Question

Find \(0.25 \underline{25}\) as a quotient of integers.

Short answer

The repeating decimal \(0.25\underline{25}\) can be expressed as a quotient of integers as follows: \(0.25 \underline{25} = \frac{25}{99}\).

Step-by-step solution

Define the variable and assign the repeating decimal formula

Let's define the variable x to be the given repeating decimal: \(x = 0.25\underline{25}\)

Multiply x by an appropriate power of 10 to isolate the repeating decimal part

We'll multiply x by 100 to isolate the repeating part in the decimal: \(100x = 25.\underline{25}\)

Subtract x from the equation in step 2

Next, we'll subtract x from the result in step 2 to remove the repeating decimal portion: \(100x - x = 25.\underline{25} - 0.25\underline{25}\) This results in: \(99x = 25\)

Solve for x

Now we can solve for x by dividing both sides of the equation by 99: \(x = \frac{25}{99}\)

Write the final answer

So, \(0.25 \underline{25}\) can be expressed as a quotient of integers like this: \(0.25 \underline{25} = \frac{25}{99}\)

Key concepts

Quotient of Integers

A repeating decimal like \(0.25\underline{25}\) can seem confusing at first. However, we can express it as a fraction, or a quotient of two integers. A quotient of integers simply means one integer divided by another, resulting in a fraction. In this case, the fraction represents the repeating decimal exactly.

To find this quotient, a common method is to use algebraic manipulation. You essentially set up an equation where you solve for the unknown variable that represents the decimal. Through well-placed multiplications and subtractions, you isolate the repeating portion and express the number as a fraction. This method works for any repeating decimal and helps convert it into a form of a quotient.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate a variable or solve for an unknown. In the case of repeating decimals, this strategy is extremely useful. Here's how it works for our specific example of \(0.25\underline{25}\):

• We start by assigning the repeating decimal to a variable: \(x = 0.25 \underline{25}\).
• Next, multiply by a power of 10 to shift the decimal point to a position where the repeating digits line up: in this case, \(100x = 25.25\underline{25}\).
• The key step is subtracting the original \(x\) from \(100x\). This cancels out the repeating decimals, leaving us with a simple equation: \(99x = 25\).
• Finally, solve for \(x\) by dividing both sides by 99, giving us \(x = \frac{25}{99}\).

By following these steps, the repeating part is effectively eliminated, making it easier to express the repeating decimal as a fraction.

Decimal Representation

Decimal representation is the way numbers are displayed in the base-10 numeral system. Repeating decimals are a special case where a sequence of digits repeats indefinitely.

For \(0.25\underline{25}\), the digits '25' repeat. When notated with a line (\(\underline{25}\)), it indicates that the pattern goes on without end. This form of representation is common for fractions where the denominator does not divide the numerator evenly.

Converting a repeating decimal to a fraction helps us understand its exact value, rather than its approximate decimal form. Recognizing patterns in decimal representation allows us to translate the number into a more manageable and exact form, namely a quotient of integers.

Rational Numbers

Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero. Every repeating decimal is a rational number because it can always be written as such a quotient.

In our example, \(0.25\underline{25}\) translates into the rational number \(\frac{25}{99}\). This showcases the beauty of numbers, where even an endless stream of digits can be captured within the simplicity of a fraction. It's important to note that rational numbers include both terminating and repeating decimals.

• Terminating decimals come to an end after a finite number of digits.
• Repeating decimals continue indefinitely but follow a repetitive pattern.

Understanding rational numbers as quotients of integers helps you see the connections between their decimal forms and the underlying fractions, simplifying complex-looking numbers into straightforward expressions.