Express 8/11 as a repeating decimal. Mark the cycle of repeating digits with a bar.(1 point)
To express 8/11 as a repeating decimal, divide 8 by 11:
0.727272...
The cycle of repeating digits is "72", therefore the decimal representation of 8/11 is 0.72 with the cycle of repeating digits marked with a bar.
Convert 0.72¯¯¯¯¯ to a rational number in simplest form.(1 point)
To convert 0.72¯¯¯¯¯ to a rational number in simplest form, we can use a variable to represent the repeating decimal.
Let x = 0.72¯¯¯¯¯
Multiplying both sides of this equation by 100 (to shift the decimal two places to the right), we get:
100x = 72.7272¯¯¯¯¯
Next, we can subtract x from 100x:
100x - x = 72.7272¯¯¯¯¯ - 0.72¯¯¯¯¯
99x = 72
Finally, solve for x to find the rational number:
x = 72/99
Simplifying the fraction, we can divide the numerator and denominator by their greatest common divisor, which is 9:
x = 8/11
Therefore, 0.72¯¯¯¯¯ is equivalent to the rational number 8/11 in simplest form.
Question
Which of the following is a rational number between 5 and 5 1/4?(1 point)
Responses
5 2/5
5 Start Fraction 2 over 5 end fraction
5 1/7
5 Start Fraction 1 over 7 end fraction
26−−√
start root 26 end root
5 1/3
5 Start Fraction 1 over 3 end fraction
What is a rational number with a denominator of 7 that is between 7–√ and 8–√. Write your answer as an improper fraction.(1 point)
To find a rational number with a denominator of 7 that is between 7–√ and 8–√, we can start by rationalizing the denominators. Let's rationalize both numbers:
7–√ = 7–√ * (√7+√7)/(√7+√7) = (7√7 + 7)/7 = √7 + 1
8–√ = 8–√ * (√7+√7)/(√7+√7) = (8√7 + 8)/7 = (8√7)/7 + 8/7
Now, to find a rational number between these two expressions, we can take their average:
(√7 + 1 + (8√7)/7 + 8/7)/2
Simplifying the numerator, we have (√7 + (8√7)/7 + 1 + 8/7)/2.
Now, let's simplify this expression to obtain the final answer:
(7√7 + 8√7 + 7 + 8)/14
Combining like terms in the numerator, we have:
(15√7 + 15)/14
Therefore, the rational number with a denominator of 7 that is between 7–√ and 8–√ is (15√7 + 15)/14 as an improper fraction.
To determine which of the options is a rational number between 5 and 5 1/4, we need to figure out if each option can be expressed as a fraction (rational number).
1) 5 2/5: This can be written as the improper fraction 27/5, which is a rational number. However, it is larger than 5 1/4, so it is not between the given range.
2) 5 2/5: This is the same option as above and is not between 5 and 5 1/4.
3) 5 1/7: This can be written as the improper fraction 36/7, which is a rational number. However, it is larger than 5 1/4, so it is not between the given range.
4) 5 1/3: This can be written as the improper fraction 16/3, which is a rational number. However, it is larger than 5 1/4, so it is not between the given range.
Out of the given options, none of them fall between 5 and 5 1/4.