Is 7 x = 1 then find the decimal expansion of x
The Correct Answer and Explanation is:
To solve the equation:
7x = 1
We divide both sides by 7: x=17x = \frac{1}{7}
Now, let’s find the decimal expansion of 17\frac{1}{7}: 17=0.142857‾\frac{1}{7} = 0.\overline{142857}
This means the decimal repeats every 6 digits: 0.142857142857142857…
Explanation
To understand why 17=0.142857‾\frac{1}{7} = 0.\overline{142857}, we can use long division and explore the nature of repeating decimals.
First, consider what dividing 1 by 7 means: you are distributing 1 equally among 7 parts. Since 1 is smaller than 7, the result will be less than 1, i.e., a decimal number.
Using long division:
- 7 goes into 10 once (1 × 7 = 7), remainder 3 → write down 1
- Bring down a 0 → 7 into 30 = 4 times (4 × 7 = 28), remainder 2 → write 4
- Bring down a 0 → 7 into 20 = 2 times (2 × 7 = 14), remainder 6 → write 2
- 7 into 60 = 8 (8 × 7 = 56), remainder 4 → write 8
- 7 into 40 = 5 (5 × 7 = 35), remainder 5 → write 5
- 7 into 50 = 7 (7 × 7 = 49), remainder 1 → write 7
Now we’re back to the original situation: remainder 1, meaning the process repeats.
So, 17\frac{1}{7} has a repeating cycle of six digits: 142857.
This is an example of a repeating decimal, which occurs whenever the denominator of a fraction (in lowest terms) has prime factors other than 2 or 5. Since 7 is a prime other than 2 or 5, 17\frac{1}{7} cannot terminate and must repeat.
Interestingly, 142857 is a cyclic number. Multiplying it by numbers 1 to 6 produces cyclic permutations of itself:
- 1 × 142857 = 142857
- 2 × 142857 = 285714
- 3 × 142857 = 428571
- …and so on.
Thus, the decimal expansion of xx in the equation 7x=17x = 1 is: x=0.142857‾\boxed{x = 0.\overline{142857}}
