Question

Write each decimal number as a percent. $$1 . \overline{63}$$

Short answer

Answer: \(163\frac{7}{11}\%\).

Step-by-step solution

Identify the repeating part of the decimal number

First, observe that \(1 . \overline{63}\) has a repeating part: \(63\). This means that the number can be expressed as the mixed number \(1 + 0.\overline{63}\).

Convert the repeating decimal to a fraction

Let \(x = 0.\overline{63}\). To convert this repeating decimal to a fraction, we can multiply both sides of the equation by 100: $$100x = 63.\overline{63}.$$ Next, subtract \(x\) from \(100x\): $$100x - x = 63.\overline{63} - 0.\overline{63}.$$ Simplifying this gives: $$99x = 63.$$ Now, divide both sides by 99: $$x = \frac{63}{99}.$$ Reduce the fraction to its simplest form by dividing both the numerator and denominator by their greatest common divisor, which is 9: $$x = \frac{7}{11}.$$ Now, our number can be written as \(1 + \frac{7}{11}\).

Convert the fraction to a percentage

To convert the mixed number \(1 + \frac{7}{11}\) to a percentage, first add the whole number and the fraction: $$1 + \frac{7}{11} = \frac{11}{11} + \frac{7}{11} = \frac{18}{11}.$$ Next, to express the fraction as a percent, multiply it by 100: $$\frac{18}{11} \times 100\% = \frac{1800}{11}\%.$$ Finally, we can express the result as a mixed number, which is more easily understood: $$\frac{1800}{11}\% = 163\frac{7}{11}\%.$$ So, the decimal number $$1 . \overline{63}$$ can be written as $$163\frac{7}{11}\%$$.

Key concepts

Repeating Decimals

Understanding repeating decimals is a foundational skill for converting such numbers into fractions and percents. A repeating decimal is a decimal number that has a digit or a group of digits after the decimal point that repeat indefinitely. For instance, in the number \(1 . \overline{63}\), the numbers 63 are the repeating digits, often denoted by a bar over the digits to indicate the repeat. The challenge with repeating decimals is to express them as precise fractions since our decimal system doesn't easily accommodate infinite repeats. The trick is to create an equation where the repeating decimal is represented by a variable (like \(x\)), which is then manipulated to reveal its fractional equivalent. This process is quite ingenious—by multiplying the variable by a power of 10 so that the repeating digits align, and then subtracting the original from the multiplied variable, you eliminate the repeating part, leaving you with a solvable equation.

Percent Conversion

Percent conversion is turning a number (usually a fraction or a decimal) into a percentage. The term 'percent' means 'per hundred', so converting to a percentage is essentially finding out how many parts of a hundred the number is equivalent to. The standard method involves multiplying the number by 100 and adding the percent symbol (%). For decimals, the process is as straightforward as shifting the decimal point two places to the right, but things get a little more involved with mixed numbers or repeating decimals. With the mixed number \(1 + \frac{7}{11}\) derived from the repeating decimal \(1 . \overline{63}\), you're dealing with a value already greater than 1, which corresponds to 100% and then some. It’s essential to convert any whole numbers to the equivalent 'per hundred' form before addressing the fractional part to provide an accurate percentage representation.

Fractions to Percents

Now we delve into the transition from fractions to percents, which is an extension of the percent conversion process. Since a percent is a fraction with a denominator of 100, converting is a matter of scaling the fraction up or down. For example, \(\frac{18}{11}\) cannot be directly represented out of 100, so you multiply the top and bottom of the fraction by the same number that would lead to a denominator of 100. In this case, though, the most convenient route is to simply multiply by 100 to find the percentage, knowing that the result won't be a neat whole number, hence \(\frac{1800}{11}\%\). A student must be comfortable with the idea that percentages can be larger than 100 and can also be given in mixed number form, which can sometimes make them easier to understand and visualize.

Simplifying Fractions

The last important step is simplifying fractions, which makes it easier to work with them and understand their value. Simplifying, or reducing, a fraction means finding an equivalent fraction where the numerator (top number) and the denominator (bottom number) are as small as possible. This is achieved by dividing both the top and bottom by their greatest common divisor. In our example, the fraction \(\frac{63}{99}\) is simplified by acknowledging that 9 is the highest number that divides evenly into both 63 and 99. Dividing both by 9 simplifies the fraction to \(\frac{7}{11}\), which is in its simplest form. This step is vital not just to make the math look nicer, but to help you easily convert the fraction into other forms, like percents or different fractions, and for better practical comprehension in comparisons and other mathematical operations.