Problem 72 - Introductory Algebra for College Students

Question

In Exercises \(29-72,\) use the order of operations to simplify each expression. $$\frac{\frac{17}{25}}{\frac{3}{5}-4} \div \frac{1}{5}+\frac{1}{2}$$

Step-by-Step Solution

Answer

-\frac{9}{2} or -4.5

Step 1: Division

Start the operation from left to right. The first operation will be the division of fractions which is equivalent to multiplication by the reciprocal of the divisor: \(\frac{\frac{17}{25}}{\frac{3}{5}-4} \div \frac{1}{5}+\frac{1}{2}\) becomes \(\frac{\frac{17}{25}}{\frac{3}{5}-4} \times 5 +\frac{1}{2}\).

Step 2: Calculation of Inner Fraction

Now, simplify the fraction inside, which contains a fraction and a whole number: \( \frac{\frac{3}{5}-4}{1} = \frac{3-20}{5} = \frac{-17}{5} = - \frac{17}{5} \). So, the expression becomes \(\frac{\frac{17}{25}}{- \frac{17}{5}} \times 5 + \frac{1}{2} \).

Step 3: Fraction Division

Now perform a fraction division which is multiplication by the reciprocal of the divisor: \( \frac{\frac{17}{25}}{- \frac{17}{5}} \) becomes \(\frac{17}{25} \times -\frac{5}{17},\) which simplifies to \(-1\) . So the expression becomes \(-1 \times 5 + \frac{1}{2}= -5 + \frac{1}{2} \)

Step 4: Addition

Finally, add \(-5\) to \(\frac{1}{2}\) which gives \(-5 + \frac{1}{2}= -\frac{9}{2}\)

Key Concepts

Simplifying Expressions

Understanding how to simplify mathematical expressions is crucial in solving complex problems. Simplifying an expression often involves combining like terms, applying the order of operations, and reducing fractions to their simplest form. The order of operations, remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right), plays a pivotal role in simplifying correctly.

It's important to tackle operations inside parentheses or fractions first, then move on to exponents, followed by multiplication or division, and finally addition or subtraction. For instance, given an expression like \[\frac{\frac{17}{25}}{\frac{3}{5}-4} \div \frac{1}{5}+\frac{1}{2}\], we first focus on the numerator and the denominator inside the complex fraction before addressing the division by \(\frac{1}{5}\) and the addition of \(\frac{1}{2}\).

Improving simplification skills involves practice in recognizing patterns and combining operations efficiently. This helps in breaking down seemingly daunting expressions into manageable parts.

Fraction Division

When dividing fractions, the key is to multiply by the reciprocal of the fraction by which you are dividing. To illustrate, let's consider the fraction division \(\frac{\frac{17}{25}}{- \frac{17}{5}}\).

You can transform this division into multiplication as follows: \(\frac{17}{25} \times -\frac{5}{17}\). To get the reciprocal of the second fraction, simply flip its numerator and denominator.

Steps to Divide Fractions:

• Find the reciprocal (invert the numerator and the denominator) of the fraction that you're dividing by.
• Multiply the first fraction by the reciprocal of the second fraction.
• If necessary, simplify the resulting fraction to its lowest terms.

Following these steps leads to a simplified multiplication process, ultimately yielding the answer with less confusion and reduced errors.

Arithmetic Operations

Arithmetic operations are the foundation of all mathematics. These operations include addition, subtraction, multiplication, and division. Mastery of these four basic operations is essential for solving various mathematical problems, from simple equations to complex expressions involving multiple steps and terms.

For example, after simplifying fraction division, you may need to add or subtract fractions or whole numbers, as we see in the expression \( -1 \times 5 + \frac{1}{2}\). This involves understanding how to handle negative numbers during multiplication and adding fractions by finding a common denominator. Here, the multiplication results in \( -5 \), and we add \(\frac{1}{2}\) to get \( -\frac{9}{2}\).

When performing mixed arithmetic operations, it's crucial to follow the order of operations to arrive at the correct result. By refining skills in addition, subtraction, multiplication, and division, students can approach mathematical problems with greater confidence.