How many digits are in the repeating cycle of 17/27?
To find the repeating cycle of the fraction 17/27, we need to perform long division.
____
27|17.00
16
---
100
81
--
190
189
---
100
We see that the remainder is 100, which is the same as the dividend when multiplied by 10.
___
27|100.00
81
---
190
189
---
100
Again, we see that the remainder is 100. So, the process will repeat.
Since the remainder is the same after every division, the repeating cycle in 17/27 is the number 100.
Therefore, the repeating cycle of 17/27 has 3 digits.
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:
____
13| 7.000
6
---
10
9
--
10
9
--
10
9
The remainder is 10, which is the same as the dividend when multiplied by 10.
____
13| 7.000
6
---
10
9
--
10
9
--
10
9
Again, the remainder is 10, which indicates that the digits will start repeating.
Hence, 7/13 as a repeating decimal is 0.538461 with the digits 538461 repeating. To indicate the repeating digits, we put a bar over them, so the repeating decimal form of 7/13 is 0.538461
Convert 0.8
to a rational number.
To convert 0.8 to a rational number, we can rewrite it as a fraction by placing it over a power of 10.
0.8 can be written as 8/10 because 0.8 is equivalent to 8 tenths.
Next, we can simplify 8/10 by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
8 ÷ 2 = 4
10 ÷ 2 = 5
Therefore, 0.8 as a rational number is 4/5.
Convert 0.8¯¯¯
to a rational number.
To convert 0.8¯¯¯ to a rational number, we need to recognize that the bar indicates a repeating decimal pattern. Let's call the number 0.8x, where x represents the repeating pattern.
To eliminate the decimal, we can multiply both sides of the equation by 10, since there is only one repeating digit after the decimal point.
10 * 0.8x = 8.x
Next, we subtract the original equation from the new equation:
10 * 0.8x - 0.8x = 8.x - 0.8x
Expanding these expressions, we have:
8x = 8.x - 0.8x
8x = 8.8x - 0.8x
8x = 8x
We have obtained an equation where the variable x cancels out. This indicates that x does not depend on any specific value and that the repeating pattern 0.8¯¯¯ can be converted to a rational number without the need for any specific value for x.
Therefore, 0.8¯¯¯ as a rational number is equal to 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, we can set x equal to 0.82¯¯¯¯¯ and multiply both sides of the equation by an appropriate power of 10. Since there are two repeating digits after the decimal point, we would multiply both sides of the equation by 100:
100x = 82.82¯¯¯¯¯
This multiplication allows us to shift the decimal point to eliminate the repeating pattern.
Therefore, to convert 0.82¯¯¯¯¯ to a rational number, we would multiply both sides of the equation by 100.
How many digits long could the repeating cycle of a repeating decimal be if the denominator of the rational number is 7?
To determine the length of the repeating cycle of a repeating decimal with a denominator of 7, we need to find the smallest positive integer n for which 10^n - 1 is divisible by 7. This is because when we perform long division to convert the fraction to a decimal, we are essentially dividing the numerator (which is 1) by the denominator (which is 7), then multiplying by 10^n, and getting the remainder at each step.
Checking the values of 10^n - 1 mod 7 for values of n:
n = 1: 10^1 - 1 = 10 - 1 = 9 (not divisible by 7)
n = 2: 10^2 - 1 = 100 - 1 = 99 (not divisible by 7)
n = 3: 10^3 - 1 = 1000 - 1 = 999 (not divisible by 7)
n = 4: 10^4 - 1 = 10000 - 1 = 9999 (not divisible by 7)
n = 5: 10^5 - 1 = 100000 - 1 = 99999 (not divisible by 7)
n = 6: 10^6 - 1 = 1000000 - 1 = 999999 (divisible by 7)
We see that for n = 6, 10^n - 1 is divisible by 7. Therefore, the repeating cycle of a repeating decimal with a denominator of 7 can have at most 6 digits.
Convert 0.27¯¯¯¯¯
to a rational number in simplest form.
