Using a number line, what whole number and rational number to the nearest half is 105−−−√ between?(1 point)
Responses
between 10.5 and 11
between 10.5 and 11
between 11 and 11.5
between 11 and 11.5
between 9.5 and 10
between 9.5 and 10
between 10 and 10.5
43 answers
between 10 and 10.5
Using rational approximations, what statement is true?(1 point)
Responses
48−−√>36−−√
start root 48 end root greater than start root 36 end root
49−−√>7
start root 49 end root greater than 7
48−−√<36−−√
start root 48 end root less than start root 36 end root
49−−√<7
Responses
48−−√>36−−√
start root 48 end root greater than start root 36 end root
49−−√>7
start root 49 end root greater than 7
48−−√<36−−√
start root 48 end root less than start root 36 end root
49−−√<7
start root 49 end root less than 7
Using rational approximations, what statement is true?(1 point)
Responses
71 −−−√<61−−√
start root 71 end root less than start root 61 end root
81−−√>9
start root 81 end root greater than 9
71−−√>61−−√
start root 71 end root greater than start root 61 end root
81−−√<9
Responses
71 −−−√<61−−√
start root 71 end root less than start root 61 end root
81−−√>9
start root 81 end root greater than 9
71−−√>61−−√
start root 71 end root greater than start root 61 end root
81−−√<9
start root 81 end root less than 9
Using rational approximations, what statement is true?(1 point)
Responses
12 −−−√ >π
start root 12 end root greater than pi
16−−√>4
start root 16 end root greater than 4
12 −−−√<π
start root 12 end root less than pi
16 −−−√<4
Responses
12 −−−√ >π
start root 12 end root greater than pi
16−−√>4
start root 16 end root greater than 4
12 −−−√<π
start root 12 end root less than pi
16 −−−√<4
start root 16 end root less than 4
Using a number line, what is the approximate value of 19−−√?(1 point)
Responses
4.5
4.5
4.13
4.13
4.25
4.25
4.38
Responses
4.5
4.5
4.13
4.13
4.25
4.25
4.38
4.38
Using a number line, what is the approximate number value of 103−−−√? Provide your answer in decimal form, rounded to the nearest hundredth.(1 point)
103−−−√ is approximately
103−−−√ is approximately
10.15
A government meets to discuss policies about controlling pollution. Which is the most appropriate type of information for the government to use to make decisions?(1 point)
Responses
personal anecdotes from citizens
personal anecdotes from citizens
stories from other governments
stories from other governments
guesses about the impact of pollutants
guesses about the impact of pollutants
research reports containing scientific data
research reports containing scientific data
Responses
personal anecdotes from citizens
personal anecdotes from citizens
stories from other governments
stories from other governments
guesses about the impact of pollutants
guesses about the impact of pollutants
research reports containing scientific data
research reports containing scientific data
research reports containing scientific data
A patient reports side effects from a medication. A government committee learns about the side effects and decides the medication is not safe for public use. Did the committee use science to make the decision?(1 point)
Responses
Yes; the committee used research.
Yes; the committee used research.
No; committees use rules, not science.
No; committees use rules, not science.
Yes; all medicine is science.
Yes; all medicine is science.
No; the committee used an anecdote.
No; the committee used an anecdote.
Responses
Yes; the committee used research.
Yes; the committee used research.
No; committees use rules, not science.
No; committees use rules, not science.
Yes; all medicine is science.
Yes; all medicine is science.
No; the committee used an anecdote.
No; the committee used an anecdote.
No; the committee used an anecdote.
Which type of research might result in a law that restricts hunting of a particular species?(1 point)
Responses
research showing a decrease in the popularity of hunting
research showing a decrease in the popularity of hunting
research showing the species is harmful
research showing the species is harmful
research showing the species is endangered in other geographical areas
research showing the species is endangered in other geographical areas
research showing an increase in the population of the species
Responses
research showing a decrease in the popularity of hunting
research showing a decrease in the popularity of hunting
research showing the species is harmful
research showing the species is harmful
research showing the species is endangered in other geographical areas
research showing the species is endangered in other geographical areas
research showing an increase in the population of the species
research showing the species is endangered in other geographical areas
Which statement best explains why a rare disease would get more research funding than a more common disease?(1 point)
Responses
Sufferers of the rare disease ran a public awareness campaign.
