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Chapter 13: Problem 65

Evaluate each of the following. \(\frac{n}{2}(3+n),\) for \(n=11\)

Short Answer

Expert verified
77

Step by step solution

01

- Substitute the given value of n

Replace the variable \(n\) in the expression \( \frac{n}{2}(3+n) \) with the given value, \(n = 11\). The expression now becomes \( \frac{11}{2}(3+11) \).
02

- Simplify inside the parentheses

Calculate the value inside the parentheses \( (3+11) \). Since \( 3+11 = 14 \), the expression is now \( \frac{11}{2}(14) \).
03

- Multiply the values

Multiply \( \frac{11}{2} \) by \( 14 \). This can be done as follows: \( \frac{11 \times 14}{2} \). Simplify the multiplication first which gives \( \frac{154}{2} \).
04

- Simplify the fraction

Divide \( 154 \) by \( 2 \). The result is \( 77 \). Therefore, \( \frac{154}{2} = 77 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Substitution
When solving algebraic expressions, the first important concept is substitution. This involves replacing a variable with its given value. For example, in the expression \( \frac{n}{2}(3+n) \) and the value \( n = 11 \), you substitute 11 in place of \( n \). This turns the expression into \( \frac{11}{2}(3+11) \). This makes it easier to work with concrete numbers rather than unknown variables.
Simplification
After substitution, the next step is simplification. Simplification means reducing the expression to its simplest form. You start by calculating the values inside parentheses. In this example: \( 3 + 11 = 14 \). The expression then becomes \( \frac{11}{2}(14) \). It is much easier to understand and work with this new form, as it involves simpler arithmetic operations.
Multiplication
Next, let's talk about multiplication. Multiplication in algebra can sometimes involve fractions. Here, you have \( \frac{11}{2}(14) \). To multiply, you can multiply the numerator of the fraction by the number outside the parentheses: \( \frac{11 \times 14}{2} \). This results in \( \frac{154}{2} \). Multiplying first helps in breaking down complex expressions.
Fractions
Lastly, we deal with fractions. Fraction simplification involves dividing the numerator by the denominator. Here, \( \frac{154}{2} \) simplifies by dividing 154 by 2. The result is 77. Simplifying fractions allows you to get the final, most simplified answer. Understanding how to work with and simplify fractions is crucial in solving algebraic problems efficiently. Simplify step-by-step, and the process becomes much more manageable.

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