Problem 97 - Prealgebra

Question

Simplify the given expression. \(\frac{-9.5 \cdot 1.6-3.7}{-3.6}\)

Step-by-Step Solution

Answer

The simplified expression is 5.25.

Step 1: Calculate the Numerator

First, solve the expression in the numerator: \(-9.5 \cdot 1.6 - 3.7\).First, calculate the product:\(-9.5 \cdot 1.6 = -15.2\).Then subtract 3.7 from -15.2:\(-15.2 - 3.7 = -18.9\).

Step 2: Simplify the Fraction

Now, substitute the simplified numerator into the overall expression:\(\frac{-18.9}{-3.6}\).

Step 3: Divide the Numerator by the Denominator

Perform the division of the numerator by the denominator:\(-18.9 \div -3.6 = 5.25\).

Step 4: Express the Final Result

Since dividing two negative numbers results in a positive number, the simplified value of the expression is 5.25.

Key Concepts

Numerators and Denominators

In mathematical expressions, especially fractions, understanding numerators and denominators is crucial. The numerator is the top part of a fraction, indicating how many parts of a whole you are considering. The denominator is the bottom part, representing the total number of equal parts into which the whole is divided. For instance, in the fraction \( \frac{3}{4} \), 3 is the numerator and 4 is the denominator.

When simplifying expressions, it's essential to simplify the numerator and the denominator separately before dividing them. This helps in achieving an accurate result without errors. For the exercise given, the numerator involves performing operations like multiplication and subtraction before dividing by the denominator.

Remember:

• Simplify each component (numerator and denominator) separately.
• Solve any arithmetic calculations within the numerator or denominator first before dividing or further simplifying.

Learning to manage numerators and denominators effectively builds a strong foundation in tackling more complex problems involving fractions.

Multiplication and Division

Multiplication and division are two of the most fundamental operations in mathematics. They are closely related and often seen in expressions that require simplification. In the exercise, the expression is tackled first by performing multiplication within the numerator before then moving on to division.

When you multiply numbers, you're essentially adding a number to itself a specified number of times. For example, when calculatin \( -9.5 \cdot 1.6 \), you're scaling -9.5 by 1.6 times. This gives the product -15.2, emphasized in handling the negative values right from the start.

Once the multiplication and any necessary subtraction in the numerator is completed, division comes into play. Division is the process of breaking down a number into a specified number of parts. With the fraction \( \frac{-18.9}{-3.6} \), dividing the two results in 5.25.

• Always perform multiplication first when simplifying expressions.
• Ensure all arithmetic operations in the numerator are completed before proceeding to division with the denominator.

Both operations play a significant role in breaking down complex expressions into more manageable solutions.

Negative Numbers

Negative numbers often confuse students when simplifying expressions, but understanding how they work can make the process much easier. When dealing with negative numbers, remember they denote values less than zero and are represented with a minus (-) sign.

In our given exercise, negatives appear in both the numerator and denominator. It's critical to remember that when multiplying or dividing two negative numbers, the result is positive. This is because the negatives cancel each other out. For instance, multiplying -9.5 by 1.6 results in -15.2—negative because you're multiplying a negative by a positive, yielding the product with a negative sign.

When dividing the numerator -18.9 by the denominator -3.6, both are negative, resulting in a positive outcome, leading to our final value of 5.25.

• Keep track of negative signs during arithmetic operations.
• Remember: in multiplication/division of negatives, two negatives result in a positive.

Practicing with negative numbers can enhance overall confidence in mathematics, especially in complex problem-solving scenarios.