but it is quite simple if broken down into steps. Following the division operation, now we multiply the resulting fraction \( \frac{6}{17} \) by 15.
When multiplying a fraction by a whole number, treat 15 as \( \frac{15}{1} \).
This makes it easier to apply the rule of multiplying fractions:

• Multiply the numerators: \( 6 \times 15 \).
• Multiply the denominators: \( 17 \times 1 \).

The multiplication of the numerators, \( 6 \times 15 = 90 \), results in the fraction \( \frac{90}{17} \).
This methodology helps to maintain accuracy and ensures that fractions remain in rational form, making the final answer simple yet accurate.

After performing the arithmetic operations, simplifying the fraction is often the final step. This involves reducing the fraction to its lowest terms.
In our case, the fraction \( \frac{90}{17} \) is examined for further simplification.
To simplify, check for any common factors between the numerator (90) and the denominator (17).

• Consider if the denominator divides the numerator cleanly.
• Check for any common factors other than 1.

Since 17 is a prime number, its only factors are 1 and itself, and it does not divide 90.
This means \( \frac{90}{17} \) is already in its simplest form.
Understanding when a fraction reaches its irreducible state is crucial, as it confirms that the solution is as simplified as possible. This makes your answer precise and mathematically sound.