Problem 42 - Basic College Mathematics

Question

Multiply. $$ \frac{28}{45} \cdot \frac{3}{2} \quad[2.6 \mathrm{a}] $$

Step-by-Step Solution

Answer

The product of \(\frac{28}{45} \cdot \ \frac{3}{2}\) is \(\frac{14}{15}\).

Step 1: Multiply the Numerators

To solve the multiplication of two fractions, first multiply the numerators (the top parts of the fractions). For the fractions \(\frac{28}{45}\) and \(\frac{3}{2}\), the numerators are 28 and 3. Multiply them: \ 28 \times 3 = 84.

Step 2: Multiply the Denominators

Next, multiply the denominators (the bottom parts of the fractions). For the fractions \(\frac{28}{45}\) and \(\frac{3}{2}\), the denominators are 45 and 2. Multiply them: \ 45 \times 2 = 90.

Step 3: Write the Resulting Fraction

After multiplying the numerators and denominators, write the resulting fraction: \(\frac{84}{90}\).

Step 4: Simplify the Fraction

To simplify the fraction, find the greatest common divisor (GCD) of 84 and 90, which is 6. Then, divide both the numerator and the denominator by the GCD: \(\frac{84 \: \ 6}{90 \: \ 6} = \frac{14}{15}\).

Key Concepts

Understanding Fractions

Fractions are a way to represent parts of a whole. A fraction consists of two main parts: the numerator and the denominator. The numerator is the number on top and indicates how many parts we have. The denominator is the number on the bottom and shows how many equal parts the whole is divided into. For example, in the fraction \(\frac{3}{4}\), 3 is the numerator and 4 is the denominator. This fraction tells us we have 3 out of 4 equal parts. Fractions can represent real-life scenarios like slicing a pizza or measuring ingredients.

The Role of Numerators

When working with fractions, the numerator is crucial. It's the number that shows how many parts of the whole you have. In the multiplication of fractions, you multiply the numerators together. For instance, when multiplying \(\frac{28}{45}\) by \(\frac{3}{2}\), you first focus on the numerators 28 and 3. We multiply these two numbers as 28 x 3, which gives us 84. This new value becomes the numerator of the resulting fraction. Always remember to handle the numerators first.

The Role of Denominators

The denominator in a fraction tells us how many equal parts the whole is divided into. When multiplying fractions, multiplying the denominators is necessary. Consider the fractions \(\frac{28}{45}\) and \(\frac{3}{2}\). The denominators are 45 and 2. To proceed, we multiply these denominators, 45 x 2, which equals 90. The result becomes the denominator of the new fraction, \(\frac{84}{90}\). Always ensure both denominators are multiplied properly to get an accurate resulting fraction.

Simplifying Fractions

Simplifying fractions makes them easier to work with and understand. Simplification involves reducing the fraction to its lowest terms. To do this, find the greatest common divisor (GCD) of the numerator and denominator. In our example, with \(\frac{84}{90}\), the GCD is 6. Divide both the numerator and the denominator by this GCD. So, \(\frac{84 \:\div \ 6}{90 \:\div \ 6} \ = \ \frac{14}{15}\). Now, the fraction is simplified. Always simplify fractions if possible to make calculations easier and results clearer.