Convert $3.\overline{5}\dot{3}$ into

To convert the repeating decimal 3. 5 3 into a fraction, define x = 3. 5 3 and multiply by 100 to eliminate the repeating part. This results in the equation 99 x = 350 , which simplifies to x = 99 350 ​ . Therefore, the repeating decimal is equal to the fraction 99 350 ​ .
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Answered by Anonymous | 2025-07-04

Let x = 3. 53 .
Multiply by 100: 100 x = 353. 53 .
Subtract x from 100 x : 99 x = 350 .
Solve for x : x = 99 350 ​ ​ .

Explanation

Understanding the Problem We are asked to convert the repeating decimal 3. 53 into a fraction. This means we need to express the number 3.535353... in the form q p ​ , where p and q are integers and q  = 0 .

Setting up the Equation Let x = 3. 53 = 3.535353... . To eliminate the repeating part, we multiply x by 100, since the repeating block '53' has a length of 2. This shifts the decimal point two places to the right: 100 x = 353.535353... .

Eliminating the Repeating Part Now, we subtract x from 100 x to eliminate the repeating decimal part: 100 x − x = 353.535353... − 3.535353... . This simplifies to 99 x = 350 .

Solving for x To solve for x , we divide both sides of the equation 99 x = 350 by 99: x = 99 350 ​ .

Final Answer Therefore, the repeating decimal 3. 53 is equal to the fraction 99 350 ​ .


Examples
Repeating decimals are commonly encountered when dealing with measurements or calculations that don't result in whole numbers. For example, when dividing 10 by 3, you get 3. 3 . Converting repeating decimals to fractions allows for more precise calculations, especially in fields like engineering or finance where accuracy is crucial. Imagine you're calculating the area of a rectangular garden plot and one side measures 3. 53 meters. Converting this to 99 350 ​ meters allows you to find the exact area without rounding errors.

Answered by GinnyAnswer | 2025-07-04