Problem 21 - Algebra 1

Question

Evaluate the expression for the given value of the variable. $$\frac{4}{5} \div n+13 \text { when } n=\frac{1}{5}$$

Step-by-Step Solution

Answer

The evaluation of the expression for the given value of the variable is \( 33 \)

Step 1: Substitution

Substitute the value of \( n \) into the equation. So, the equation becomes \( \frac{4}{5} \div \frac{1}{5} +13 \)

Step 2: Performing Division

Remember that dividing by a fraction is the same as multiplying by its reciprocal. So, the equation becomes \( 4 \times 5 + 13 \) which simplifies further to \( 20 + 13 \)

Step 3: Performing Addition

Finally, perform the addition to get the final evaluation which is \( 33 \)

Key Concepts

Substitution in Algebraic Expressions

The concept of substituting values in algebraic expressions is foundational in performing arithmetic in algebra. It involves replacing the variables with the given numbers or expressions.

Let's explore this with the sample expression \( \frac{4}{5} \div n + 13 \) where we need to evaluate it for \( n = \frac{1}{5} \). Here's the step-by-step process of substitution:

• First, identify the variable. In our expression, the variable is \( n \).
• Next, replace the variable with the given value. Hence, the expression when \( n = \frac{1}{5} \) transforms to \( \frac{4}{5} \div \frac{1}{5} + 13 \).

Through substitution, algebraic expressions become much simpler to handle as they are turned into numerical expressions that are ready for arithmetic operations.

Dividing by a Fraction

When faced with the task of dividing by a fraction, one effective method is to multiply by the reciprocal of the fraction.

The reciprocal of a fraction is created by swapping its numerator and denominator. For instance, the reciprocal of \( \frac{1}{5} \) is \( \frac{5}{1} \), or simply \( 5 \).

Applying this concept to our substituted expression \( \frac{4}{5} \div \frac{1}{5} \), it changes to
\( \frac{4}{5} \times \frac{5}{1} \) or \( 4 \times 5 \), because dividing by \( \frac{1}{5} \) is the same as multiplying by its reciprocal, \( 5 \). This simplification is a key step that allows students to move forward with the arithmetic process.

Performing Arithmetic Operations

Once we have a clear numerical expression after substitution and division by a fraction, we can proceed to executing arithmetic operations. In our case, we have simplified our expression to \( 4 \times 5 + 13 \).

Now, we perform the multiplication and addition. First, we calculate \( 4 \times 5 \), which gives us \( 20 \). Then, we add \( 13 \) to this product, resulting in \( 20 + 13 = 33 \).

This final number, \( 33 \), is the answer we are looking for. It is the result of correctly applying the arithmetic operations of multiplication and addition in the given order. Performing these operations step by step is crucial to ensure accuracy in evaluating algebraic expressions.