Problem 89 - Prealgebra

Question

In Exercises 89-100, simplify the given expression. \(\frac{7.5 \cdot 7.1-19.5}{0.54}\)

Step-by-Step Solution

Answer

The simplified expression is 62.5.

Step 1: Multiply the Numerator

The expression given is \( \frac{7.5 \cdot 7.1 - 19.5}{0.54} \). First, we need to calculate the product in the numerator: \( 7.5 \cdot 7.1 \). Performing this multiplication gives \( 53.25 \).

Step 2: Subtract in the Numerator

With the product from Step 1, we subtract 19.5 from 53.25. So the calculation is \( 53.25 - 19.5 = 33.75 \).

Step 3: Divide by the Denominator

Now take the result from Step 2, which is 33.75, and divide it by the denominator 0.54. Performing this division: \( \frac{33.75}{0.54} = 62.5 \).

Key Concepts

Numerator

In fractions, the numerator is the number that sits above the fraction bar. It indicates how many parts of a whole are being considered. In our example expression \(\frac{7.5 \cdot 7.1 - 19.5}{0.54}\), the entire operation \(7.5 \cdot 7.1 - 19.5\) makes up the numerator. Here, the numerator consists of a multiplication followed by a subtraction operation. The number atop tells us the value of the portion that needs to be divided by the denominator. Remember:

• The numerator can be a single number, or it can be an expression like in this case.
• Operations within the numerator should be completed before dividing by the denominator.

Understanding how to properly handle and compute the expression within the numerator is crucial for simplifying the entire fraction.

Denominator

The denominator is the number located beneath the fraction bar. It tells us the total number of equal parts into which the numerator is being divided. When simplifying fractions, the denominator is critical because it determines how the group's entirety is split. In our case, the denominator is the single number 0.54. Here's what to keep in mind about denominators:

• The denominator should never be zero. Division by zero is undefined.
• The size of the denominator affects the size of the fraction's value - larger denominators mean smaller values when the numerator remains the same.

In the given exercise, the denominator 0.54 shows how we divide the evaluated numerator, guiding us to the final simplified result.

Multiplication

Multiplication is a mathematical operation that combines groups of equal sizes. In the context of fractions, multiplying values in the numerator or the denominator affects the entire fraction. Let's break down the multiplication in our exercise:

• Here, we have \(7.5 \cdot 7.1\) within the numerator part of the fraction. This multiplication gives us 53.25.
• Always solve multiplication problems in the numerator before moving to other operations like subtraction.

By multiplying these numbers correctly, you're ensuring that the computations for the subsequent steps are accurate.

Division

Division involves splitting a number into specified parts or groups. This operation is fundamental when simplifying fractions, as it determines the fraction's outcome. Let's explore the division concerning our exercise:

• We take the resulting numerator 33.75 and divide it by the denominator 0.54 to achieve the simplified result of 62.5.
• The division step is the last operation after any multiplications and subtractions have been completed in the numerator.
• It's essential to carry out division with precision to avoid errors in simplification.

Ultimately, division allows us to see how many times the denominator fits inside the numerator, completing the process of fraction simplification.