Repeating Decimals Calculation: 2.&overline{36} + 3.&overline{21} - 1.&overline{05}

This question requires us to perform addition and subtraction operations on numbers expressed as repeating decimals. The goal is to find the value of the expression $2.\overline{36} + 3.\overline{21} - 1.\overline{05}$.

Understanding Repeating Decimals

A repeating decimal is a decimal number where a sequence of digits repeats infinitely. The bar over the digits signifies the repeating block.

• $2.\overline{36}$ means $2.363636...$
• $3.\overline{21}$ means $3.212121...$
• $1.\overline{05}$ means $1.050505...$

Converting Repeating Decimals to Fractions

To perform the calculation accurately, we first convert each repeating decimal into its equivalent fraction form. We use algebra for this conversion.

1. Converting $2.\overline{36}$ to a Fraction: Let the number be $x$. So, $x = 2.\overline{36}$. Since two digits repeat, multiply by $100$: $100x = 236.\overline{36}$. Subtract the first equation from the second: $100x - x = 236.\overline{36} - 2.\overline{36}$ $99x = 234$ Therefore, $x = \frac{234}{99}$.
2. Converting $3.\overline{21}$ to a Fraction: Let the number be $y$. So, $y = 3.\overline{21}$. Multiply by $100$: $100y = 321.\overline{21}$. Subtract the first equation from the second: $100y - y = 321.\overline{21} - 3.\overline{21}$ $99y = 318$ Therefore, $y = \frac{318}{99}$.
3. Converting $1.\overline{05}$ to a Fraction: Let the number be $z$. So, $z = 1.\overline{05}$. Multiply by $100$: $100z = 105.\overline{05}$. Subtract the first equation from the second: $100z - z = 105.\overline{05} - 1.\overline{05}$ $99z = 104$ Therefore, $z = \frac{104}{99}$.

Performing the Arithmetic Calculation

Now substitute these fractions back into the original expression:

Expression = $x + y - z = \frac{234}{99} + \frac{318}{99} - \frac{104}{99}$

Since all fractions have the same denominator ($99$), we can combine the numerators:

Expression = $\frac{234 + 318 - 104}{99}$

First, add the positive terms: $234 + 318 = 552$.

Next, subtract the negative term: $552 - 104 = 448$.

So, the expression equals $\frac{448}{99}$.

Converting the Result Back to a Repeating Decimal

The final step is to convert the resulting fraction $\frac{448}{99}$ back into a repeating decimal.

To do this, divide the numerator ($448$) by the denominator ($99$):

$\frac{448}{99} = 448 \div 99$

Performing the division:

$448 \div 99 = 4$ with a remainder of $52$. ($4 \times 99 = 396$; $448 - 396 = 52$).

This can be written as a mixed number: $4 \frac{52}{99}$.

The fractional part $\frac{52}{99}$ converts to the repeating decimal $0.525252...$, which is represented as $0.\overline{52}$.

Therefore, the value of the expression is $4.\overline{52}$.