Sufferers of the rare disease ran a public awareness campaign.
Diseases that affect more people tend to get more funding.
Diseases that affect more people tend to get more funding.
Corporations refuse to fund research for rare diseases.
Corporations refuse to fund research for rare diseases.
Organizations are concerned about the results of the research into the rare disease.
Responses
Sufferers of the rare disease ran a public awareness campaign.
Sufferers of the rare disease ran a public awareness campaign.
Diseases that affect more people tend to get more funding.
Diseases that affect more people tend to get more funding.
Corporations refuse to fund research for rare diseases.
Corporations refuse to fund research for rare diseases.
Organizations are concerned about the results of the research into the rare disease.
Diseases that affect more people tend to get more funding.
Which of the following geographical locations share borders with Spain?
(1 point)
Responses
Portugal, France, and the Atlantic Ocean
Portugal, France, and the Atlantic Ocean
France, England, and the Pacific Ocean
France, England, and the Pacific Ocean
Italy, France, and the Mediterranean Sea
Italy, France, and the Mediterranean Sea
(1 point)
Responses
Portugal, France, and the Atlantic Ocean
Portugal, France, and the Atlantic Ocean
France, England, and the Pacific Ocean
France, England, and the Pacific Ocean
Italy, France, and the Mediterranean Sea
Italy, France, and the Mediterranean Sea
Portugal, France, and the Atlantic Ocean
How many digits are in the repeating cycle of 17/27?
The fraction 17/27 can be expressed as a repeating decimal. To determine the length of the repeating cycle, we can perform long division:
```
0.629629...
------------
27 | 17.000000...
16.2
-----
78
-54
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240 (repeating cycle starts repeating here)
...
```
From the long division, we can see that the repeating cycle starts after the decimal point and the digits 240 repeat continuously. Therefore, the repeating cycle of 17/27 has a length of 3 digits: 240.
Note: Sometimes, the repeating cycle may not start immediately after the decimal point, so additional calculations may be required to determine the exact length of the cycle. However, in this case, it is clear that the cycle starts right after the decimal point.
```
0.629629...
------------
27 | 17.000000...
16.2
-----
78
-54
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240
-216
----
240 (repeating cycle starts repeating here)
...
```
From the long division, we can see that the repeating cycle starts after the decimal point and the digits 240 repeat continuously. Therefore, the repeating cycle of 17/27 has a length of 3 digits: 240.
Note: Sometimes, the repeating cycle may not start immediately after the decimal point, so additional calculations may be required to determine the exact length of the cycle. However, in this case, it is clear that the cycle starts right after the decimal point.
Which of the following rational numbers can be expressed as a repeating decimal: 7/8, 5/12, 17/25, or 31/40?
To determine which rational numbers can be expressed as repeating decimals, we need to see if the denominators have any prime factors other than 2 and 5 (since decimal representations with repeating patterns are only possible when the denominator is in the form of 2^n * 5^m).
Among the given options:
- 7/8: The denominator 8 can be expressed as 2^3, which only has a factor of 2. So, 7/8 can be expressed as a terminating decimal.
- 5/12: The denominator 12 can be expressed as 2^2 * 3^1, which has factors of 2 and 3. Therefore, 5/12 can be expressed as a repeating decimal.
- 17/25: The denominator 25 can be expressed as 5^2, which only has a factor of 5. So, 17/25 can be expressed as a terminating decimal.
- 31/40: The denominator 40 can be expressed as 2^3 * 5^1, which has factors of both 2 and 5. Therefore, 31/40 can be expressed as a repeating decimal.
So, the rational numbers that can be expressed as repeating decimals are 5/12 and 31/40.
Among the given options:
- 7/8: The denominator 8 can be expressed as 2^3, which only has a factor of 2. So, 7/8 can be expressed as a terminating decimal.
- 5/12: The denominator 12 can be expressed as 2^2 * 3^1, which has factors of 2 and 3. Therefore, 5/12 can be expressed as a repeating decimal.
