Problem 40 - Fundamentals of Mathematics

Question

For the following problems, find each part without using a diagram. $$\frac{2}{3}\( of \)\frac{9}{10}$$

Step-by-Step Solution

Answer

The product is \( \frac{3}{5} \).

Step 1: Understand the Problem

We are tasked with finding the product of two fractions: \( \frac{2}{3} \times \frac{9}{10} \). This is a multiplication problem involving fractions.

Step 2: Multiply the Numerators

To multiply fractions, we first multiply the numerators (the top numbers) of each fraction. The numerators here are 2 and 9.\[2 \times 9 = 18\]

Step 3: Multiply the Denominators

Next, we multiply the denominators (the bottom numbers) of each fraction. The denominators here are 3 and 10.\[3 \times 10 = 30\]

Step 4: Form the New Fraction

Now that we have multiplied the numerators and the denominators, we can form a new fraction which is the result of these products:\[\frac{18}{30}\]

Step 5: Simplify the Fraction

Finally, we simplify \( \frac{18}{30} \) by finding the greatest common divisor (GCD) of 18 and 30, which is 6. Divide both the numerator and the denominator by their GCD:\[\frac{18 \div 6}{30 \div 6} = \frac{3}{5}\] The simplified fraction is \( \frac{3}{5} \).

Key Concepts

Simplifying Fractions

When working with fractions, simplifying them to their simplest form can make calculations easier. Simplifying a fraction means finding a new fraction that is equivalent to the original, but with smaller numbers in both the numerator and the denominator. This process involves dividing the numerator and the denominator by the same number, which is called their greatest common divisor (GCD).

• To simplify a fraction like \( \frac{18}{30} \), we start by identifying the GCD.
• Both 18 and 30 can be divided by 6, so \( \frac{18}{30} \) simplifies to \( \frac{3}{5} \).

Remember that simplifying fractions doesn’t change their value, only their appearance. Always simplify your final answer to ensure it's in the simplest form possible.

Greatest Common Divisor

The greatest common divisor (GCD) is an essential concept when working with fractions. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. To find the GCD, you can use several methods like listing factors, prime factorization, or the Euclidean algorithm.

• Using listing, for example, you would list all factors of both numbers and find the largest one they have in common.
• For 18 and 30, the factors of 18 are 1, 2, 3, 6, 9, 18, and the factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30. The largest factor they share is 6.

Once the GCD is identified, use it to divide the numerator and the denominator of the fraction to simplify it. This step is crucial to ensure your fraction answers are neat and most manageable.

Numerators and Denominators

When multiplying fractions, understanding numerators and denominators is key. The numerator is the top number of a fraction and indicates how many parts of a whole are being considered. The denominator is the bottom number, indicating into how many parts the whole is divided.

• In \( \frac{2}{3} \), the numerator is 2 and the denominator is 3.
• In \( \frac{9}{10} \), the numerator is 9 and the denominator is 10.

To multiply fractions, multiply the numerators together and the denominators together. So, calculating \( \frac{2}{3} \times \frac{9}{10} \) involves multiplying the numerators, 2 and 9, to get 18, and the denominators, 3 and 10, to get 30, resulting in \( \frac{18}{30} \). Always remember to simplify afterwards.