Problem 55 - Algebra 1: Concepts and Skills

Question

Simplify. $$ \left(-\frac{3}{4}\right)\left(\frac{3 y}{-5}\right) $$

Step-by-Step Solution

Answer

The simplified form of \(-\frac{3}{4} * \frac{3y}{-5}\) is \(\frac{9y}{20}\)

Step 1: Identify Numerators and Denominators

The fractions being multiplied are \(-\frac{3}{4}\) and \(\frac{3y}{-5}\). Therefore, the numerators are -3 and 3y, and the denominators are 4 and -5.

Step 2: Multiply the Numerators

Multiply the numerators. The product of -3 and 3y is -9y.

Step 3: Multiply the Denominators

Next, multiply the denominators. The product of 4 and -5 is -20.

Step 4: Combine the Results

Since the fractions are being multiplied, combine the results of the multiplication into one fraction. This results in the fraction \(\frac{-9y}{-20}\)

Step 5: Simplify the fraction

To simplify the fraction, note that the product of two negative numbers is always positive, so the fraction simplifies to \(\frac{9y}{20}\)

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential math skill that involves reducing a fraction to its simplest form. A fraction is not in its simplest form when both the numerator (the top part) and the denominator (the bottom part) have common factors other than one. By cancelling out these common factors, you can simplify the fraction. This makes it easier to understand and use in further calculations.

For example, consider the fraction \(\frac{6}{8}\). Both numbers have 2 as a common factor. Dividing both by 2, we get \(\frac{3}{4}\), which is the simplified form.

This process helps in making math operations, like addition or multiplication of fractions, more straightforward and less prone to errors. The same principle is applied when dealing with more complex fractions involving variables, like \(\frac{9y}{20}\). Here, simplifying means making sure no further cancellation can be done.

Numerators and Denominators

Understanding numerators and denominators is crucial when working with fractions. In a fraction, the numerator is the top number, and the denominator is the bottom number. They indicate how many parts of a whole we are dealing with.

In our problem, the fractions \(-\frac{3}{4}\) and \(\frac{3y}{-5}\), have numerators -3 and 3y, and denominators 4 and -5, respectively. The numerator tells us how many parts we consider, while the denominator tells us how many equal parts make up a whole.

This division of a whole allows calculations such as multiplication or addition of fractions to represent real-world situations accurately, such as dividing a pizza among friends or measuring ingredients for a recipe. Recognizing numerators and denominators correctly helps avoid mistakes in mathematical calculations.

Negative Numbers in Fractions

Working with negative numbers in fractions can seem intimidating, but understanding a few simple rules can make it easy. A negative sign can be applied to either the numerator or the denominator, or it could be in front of the fraction as a whole.

In our exercise, we saw fractions like \(-\frac{3}{4}\) and \(\frac{3y}{-5}\). Both fractions carried negative signs. When multiplying fractions, even with negative numbers, the negative sign affects the overall product.

The rule is straightforward:

• A negative multiplied by a negative results in a positive.
• A negative multiplied by a positive results in a negative.

The fraction \(\frac{-9y}{-20}\) simplifies to \(\frac{9y}{20}\) because the negative signs cancel out each other, resulting in a positive fraction. Remembering this rule helps in handling calculations involving negative numbers with confidence.