- 17/25: The denominator 25 can be expressed as 5^2, which only has a factor of 5. So, 17/25 can be expressed as a terminating decimal.
- 31/40: The denominator 40 can be expressed as 2^3 * 5^1, which has factors of both 2 and 5. Therefore, 31/40 can be expressed as a repeating decimal.
So, the rational numbers that can be expressed as repeating decimals are 5/12 and 31/40.
Express 7/13 as a repeating decimal. Mark the repeating digits with a bar.
To express 7/13 as a repeating decimal, we perform long division:
```
0.538461538461538...
--------------------
13 | 7.000000000000000...
65
-----
35
-26
----
90
-78
----
120
-117
-----
30
-26
-----
40
-39
-----
10
-9
----
10
-9
----
10
-9
----
...
```
The repeating cycle in the decimal representation of 7/13 is "538461", with the digits 538461 repeating infinitely. Therefore, we can express 7/13 as 0.538461 with the repeating digits marked with a bar: 0.538461̅.
```
0.538461538461538...
--------------------
13 | 7.000000000000000...
65
-----
35
-26
----
90
-78
----
120
-117
-----
30
-26
-----
40
-39
-----
10
-9
----
10
-9
----
10
-9
----
...
```
The repeating cycle in the decimal representation of 7/13 is "538461", with the digits 538461 repeating infinitely. Therefore, we can express 7/13 as 0.538461 with the repeating digits marked with a bar: 0.538461̅.
Convert 0.8¯¯¯ to a rational number.
To convert the repeating decimal 0.8¯¯¯ to a rational number, we can assign a variable (let's say x) to the repeating part of the decimal, which in this case is 8.
Since the repeating digit appears once after the decimal point, we multiply both sides of the equation by 10 to shift the decimal point one place to the right:
10x = 8.8¯¯¯
Next, we subtract the original equation from this new equation to eliminate the repeating part:
10x - x = 8.8¯¯¯ - 0.8¯¯¯
9x = 8
Finally, divide both sides of the equation by 9 to solve for x:
x = 8/9
Therefore, 0.8¯¯¯ is equivalent to the rational number 8/9.
Since the repeating digit appears once after the decimal point, we multiply both sides of the equation by 10 to shift the decimal point one place to the right:
10x = 8.8¯¯¯
Next, we subtract the original equation from this new equation to eliminate the repeating part:
10x - x = 8.8¯¯¯ - 0.8¯¯¯
9x = 8
Finally, divide both sides of the equation by 9 to solve for x:
x = 8/9
Therefore, 0.8¯¯¯ is equivalent to the rational number 8/9.
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, you would set x equal to 0.82¯¯¯¯¯ and then multiply both sides of the equation by 100, since there are two repeating digits (82) after the decimal point.
whats the answer tho
Apologies for the confusion. Let's proceed with the calculation.
If we set x = 0.82¯¯¯¯¯, we need to multiply both sides of the equation by a suitable power of 10 to eliminate the decimal part. Since there is a repeating block of two digits (82), we would multiply by 100.
100x = 82.82¯¯¯¯¯
Next, we subtract the original equation from this new equation to eliminate the repeating part:
100x - x = 82.82¯¯¯¯¯ - 0.82¯¯¯¯¯
99x = 82
Finally, we divide both sides of the equation by 99 to solve for x:
x = 82/99
Therefore, 0.82¯¯¯¯¯ is equivalent to the rational number 82/99.
If we set x = 0.82¯¯¯¯¯, we need to multiply both sides of the equation by a suitable power of 10 to eliminate the decimal part. Since there is a repeating block of two digits (82), we would multiply by 100.
