Problem 104 - Prealgebra

Question

Given \(a=-10.79, b=3.94, c=-3.2\), and \(d=-8.11\), evaluate and simplify the following expression: $$ \frac{a-b}{c-d} $$

Step-by-Step Solution

Answer

The expression simplifies to approximately -3.00.

Step 1: Substitute the Values into the Expression

First, substitute the given values of the variables into the expression \( \frac{a-b}{c-d} \). This will give us: \[ \frac{-10.79 - 3.94}{-3.2 - (-8.11)} \]

Step 2: Simplify the Numerator

Perform the subtraction in the numerator. Calculate \(-10.79 - 3.94\): \(-10.79 - 3.94 = -14.73\). So, the expression becomes \(\frac{-14.73}{-3.2 - (-8.11)}\).

Step 3: Simplify the Denominator

Simplify the denominator by calculating \(-3.2 - (-8.11)\). Remember that subtracting a negative number is the same as adding its positive, so \(-3.2 + 8.11 = 4.91\). Now the expression is \(\frac{-14.73}{4.91}\).

Step 4: Perform the Division

Finally, perform the division. Divide \(-14.73\) by \(4.91\): \(\frac{-14.73}{4.91} \approx -3.00\).

Key Concepts

Substitution

Substitution is like putting on a name tag. We take the letters that stand for numbers (our variables) and replace them with the actual numbers given. Here’s a quick way to understand it: imagine you have a recipe, and instead of saying “add a bit of this or a bit of that,” you say exactly what ingredient and how much to add.
In our exercise, substitute the values to change the letters in the expression \( \frac{a-b}{c-d} \) into numbers:

• Replace \(a\) with \(-10.79\)
• Replace \(b\) with \(3.94\)
• Replace \(c\) with \(-3.2\)
• Replace \(d\) with \(-8.11\)

Thus, the expression becomes \( \frac{-10.79 - 3.94}{-3.2 - (-8.11)} \). Substitution makes sure we’re working with the right numbers right from the start.

Simplification

Simplification in math means making a problem as easy to solve as possible by reducing unnecessary complexity. Basically, it's tidying things up. When we simplify an expression, we combine and reduce terms without changing the expression's value.
In the numerator \(-10.79 - 3.94\), simplify it by directly calculating the subtraction which gives us \(-14.73\).

• Numerator simplification: \( -10.79 - 3.94 = -14.73\)

In the denominator \(-3.2 - (-8.11)\), remember that subtracting a negative is like adding. So, simplify it to \(4.91\).

• Denominator simplification: \( -3.2 - (-8.11) = -3.2 + 8.11 = 4.91\)

Now, our original expression looks simpler: \( \frac{-14.73}{4.91} \). Simplification helps us see the math more clearly.

Arithmetic Operations

Arithmetic operations include a range of basic calculations, mainly addition, subtraction, multiplication, and division. These are the building blocks for all math solutions.
The expression \( \frac{-14.73}{4.91} \) calls for division, which is an arithmetic operation. After substitution and simplification, the division is the next step to find the result. Let’s take each calculation one by one:

• Subtraction in the numerator: \( -10.79 - 3.94 \) was performed to get \(-14.73\).
• Subtraction in the denominator where we actually add: \( -3.2 + 8.11 \) resulting in \(4.91\).
• Finally, divide \(-14.73\) by \(4.91\) to perform the last arithmetic operation in our equation.

These operations help in gradually breaking down and solving the expression.

Negative Numbers

Negative numbers are less than zero and appear often in real life, like temperatures below freezing or debts. Handling negatives correctly is key in prealgebra.
Here's how negative numbers played a role in our exercise:

• When subtracting \(3.94\) from \(-10.79\), both numbers were negative which emphasized their effects in calculations.
• In the denominator, handling negatives was a critical point: subtracting \(-8.11\) effectively turned into adding \(8.11\).
• When performing \( \frac{-14.73}{4.91} \), understanding that the result of dividing a negative by a positive is negative was crucial.

Managing negative numbers efficiently helps prevent errors and ensures accuracy in math.