How do you convert recurring decimals to fractions?

Converting recurring decimals to fractions is a process in mathematics that involves converting numbers with infinite repeating decimals into fractions. This process is useful for simplifying and representing repeating decimals in a more concise and understandable form.

One method of converting recurring decimals to fractions involves using algebraic equations. Let's take an example to illustrate this process. Consider the repeating decimal 0.666..., where the digit 6 repeats infinitely.

Step 1: Let x represent the repeating decimal.

Step 2: Multiply both sides of the equation by 10 to eliminate the repeating decimal. This gives us 10x = 6.666... .

Step 3: Subtract the original equation from the new equation to eliminate the decimal point. This gives us 10x - x = 6.666... - 0.666..., which simplifies to 9x = 6.

Step 4: Divide both sides of the equation by 9 to solve for x. This gives us x = 6/9, which can be further simplified to x = 2/3.

Therefore, the recurring decimal 0.666... is equivalent to the fraction 2/3. This process can be applied to any repeating decimal to convert it into its corresponding fraction.

It's important to note that not all decimals can be converted into exact fractions. Some decimals, such as √2 or π (pi), are considered irrational numbers and cannot be expressed as a fraction with whole numbers. In these cases, the decimal is said to be non-terminating and non-recurring.

In conclusion, converting recurring decimals to fractions is a method used to simplify and represent repeating decimals in a more concise form. By using algebraic equations, it is possible to convert a repeating decimal into its equivalent fraction.

How do you convert a recurring decimal into a fraction?

To convert a recurring decimal into a fraction, you need to follow a few steps. First, identify the repeating pattern in the decimal. This pattern is usually represented by a set of digits that repeat continuously. For example, in the decimal 0.333..., the digit 3 repeats indefinitely.

Next, let's denote the repeating pattern as x. To highlight the repetition, we can write the decimal as a sum of two terms, one that represents the non-repeating part (let's call it a) and another that represents the repeating part (x). So the decimal 0.333... can be written as a + x.

Now comes the crucial step. We need to isolate the repeating pattern in the decimal. To do this, we multiply the entire expression a + x by a power of 10 that eliminates the decimal places in the repeating part. Since there is only one digit repeating in our example (3), we multiply by 10. If there were two digits repeating, we would multiply by 100, and so on.

After multiplying by 10, the decimal expression changes to 10a + 10x. Now, let's subtract the original expression a + x from this new expression. This will eliminate the decimal places in the repeating part, as x - x = 0. The expression becomes 10a + 10x - a - x = 9a + 9x.

Now that the repeating part has been eliminated, we can solve for x. We can factor out 9 from the expression, giving us 9(a + x) = 9a + 9x = 9a. Simplifying further, we get x = 9a.

Finally, to convert the decimal into a fraction, we write the expression a + x = a + 9a as a fraction. Combining the terms, we get 10a = 9a. Dividing both sides of the equation by 9, we find that a = 1.

Therefore, the original decimal 0.333... can be written as the fraction 1/3. This process works for other recurring decimals as well, helping us convert them into fractions. Understanding and applying this method will enable you to solve problems involving recurring decimals with ease and accuracy.

How do you turn 0.33333 into a fraction?

Turning 0.33333 into a fraction is a process that involves manipulating the decimal value to express it as a fraction. To begin, we need to understand that the decimal value 0.33333 represents the number three-thousandths (3/1000). Converting this decimal into a fraction will help us represent it in a simpler form.

One way to turn 0.33333 into a fraction is by recognizing that the decimal can be expressed as a repeating decimal. In this case, the digits 3 in the decimal repeat indefinitely. To represent this as a fraction, we need to identify the pattern and create an equation.

Let's assign a variable to the repeating decimal, say x. Since the digit 3 repeats, we can write it as:

x = 0.33333

If we multiply both sides of the equation by 10, we can shift the decimal point one place to the right:

10x = 3.33333

Now, we subtract the original equation from this new equation to eliminate the repeating decimal:

10x - x = 3.33333 - 0.33333

This simplifies to:

9x = 3

To isolate x, the variable, we divide both sides of the equation by 9:

x = 3/9

And finally, we can simplify the fraction by dividing the numerator and denominator by their greatest common divisor, which is 3:

x = 1/3

Hence, we have successfully turned the repeating decimal 0.33333 into the simplified fraction 1/3.

What is the shortcut to convert repeating decimal to fraction?

When dealing with repeating decimals, it can be quite time-consuming to convert them into fractions, especially if the repeating pattern is long. However, there is a shortcut method that simplifies this process and allows you to obtain the fraction quickly.

The first step in this shortcut method is to identify the repeating pattern in the decimal. Once you have identified the pattern, write it down without the dots or bars above the repeating digits. For example, if the decimal is 0.333..., the repeating pattern is 3. Remove the dots or bars and write 3 as the repeating pattern.

Next, determine the number of digits in the repeating pattern. Count the digits in the pattern you just wrote down. In this case, there is only one digit, which is 3.

Now, you need to construct the numerator of the fraction. Write down the repeating pattern as the numerator. For example, write 3 as the numerator.

Finally, construct the denominator of the fraction. The denominator is determined by the number of 9s in the repeating pattern, followed by as many zeros as there are digits in the repeating pattern. In this case, since the repeating pattern has only one digit (3), write 9 followed by one zero. Therefore, the denominator would be 90.

Putting it all together, the decimal 0.333... can be converted to the fraction 3/90. However, we can simplify this fraction further. Both the numerator and the denominator have a common factor of 3. By dividing both the numerator and the denominator by 3, the fraction simplifies to 1/30.

This shortcut method allows you to quickly convert repeating decimals to their corresponding fractions without much hassle. It is especially useful when dealing with longer repeating patterns, as it saves time and reduces the chances of errors.

How do you write 0.25 repeating as a fraction?

Writing 0.25 repeating as a fraction involves understanding the concept of repeating decimals and converting them into rational numbers. When we say that 0.25 is repeating, we mean that the decimal 0.25 is followed by an infinite number of 25s.

To represent this as a fraction, we first need to identify the pattern in the decimal. In the case of 0.25 repeating, we see that the decimal repeats the number 25. Therefore, we can represent this decimal as follows:

0.25 repeating = 0.252525...

We can express the decimal as an infinite geometric series. Let's denote the decimal as "x":

x = 0.252525...

If we multiply the decimal by 100, we shift the entire decimal two places to the right:

100x = 25.252525...

Now, if we subtract the original decimal "x" from the shifted decimal "100x," we eliminate the repeating parts:

100x - x = 25.252525... - 0.252525...

This simplifies to:

99x = 25

To solve for "x," we divide both sides by 99:

x = 25/99

Therefore, 0.25 repeating as a fraction is 25/99. This fraction represents the decimal accurately and completely.