0.16666666666 I need to know what this number is a fraction.
The Correct Answer and Explanation is:
The decimal number 0.16666666666… is a repeating decimal, where the digit 6 repeats infinitely. This number can be exactly expressed as the fraction 1/6.
Explanation
Decimals that repeat in a consistent pattern can often be written as exact fractions. In this case, the decimal 0.16666666666… (with the 6 repeating forever) is a classic example of a repeating decimal that corresponds to a simple rational number.
Let’s denote the number as:
x = 0.1666666666…
We notice the repeating part is just the digit 6, and it starts immediately after the digit 1 in the tenths place. Here’s how we can convert this to a fraction step by step.
Step 1: Let x = 0.1666666666…
We want to eliminate the repeating part. To do that, we multiply x by 10 so the repeating part lines up with itself:
10x = 1.6666666666…
Now subtract the original x from this equation:
10x – x = 1.6666666… – 0.1666666…
That gives:
9x = 1.5
Now divide both sides by 9:
x = 1.5 / 9
We can simplify 1.5 as 3/2:
x = (3/2) / 9 = 3 / 18 = 1 / 6
So, the decimal 0.1666666666… = 1/6
Why Does This Work?
Repeating decimals occur because of division. When a number doesn’t divide evenly, the remainder repeats. In this case, 1 ÷ 6 = 0.1666… because 6 doesn’t divide 1 evenly and creates a repeating cycle.
This is one way to define rational numbers: any number that can be expressed as a ratio of two integers. Repeating decimals are always rational, and converting them into fractions involves algebraic manipulation like the example above.
So the decimal 0.1666666666… is exactly equal to the fraction 1/6.
