Problem 8 - Fundamentals of Mathematics

Question

Find the value of each expression. $$\frac{10 a}{3 b}+\frac{4 b}{2}, \text { if } a=-6, \text { and } b=2$$

Step-by-Step Solution

Answer

The value of the expression is -6.

Step 1: Substitute variables with values

Start by replacing the given variables in the expression with their specified values. Here, substitute \( a = -6 \) and \( b = 2 \). The expression becomes \( \frac{10 \times (-6)}{3 \times 2} + \frac{4 \times 2}{2} \).

Step 2: Simplify each fraction

First, simplify the fraction \( \frac{10 \times (-6)}{3 \times 2} \). Calculate the numerator \( 10 \times (-6) = -60 \) and the denominator \( 3 \times 2 = 6 \). So, \( \frac{-60}{6} = -10 \). Likewise, for the second part \( \frac{4 \times 2}{2} \), the numerator \( 4 \times 2 = 8 \) with the denominator 2 gives \( \frac{8}{2} = 4 \).

Step 3: Add the simplified values

Combine the simplified fractions from the previous step. Thus, \( -10 + 4 \). Simplifying this gives \( -10 + 4 = -6 \).

Key Concepts

Substitution

Substitution is often the first step in evaluating algebraic expressions. It involves replacing the variables in an expression with their given numerical values. This helps in transforming an abstract expression into a more concrete form, ready for computation.
For example, if you have an expression like \(\frac{10a}{3b}+\frac{4b}{2}\), and you know \(a = -6\) and \(b = 2\), substitution is where you put \(a\) and \(b\) into the equation. The expression becomes \(\frac{10 \times (-6)}{3 \times 2} + \frac{4 \times 2}{2}\).

• Always carefully replace every instance of the variable with its given value.
• Be mindful of any operations you need to perform, like multiplication in this case.
• Substitution is a simple but crucial step to prepare your expression for further simplification.

Fraction Simplification

Fraction simplification involves making the fraction as simple as possible by reducing it to its lowest terms. This is done by dividing both the numerator and the denominator by their greatest common divisor.
In the expression \(\frac{10 \times (-6)}{3 \times 2}\), simplification is used to break it down to \(-10\). Here's how you simplify fractions:

• First, calculate the products: the numerator \(10 \times (-6) = -60\), and the denominator \(3 \times 2 = 6\).
• Next, divide the numerator by the denominator: \(\frac{-60}{6} = -10\).

Repeating these steps with the fraction \(\frac{4 \times 2}{2}\) results in a simplified value of \(4\). Fraction simplification allows for a clearer and easier computation in further steps.

Integer Operations

Integer operations come into play when adding, subtracting, multiplying, or dividing whole numbers. These operations are fundamental in bringing simplified expressions down to their final values.
After simplifying the fractions from the given problem, you have \(-10 + 4\). This is where your integer addition skills are utilized. Follow these steps to solve the operation:

• Look at the signs: \(-10\) is negative, and \(4\) is positive.
• When adding integers with different signs, subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.
• For \(-10 + 4\), subtract \(4\) from \(10\), which gives \(6\), and since \(10\) is the larger number and negative, the result is \(-6\).

Mastering integer operations simplifies complex expressions and efficiently finds the solution.