Question
To convert 0.264¯¯¯¯¯¯¯¯ to a rational number, you would set x equal to 0.264¯¯¯¯¯¯¯¯ and then multiply both sides of the equation by what number?(1 point)
Responses
999
999
1,000
1,000
10
10
100
Responses
999
999
1,000
1,000
10
10
100
Answers
GPT-4o mini
To convert the repeating decimal \( 0.264\overline{6} \) to a rational number, we can set \( x \) equal to the decimal:
\[
x = 0.2646666\ldots
\]
To eliminate the repeating part, we need to determine how many digits are in the repeating cycle. The repeating part ("6") has 1 digit. We also consider the non-repeating part ("264") which has 3 digits before the repeating part starts.
Thus, we can multiply both sides by \( 10^4 \) (which shifts the decimal point 4 places to the right) because \( 3 \) digits of the non-repeating part plus \( 1 \) digit of the repeating part equals \( 4 \):
\[
10000x = 2646.6666\ldots
\]
This shows that the correct answer for the multiplier to eliminate the decimal and shift it appropriately is **10000**.
However, given the provided options, the closest are:
- **100** (which would work for \( 0.2\overline{666}\))
- **10** (which would work for \( 0.26\overline{6}\))
- **1,000** (which would work for \( 0.264\overline{6}\), effectively making sense only if we were just shifting the decimal by 3 places)
So while \( 10^4 = 10000 \) would perfectly eliminate the repeating decimal, if you're looking for the correct answer from the options provided related to how many digits you shift for the repeating decimal:
Since the answer is denominated in terms multipliers given, **1000** (to shift the decimal 3, almost reaching the repeating effect) would be the best guess if forced to choose from less accurate options available. However, note that fully examining would need \( 10000.\)
Thus the response you seek (from provided options): **1,000**.
\[
x = 0.2646666\ldots
\]
To eliminate the repeating part, we need to determine how many digits are in the repeating cycle. The repeating part ("6") has 1 digit. We also consider the non-repeating part ("264") which has 3 digits before the repeating part starts.
Thus, we can multiply both sides by \( 10^4 \) (which shifts the decimal point 4 places to the right) because \( 3 \) digits of the non-repeating part plus \( 1 \) digit of the repeating part equals \( 4 \):
\[
10000x = 2646.6666\ldots
\]
This shows that the correct answer for the multiplier to eliminate the decimal and shift it appropriately is **10000**.
However, given the provided options, the closest are:
- **100** (which would work for \( 0.2\overline{666}\))
- **10** (which would work for \( 0.26\overline{6}\))
- **1,000** (which would work for \( 0.264\overline{6}\), effectively making sense only if we were just shifting the decimal by 3 places)
So while \( 10^4 = 10000 \) would perfectly eliminate the repeating decimal, if you're looking for the correct answer from the options provided related to how many digits you shift for the repeating decimal:
Since the answer is denominated in terms multipliers given, **1000** (to shift the decimal 3, almost reaching the repeating effect) would be the best guess if forced to choose from less accurate options available. However, note that fully examining would need \( 10000.\)
Thus the response you seek (from provided options): **1,000**.