Problem 22 - Fundamentals of Mathematics

Question

Find \(\frac{2}{3}\) of \(\frac{1}{6}\) of \(\frac{9}{2}\)

Step-by-Step Solution

Answer

The result is \( \frac{1}{2} \).

Step 1: Calculate \( \frac{1}{6} \) of \( \frac{9}{2} \)

To find \( \frac{1}{6} \) of \( \frac{9}{2} \), multiply them together. This means: \[ \frac{1}{6} \times \frac{9}{2} = \frac{1 \times 9}{6 \times 2} = \frac{9}{12} \]. Simplify \( \frac{9}{12} \) by dividing the numerator and the denominator by their greatest common divisor, which is 3. Thus, \( \frac{9}{12} = \frac{3}{4} \).

Step 2: Calculate \( \frac{2}{3} \) of \( \frac{3}{4} \)

Now, find \( \frac{2}{3} \) of \( \frac{3}{4} \) by multiplying them: \[ \frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12} \]. Simplify \( \frac{6}{12} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 6. Therefore, \( \frac{6}{12} = \frac{1}{2} \).

Key Concepts

Simplifying Fractions

Simplifying fractions is about making a fraction as simple as possible. This means writing it in its smallest equivalent form. For instance, consider the fraction \( \frac{9}{12} \). To simplify it, you need to find a number that both 9 and 12 can be divided by evenly. That number is typically the greatest common divisor (GCD). Once the GCD is found, divide both numbers by it. Simplifying fractions helps to:

• Make calculations easier
• Provide a clearer understanding of the fraction's value
• Ensure consistency in mathematical practices

It's important to always simplify your fractions to make your answers more precise and acceptable in mathematical disciplines.

Greatest Common Divisor

The greatest common divisor (GCD) is an important concept when simplifying fractions. It refers to the largest number that can evenly divide two or more numbers. To find the GCD of two numbers like 9 and 12:

• List the factors of each number. For 9, the factors are 1, 3, and 9. For 12, they are 1, 2, 3, 4, 6, and 12.
• Identify the largest factor that is common to both lists of factors. In this case, that number is 3.

Using the GCD to simplify fractions ensures that you reduce the fraction to its most simplest form, leading to more efficient computation and easier understanding of the math involved.

Step-by-Step Solutions

Step-by-step solutions are a powerful way to tackle and understand math problems. By breaking a problem down into simpler, smaller parts, you can see exactly what operations are being done and why.Here's how it works in this exercise:1. **Multiply Each Fraction in Turn:** - Start with \( \frac{1}{6} \) of \( \frac{9}{2} \). The calculation requires multiplying the numerators and the denominators separately. This results in \( \frac{9}{12} \). - Simplify \( \frac{9}{12} \) using the GCD of 9 and 12, which is 3, to get \( \frac{3}{4} \).2. **Continue with Further Multiplications:** - Now, find \( \frac{2}{3} \) of \( \frac{3}{4} \) by multiplying the numerators and denominators. This results in \( \frac{6}{12} \). - Again, simplify \( \frac{6}{12} \) by the GCD which is 6 to find \( \frac{1}{2} \).By understanding each individual step, you'll be better prepared to approach similar problems with confidence and clarity. Always remember to check if your final fraction can be simplified further.