write the decimal number 0.83333333 as a rational number
The correct answer and explanation is:
The decimal number 0.83333333 can be expressed as a rational number in fraction form.
Step 1: Identify the repeating decimal
The decimal 0.83333333… can be seen as:
- 0.8 (which is 8 tenths), plus
- 0.03333333… (a repeating decimal)
The repeating part is the digit 3, repeating infinitely.
Step 2: Express the decimal as sum of parts
We write: 0.83333333…=0.8+0.03333333…0.83333333… = 0.8 + 0.03333333…
We already know 0.8 = 810=45\frac{8}{10} = \frac{4}{5}.
Now focus on 0.03333333…0.03333333….
Step 3: Express the repeating decimal 0.0333333…0.0333333… as a fraction
Let x=0.0333333…x = 0.0333333….
Multiply xx by 10: 10x=0.3333333…10x = 0.3333333…
We know that 0.3333…0.3333… (repeating 3) equals 13\frac{1}{3}.
So: 10x=13 ⟹ x=13010x = \frac{1}{3} \implies x = \frac{1}{30}
Step 4: Add the two parts as fractions
Recall: 0.83333333…=45+1300.83333333… = \frac{4}{5} + \frac{1}{30}
Find a common denominator, which is 30: 45=2430\frac{4}{5} = \frac{24}{30}
Add: 2430+130=2530\frac{24}{30} + \frac{1}{30} = \frac{25}{30}
Simplify the fraction: 2530=56\frac{25}{30} = \frac{5}{6}
Final answer:
0.83333333…=560.83333333… = \frac{5}{6}
Explanation:
Decimal numbers that repeat infinitely, such as 0.3333…, 0.6666…, or 0.8333333…, can be expressed exactly as fractions, also called rational numbers. A rational number is any number that can be written as the ratio of two integers pq\frac{p}{q}, where q≠0q \neq 0.
The decimal 0.83333333… consists of a non-repeating part (0.8) followed by a repeating part (0.03333…). To convert such a decimal, you break it down into these two parts and convert each to fractions separately.
The repeating decimal 0.3333… is well known to equal 13\frac{1}{3}. Since in our case, the repeating 3 starts after one decimal place (i.e., it is 0.03 repeating), multiplying by 10 shifts the decimal point right so that the repeating part becomes 0.3333…, allowing us to isolate the repeating decimal as a fraction.
Once each part is converted, adding the fractions gives the overall rational number equivalent.
In conclusion, the repeating decimal 0.8333333…0.8333333… is exactly 56\frac{5}{6}, which means the decimal is just the decimal form of that fraction.