To convert 0.27¯¯¯¯¯ to a rational number, we can set x equal to 0.27¯¯¯¯¯ and multiply both sides of the equation by an appropriate power of 10. Since there are two repeating digits after the decimal point, we would multiply both sides of the equation by 100:
100x = 27.27¯¯¯¯¯
Now, we subtract the original equation from this new equation:
100x - x = 27.27¯¯¯¯¯ - 0.27¯¯¯¯¯
99x = 27
Dividing both sides of the equation by 99, we find:
x = 27/99
This fraction can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 9:
27 ÷ 9 = 3
99 ÷ 9 = 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?
To convert 0.264¯¯¯¯¯¯¯¯ to a rational number, we can set x equal to 0.264¯¯¯¯¯¯¯¯ and multiply both sides of the equation by an appropriate power of 10. Since there are three repeating digits after the decimal point, we would multiply both sides of the equation by 1000:
1000x = 264.264¯¯¯¯¯¯¯¯
This multiplication shift the decimal point to eliminate the repeating pattern.
Therefore, to convert 0.264¯¯¯¯¯¯¯¯ to a rational number, we would multiply both sides of the equation by 1000.
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.
This can be derived from the fact that when we convert a repeating decimal to a rational number, we multiply by a power of 10 equal to the number of digits in the repeating cycle. In this case, since the repeating cycle has three digits, we would multiply by 1000.
The resulting equation would have the repeating decimal as the numerator over the denominator of 999, because 1000 - 1 = 999.
Therefore, a repeating decimal with a repeating cycle of three digits converts to a rational number with a denominator of 999.
Convert 0.6¯¯¯
to a fraction, reduced to lowest terms.
All waves carry(1 point)
Responses
• light.
light.
• particles.
particles.
• energy.
energy.
• matter.
Which of these would be the most reliable method for transmitting waves for communication?(1 point)
Responses
• The use of fiber optics to transmit a digital signal.
The use of fiber optics to transmit a digital signal.
• The use of knocking on all the doors in the neighborhood.
The use of knocking on all the doors in the neighborhood.
• The use of the postal service to mail a letter.
The use of the postal service to mail a letter.
• The use of printed newspapers and magazines.
Which of these is an advantage of digital storage of information over analog?(1 point)
Responses
• Sound can be recorded either digitally or using analog techniques.
Sound can be recorded either digitally or using analog techniques.
• Analog signals take up less bandwidth or storage space than digital.
Analog signals take up less bandwidth or storage space than digital.
• A copy of an analog file is never as good as its original.
A copy of an analog file is never as good as its original.
• Analog storage of information is continuous and captures a more pure sound.
Digital recording and transmission often requires the signal to be transmitted using a code. Which statement would suggest an advantage of this fact?(1 point)
Responses
• Coding of the signal can keep the information secret during transport.
Coding of the signal can keep the information secret during transport.
• Coding may require more time to record information.
Coding may require more time to record information.
• Coded information may require larger bandwidth to transmit.
Coded information may require larger bandwidth to transmit.
• Coding requires all users to have a method to decode information.
Consider any ocean wave. What is the relationship between an ocean wave and the energy of that wave?(1 point)
Responses
• Wave energy is directly proportional to the amplitude of the wave.
Wave energy is directly proportional to the amplitude of the wave.
• The greater the wavelength, the more kinetic energy the wave has.
The greater the wavelength, the more kinetic energy the wave has.
• Wave energy decreases as the wave moves closer to shore.
Wave energy decreases as the wave moves closer to shore.
• Wave energy increases with frequency and decreases with wavelength.
Examine the timeline of relaying information to other people over the past few hundred years.
Pony Express, US Postal Mail, Fax Machine, Digital Image Submission
Which statement explains the reasons for this progression in information delivery?
(1 point)
Responses
• The relay of information became less trustworthy as more digital technology was used.
The relay of information became less trustworthy as more digital technology was used.
• The quality of the information became less reliable as more digital technology was used.
The quality of the information became less reliable as more digital technology was used.
• The storage of information became more difficult as more digital technology was used.
The storage of information became more difficult as more digital technology was used.
• The relay of information became more reliable as digital technology was used.