Problem 6 - Prealgebra

Question

Find each of the following products. (Multiply.) $$-\frac{4}{7} \cdot \frac{7}{4}$$

Step-by-Step Solution

Answer

The product is \(-1\).

Step 1: Identify the Multiplication of Fractions

To multiply fractions, we multiply the numerators together and the denominators together. In this exercise, we are multiplying \(-\frac{4}{7}\) and \(\frac{7}{4}\).

Step 2: Multiply the Numerators

Multiply the numerators of the two fractions: \(-4\) and \(7\). The product is \(-4 \times 7 = -28\).

Step 3: Multiply the Denominators

Multiply the denominators of the two fractions: \(7\) and \(4\). The product is \(7 \times 4 = 28\).

Step 4: Simplify the Fraction

The product of the fractions is \(-\frac{28}{28}\). Simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is \(28\). The simplified fraction is \(-1\).

Key Concepts

Fractions

Fractions are a way to express numbers that are not whole numbers. They consist of two parts: a numerator and a denominator. Picture a pizza: if it's cut into 8 slices and you eat 3, you'd express how much you've eaten as a fraction: \( \frac{3}{8} \). This fraction means 3 out of 8 total slices were consumed.
In general, the numerator tells you how many parts of the whole you have. The denominator tells you how many equal parts the whole is divided into. By using fractions, you can represent quantities smaller than a whole number, which is essential for many real-world calculations.
Multiplying fractions involves multiplying the numerators together and the denominators together. This gives you a new fraction that may need to be simplified. Remember, fractions are handy because they allow us to divide something whole into smaller, more manageable pieces.

Simplifying Fractions

Simplifying fractions is about making a fraction as simple as possible. It means reducing it to its smallest-form by ensuring that both the numerator and the denominator can't be divided evenly by the same number, except for 1.
To simplify a fraction, follow these steps:

• Find the greatest common divisor (GCD) of both the numerator and the denominator. The GCD is the largest number that can divide both parts evenly.
• Divide both the numerator and the denominator by their GCD.

This process reduces the fraction to its simplest form.
For example, to simplify \( \frac{28}{28} \), notice that the GCD of both numbers is 28. Dividing both 28 and 28 by 28 gives you 1, resulting in the simplified form being \( -1 \). It's important to simplify fractions so they are easier to understand and work with.

Numerators and Denominators

The numerator and denominator are essential parts of a fraction. The numerator is the number above the line in a fraction. It shows how many parts of the whole are being considered. The denominator is the number below the line and indicates into how many equal parts the whole is divided.
For example, in the fraction \( \frac{3}{8} \), 3 is the numerator and 8 is the denominator. This tells us that we've taken 3 parts out of a total of 8.
When multiplying fractions, you must individually multiply the numerators together and the denominators together. For instance, in the fraction multiplication of \( -\frac{4}{7} \cdot \frac{7}{4} \), the numerators \(-4\) and \(7\) are multiplied first, resulting in \(-28\). Similarly, the denominators \(7\) and \(4\) are multiplied to give \(28\).
Understanding numerators and denominators is crucial to effectively and accurately working with fractions.