Question

Find the common fraction form of the repeating decimal \(0.4242 \ldots \ldots\)

Short answer

The common fraction form of the repeating decimal \(0.4242...\) is \(\frac{14}{33}\).

Step-by-step solution

Set up an equation

Let x represent the common fraction form of the repeating decimal 0.4242... . We can write this as: \(x = 0.4242...\)

Multiply the equation by a power of 10

To isolate the repeating part of the decimal, we can multiply both sides of the equation by a power of ten that has the same number of zeros as the number of repeating digits. In this case, there are two repeating digits (4 and 2), so we will multiply by 100: \(100x = 42.4242...\)

Subtract the original equation from the new equation

Now, subtract the original equation (Step 1) from the equation we obtained in Step 2: \(100x - x = 42.4242... - 0.4242...\) This simplifies to: \(99x = 42\)

Solve for x

To find the common fraction form of the repeating decimal, we just need to solve for x: \(x = \frac{42}{99}\) We can simplify the fraction by dividing both common factors of 42 and 99.

Simplify the fraction

Dividing both the numerator and the denominator by their greatest common divisor, which is 3, we get: \(x = \frac{42\div3}{99\div3} = \frac{14}{33}\) Therefore, the common fraction form of the repeating decimal 0.4242... is \(\frac{14}{33}\).

Key concepts

Converting Decimals to Fractions

Understanding how to convert a decimal to a fraction is essential for solving many math problems. To do this, we must first identify if the decimal is terminating or repeating. A terminating decimal has a finite number of digits after the decimal point, whereas a repeating decimal has one or several digits that repeat infinitely.

When we have a repeating decimal, we look for the repeating pattern, or the digit(s) that repeat. These are the digits that we use to set up our equation in the conversion process. For instance, to convert a simple repeating decimal like \(0.333...\) to a fraction, you can represent the decimal as \(\frac{3}{9}\) which simplifies to \(\frac{1}{3}\). However, the process becomes a bit more involved with decimals that have more than one repeating digit, like the example \(0.4242...\).

The trick here is to use powers of 10 to move the decimal point to the right until one full cycle of the repeating pattern aligns with the original decimal. From there, as seen in the step by step solution, we use algebra to solve for the fraction.

Repeating Decimal Representation

A repeating decimal is represented by placing a line (called a vinculum) over the digit or group of digits that repeat. In our example, the decimal \(0.4242...\) would be written with a line over the '42' to indicate it repeats indefinitely. In mathematical notation, it is sometimes represented using an ellipsis (...), which signifies that the pattern \(42\) keeps recurring.

Repeating decimals hold a unique place in the number system because, although they appear to be non-terminating, they correspond to precise fraction or rational numbers. These fractions can always be found through the method of turning a decimal into a fraction, as demonstrated in the solution.

Simplifying Fractions

Once we have a fraction, it's often necessary to simplify it. Simplification makes the fraction easier to understand and work with. To simplify a fraction, search for the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both the top and bottom of the fraction without leaving a remainder.

For the fraction \(\frac{42}{99}\), we find that both 42 and 99 can be divided by 3. After doing the division, we obtain the simplified fraction \(\frac{14}{33}\), which cannot be reduced any further because 14 and 33 share no common factors other than 1. Simplifying fractions is a fundamental skill in algebra that helps in comparing, adding, subtracting, or manipulating fractions in any mathematical context.

Algebraic Equation Manipulation

Algebraic equation manipulation is a critical skill for solving a wide range of math problems. Essentially, it involves applying mathematical operations to both sides of an equation in order to solve for an unknown variable. In our example, we multiplied both sides of the equation by 100 to align the decimal places correctly and then subtracted the original equation from this new equation.

This technique cleverly removes the decimal part, leaving us with an equation that can easily be solved for the unknown variable. Achieving mastery in manipulating equations involves practicing various operations such as addition, subtraction, multiplication, division, and even factoring to isolate and determine the value of variables. Such a fundamental understanding enables us to convert complex repeating decimals into neat fractions, as seen in the process of converting \(0.4242...\) into \(\frac{14}{33}\).