Problem 39 - Prealgebra and Introductory Algebra

Question

Simplify. $$1+\frac{1}{1+\frac{1}{1+\frac{1}{2}}}$$

Step-by-Step Solution

Answer

The simplified result of the expression \(1+\frac{1}{1+\frac{1}{1+\frac{1}{2}}}\) is \(\frac{12}{5}\).

Step 1 - Simplify the Innermost Fraction

As is the common approach in mathematics, we should start solving from innermost brackets. Here the innermost fraction is \(1 + \frac{1}{2}\). So, simplify this first: \(1 + \frac{1}{2} = \frac{3}{2}\).

Step 2 - Simplify the Next Fraction

The expression now looks like this: \(1 + \frac{1}{1+ \frac{3}{2}}\). Now, proceed to simplify the next fraction. One must remember to further simplify the denominator: \(1 + \frac{3}{2} = \frac{5}{2}\), therefore the above expression becomes \(1 + \frac{1}{\frac{5}{2}}\).

Step 3 - Simplify the Upper Fraction

We can convert the division of fractions into multiplication by taking the reciprocal of the denominator. This leads to: \(1 + 2*\frac{1}{5}\). This simplifies to \(1 + \frac{2}{5}\), which equals \(\frac{7}{5}\).

Step 4 - Simplify Final Fraction

Now, moving to the last fraction, which is now \(1 + \frac{7}{5}\), simplifying this we get \(2 + \frac{2}{5} = \frac{12}{5}\).

Key Concepts

Fraction Operations

Understanding fraction operations is essential for simplifying complex fractions, like the one given in our exercise. Fractions represent parts of a whole and when we perform operations on them, such as addition, subtraction, multiplication, or division, we follow specific rules. Addition and subtraction require a common denominator, while multiplication and division do not.

For example, when adding fractions, you need to ensure that both fractions have the same bottom number (denominator) before you combine the top numbers (numerators). If they do not, you must find a common denominator, adjust the numerators accordingly, and then add. In our exercise, this process occurs when combining whole numbers with fractions, by writing the whole number as a fraction with a denominator of 1.

Multiplication, on the other hand, is more straightforward. You simply multiply the numerators together and the denominators together. Division is the process of multiplying by the reciprocal of the fraction you are dividing by, which is crucial for simplifying nested fractions, as seen in step 3 of the exercise solution.

Reciprocal of a Fraction

The reciprocal of a fraction is what you multiply a fraction by to get the number 1. It’s simply a matter of flipping the numerator and denominator. If you have a fraction like \( \frac{a}{b} \), its reciprocal will be \( \frac{b}{a} \).

It's crucial to understand the concept of reciprocals because in fraction division, we multiply by the reciprocal of the divisor. This principle is applied in our nested fraction problem in step 3, where rather than dividing by \( \frac{5}{2} \) we multiply by its reciprocal, \( \frac{2}{5} \). Clear comprehension of reciprocals makes simplifying complex fractions with multiple division operations much easier.

Simplifying Nested Fractions

Nested fractions, also known as complex or compound fractions, have fractions within their numerators or denominators. Simplifying these requires a step-by-step approach, addressing the innermost fractions first and working outward. As with the provided exercise, we simplify \(1 + \frac{1}{2}\) before dealing with the rest of the expression.

Each level of the nested fraction is simplified by finding the reciprocal of the denominator in cases of division or by ensuring common denominators for addition or subtraction. In steps 2 and 3 of the provided solution, we observed this process. Simplifying nested fractions involves patience and attention to the order of operations to correctly simplify the complex fraction to its simplest form.