If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with what denominator? (1 point)
The repeating decimal will convert to a rational number with a denominator of 999.
How do you show the group of digits that repeat in a repeating decimal?(1 point)
To show the group of digits that repeat in a repeating decimal, you can use placing a line or bar (known as a vinculum) over the repeating part. For example, if the digits 123 repeat in a decimal, you can represent it as 0.123 with a bar over the 123: 0.ᴣ123.
How do you show the group of digits that repeat in a repeating decimal?(1 point)
Responses
by underlining them
by printing them in italics
by putting them in parentheses
by putting a bar over them
The correct answer is by putting a bar over them.
How many digits long could the repeating cycle of a repeating decimal be if the denominator of the rational number is 7?(1 point)
If the denominator of the rational number is 7, the repeating cycle of the repeating decimal could be at most 6 digits long.
This is because the maximum length of the repeating cycle in a decimal representation with a denominator of 7 will be 6 digits, ranging from 0.142857 to 0.999999.
Convert 0.27
to a rational number in simplest form
To convert 0.27 to a rational number in simplest form, we need to express it as a fraction.
Let's call the number 0.27 as x.
We can multiply x by 100 to remove the decimal:
100x = 27
Now we have the equation:
100x = 27
Dividing both sides of the equation by 100 gives us:
x = 27/100
The fraction 27/100 is not in simplest form yet, as both 27 and 100 have a common factor of 9.
We can simplify the fraction further by dividing the numerator and denominator by their greatest common divisor, which is 9:
27/100 = (27/9)/(100/9) = 3/11
Therefore, 0.27 as a rational number in simplest form is 3/11.
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)
To convert 0.264 to a rational number, we would indeed set x equal to 0.264 and then multiply both sides of the equation by 1000.
Setting x = 0.264, we have the equation:
1000x = 264
By multiplying both sides by 1000, we eliminate the decimal point and obtain the equation in integer form. This allows us to represent the rational number as a fraction.
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with what denominator? (1 point)
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with a denominator of 999.
How many digits are in the repeating cycle of 1727?(1 point)
To determine the number of digits in the repeating cycle of 1727, we can divide it by the denominator of 9.
When we perform the division, the remainder pattern will repeat until we reach a remainder of 0, which indicates that the division is complete. The length of this repeating pattern will give us the number of digits in the repeating cycle.
Dividing 1727 by 9, we have:
1727 ÷ 9 = 191 remainder 8
80 ÷ 9 = 8 remainder 8
...
...
...
The remainder pattern is 8, 8, 8, 8, ...
Therefore, the repeating cycle of 1727 is a single digit - 8. Hence, the number of digits in the repeating cycle of 1727 is 1.
Which of the following rational numbers can be expressed as a repeating decimal: 78, 512, 1725, or 3140?(1 point)
In order to determine which of the given rational numbers can be expressed as a repeating decimal, we need to examine their decimal expansions.
- 78: When we divide 78 by any power of 10 (10, 100, 1000, etc.), we get a terminating decimal. Therefore, 78 can be expressed as a terminating decimal.
- 512: Similar to 78, when we divide 512 by any power of 10, we obtain a terminating decimal. Thus, 512 can also be expressed as a terminating decimal.
- 1725: Dividing 1725 by 10, we get 172.5, which is not a terminating decimal. This indicates that 1725 cannot be expressed as a terminating decimal. However, it is possible that it can be expressed as a repeating decimal.
- 3140: Dividing 3140 by 10, we get 314, which is not a terminating decimal. Therefore, 3140 cannot be expressed as a terminating decimal, and it is worth investigating if it can be expressed as a repeating decimal.
Based on this analysis, both 1725 and 3140 could potentially be expressed as repeating decimals.
Which of the following rational numbers can be expressed as a repeating decimal 7/8,5/12,17/25,31/40
To determine which of the given rational numbers can be expressed as a repeating decimal, we need to examine their decimal equivalents.
Let's break down each fraction:
1. 7/8: When we divide 7 by 8, we get a decimal of 0.875. This is not a repeating decimal, so 7/8 cannot be expressed as a repeating decimal.
2. 5/12: Dividing 5 by 12, we get a decimal of approximately 0.416666... This decimal representation shows a repeating cycle of 6s after the decimal point. Therefore, 5/12 can be expressed as a repeating decimal.
3. 17/25: Dividing 17 by 25, we get a decimal of 0.68. This is not a repeating decimal, so 17/25 cannot be expressed as a repeating decimal.
4. 31/40: Dividing 31 by 40, we get a decimal of 0.775. Similar to 7/8, this is not a repeating decimal. Therefore, 31/40 cannot be expressed as a repeating decimal.
Based on this analysis, the rational number 5/12 can be expressed as a repeating decimal.
Express 7/13 as a repeating decimal. Mark the repeating digits with a bar.
To express 7/13 as a repeating decimal, we need to perform long division.
0.
___________
13 | 7.00000
- 6.5
_______
5
- 4.5
_______
10
- 9.1
_______
90
- 78
_______
120
- 117
_______
30
The division repeats indefinitely, with a repeating cycle of 076923.
Therefore, 7/13 as a repeating decimal is 0.538461538461... (with the 076923 part repeating) which can be represented as 0.538̄.
Convert 0.8 to a rational number.(1 point)
To convert 0.8 to a rational number, we can express it as a fraction.
Let's call the number 0.8 as x.
We can multiply x by 10 to remove the decimal:
10x = 8
Now we have the equation:
10x = 8
Dividing both sides of the equation by 10 gives us:
x = 8/10
The fraction 8/10 is not in simplest form yet, as both 8 and 10 have a common factor of 2.
We can simplify the fraction further by dividing the numerator and denominator by their greatest common divisor, which is 2:
8/10 = (8/2)/(10/2) = 4/5
Therefore, 0.8 as a rational number is 4/5.
To convert 0.82¯¯¯¯¯ to a rational number, you would set x equal to 0.82¯¯¯¯¯ and then multiply both sides of the equation by what number?
To convert 0.82¯¯¯¯¯ to a rational number, we would indeed set x equal to 0.82¯¯¯¯¯ and then multiply both sides of the equation by 100.
Setting x = 0.82¯¯¯¯¯, we have the equation:
100x = 82.82¯¯¯¯¯
By multiplying both sides by 100, we eliminate the decimal point and obtain the equation in integer form. This allows us to represent the rational number as a fraction.