100x = 82.82¯¯¯¯¯
Next, we subtract the original equation from this new equation to eliminate the repeating part:
100x - x = 82.82¯¯¯¯¯ - 0.82¯¯¯¯¯
99x = 82
Finally, we divide both sides of the equation by 99 to solve for x:
x = 82/99
Therefore, 0.82¯¯¯¯¯ is equivalent to the rational number 82/99.
its incorrect
Apologies for the incorrect response earlier. Let's reevaluate the calculation:
To convert 0.82¯¯¯¯¯ to a rational number, we can set x equal to 0.82¯¯¯¯¯ and multiply both sides of the equation by a power of 10 that eliminates the repeating part. Since there are two repeating digits (82) after the decimal point, we'll multiply by 100 to shift the decimal point two places to the right:
100x = 82.82¯¯¯¯¯
Next, we subtract the original equation from this new equation to eliminate the repeating part:
100x - x = 82.82¯¯¯¯¯ - 0.82¯¯¯¯¯
99x = 82
Finally, we divide both sides of the equation by 99 to solve for x:
x = 82/99
Therefore, 0.82¯¯¯¯¯ is equal to the rational number 82/99. I apologize for the confusion earlier, and thank you for pointing out the error.
To convert 0.82¯¯¯¯¯ to a rational number, we can set x equal to 0.82¯¯¯¯¯ and multiply both sides of the equation by a power of 10 that eliminates the repeating part. Since there are two repeating digits (82) after the decimal point, we'll multiply by 100 to shift the decimal point two places to the right:
100x = 82.82¯¯¯¯¯
Next, we subtract the original equation from this new equation to eliminate the repeating part:
100x - x = 82.82¯¯¯¯¯ - 0.82¯¯¯¯¯
99x = 82
Finally, we divide both sides of the equation by 99 to solve for x:
x = 82/99
Therefore, 0.82¯¯¯¯¯ is equal to the rational number 82/99. I apologize for the confusion earlier, and thank you for pointing out the error.
How many digits long could the repeating cycle of a repeating decimal be if the denominator of the rational number is 7?(1 point)
Responses
up to seven digits long
up to seven digits long
up to six digits long
up to six digits long
any finite length
any finite length
infinitely long
Responses
up to seven digits long
up to seven digits long
up to six digits long
up to six digits long
any finite length
any finite length
infinitely long
If the denominator of a rational number is 7, the repeating cycle of the corresponding repeating decimal can be at most six digits long. This is because when dividing by 7, there are a maximum of six possible remainders: 1, 2, 3, 4, 5, and 6. Once a remainder repeats, the division process will repeat, leading to the same sequence of digits. Therefore, the repeating cycle of a repeating decimal with a denominator of 7 can be up to six digits long.
Convert 0.27¯¯¯¯¯
to a rational number in simplest form.
to a rational number in simplest form.
To convert 0.27¯¯¯¯¯ to a rational number in simplest form, we can assign a variable (let's say x) to the repeating part of the decimal, which is 27 in this case.
Since the repeating digit appears once after the decimal point, we multiply both sides of the equation by 100 to shift the decimal point two places to the right:
100x = 27.27¯¯¯¯¯
Next, we subtract the original equation from this new equation to eliminate the repeating part:
100x - x = 27.27¯¯¯¯¯ - 0.27¯¯¯¯¯
99x = 27
Finally, we divide both sides of the equation by 99 to solve for x:
x = 27/99
Simplifying the fraction by dividing both numerator and denominator by their greatest common divisor (which is 9), we get:
x = 3/11
Therefore, 0.27¯¯¯¯¯ is equivalent to the rational number 3/11 in simplest form.
Since the repeating digit appears once after the decimal point, we multiply both sides of the equation by 100 to shift the decimal point two places to the right:
100x = 27.27¯¯¯¯¯
Next, we subtract the original equation from this new equation to eliminate the repeating part:
100x - x = 27.27¯¯¯¯¯ - 0.27¯¯¯¯¯
99x = 27
Finally, we divide both sides of the equation by 99 to solve for x:
x = 27/99
Simplifying the fraction by dividing both numerator and denominator by their greatest common divisor (which is 9), we get:
x = 3/11
Therefore, 0.27¯¯¯¯¯ is equivalent to the rational number 3/11 in simplest form.
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?
To convert 0.264¯¯¯¯¯¯¯¯ to a rational number, we can set x equal to 0.264¯¯¯¯¯¯¯¯. Since there are three repeating digits (264) after the decimal point, we would multiply both sides of the equation by 1000 to eliminate the repeating part.
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with what denominator?
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with a denominator of 999.