How many digits long could the repeating cycle of a repeating decimal be if the denominator of the rational number is 7?
any finite length
up to six digits long
up to seven digits long
infinitely long
The repeating cycle of a repeating decimal is determined by the denominator of the rational number. In this case, since the denominator is 7, the repeating cycle can be up to 6 digits long.
To see why, let's divide 1 by 7 using long division:
0.142857...
--------------
7 | 1.000000...
-7
-----
30
-28
-----
20
-14
-----
60
-56
-----
40
-35
-----
50
-49
-----
10
- 7
-----
30
...
As we can see, the repeating cycle 142857 repeats every 6 digits. Therefore, the repeating cycle of a repeating decimal with a denominator of 7 can be up to 6 digits long.
Convert 0.bar(27)
to a rational number in simplest form.
27/100
27/99
1/27
3/11
To convert 0.bar(27) to a rational number in simplest form, we can set x = 0.bar(27) and subtract x from 100x to eliminate the repeating part.
Let's multiply both sides of the equation x = 0.bar(27) by 100:
100x = 27.bar(27)
Next, we subtract the original equation x = 0.bar(27) from 100x:
100x - x = 27.bar(27) - 0.bar(27)
99x = 27
Dividing both sides of the equation by 99:
x = 27/99
So, 0.bar(27) is equal to 27/99 in simplest form.
To convert 0.bar(264) to a rational number, you would set x equal to 0.bar(264) and then multiply both sides of the equation by what number?
10
100
1,000
999
To convert 0.bar(264) to a rational number, you would set x equal to 0.bar(264) and then multiply both sides of the equation by 10.
Let's see the process:
Let x = 0.bar(264).
Multiplying both sides of the equation by 10, we get:
10x = 26.4bar(264)
Now, let's subtract the original equation x = 0.bar(264) from 10x:
10x - x = 26.4bar(264) - 0.bar(264)
9x = 26.4
Dividing both sides of the equation by 9:
x = 26.4/9
Thus, to convert 0.bar(264) to a rational number, you would multiply both sides of the equation by 10.
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with what denominator?
999
100
99
1,000
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with a denominator of 999.
In general, when the repeating cycle of a decimal has n digits, the decimal can be converted to a rational number with a denominator of 999...9, where the number of 9's is equal to the number of digits in the repeating cycle.
Which of the following types of maps shows correct shapes and directions but distorts distances or sizes?
equal-area projection
Mercator projection
Mollweide projection
Robinson projection
The correct answer is the Mercator projection. The Mercator projection accurately represents shapes and directions, but it distorts distances and sizes, particularly when moving away from the equator.
Which of the following explains why seasonal conditions vary in parts of the world?
Earth is rotating on its axis.
Earth's axis is tilted as it revolves around the sun.
Earth's orbit keeps it in line with the sun.
Earth takes 365 days to revolve around the sun
The correct answer is: Earth's axis is tilted as it revolves around the sun.
The tilt of Earth's axis is the main reason why seasonal conditions vary in different parts of the world. As Earth orbits around the sun, its axis remains tilted at an angle of about 23.5 degrees. This tilt causes different amounts of sunlight to reach different parts of the planet at different times of the year, resulting in the changing seasons.
Which of the following explains why the temperature of land near water is different than inland?
Bodies of water reflect the sun, thus changing the temperature.
Land near water is at a higher altitude, so it is closer to the sun.
The air over the water affects the air and land nearby.
Water changes land temperatures where it touches the shoreline
The correct answer is: The air over the water affects the air and land nearby.
The temperature of land near water is different than inland primarily because of the influence of the air over the water. Water has a higher specific heat than land, meaning it can absorb and retain heat energy more effectively. As a result, the air over water is cooler than the air over land. This cooler air from the water can then affect the temperature of the nearby land areas, resulting in milder and more moderate temperatures compared to inland areas.
Which of the following actions would have a positive impact on the environment? Select the two correct answers.
burning fossil fuels
cutting down trees
establishing state parks
releasing waste water into the ocean
using renewable energy sources
The two actions that would have a positive impact on the environment are:
1. Establishing state parks: Establishing state parks helps protect and preserve natural habitats and ecosystems. It provides areas for wildlife to thrive and offers opportunities for people to connect with nature.
2. Using renewable energy sources: Using renewable energy sources, such as solar or wind power, reduces dependency on fossil fuels and decreases greenhouse gas emissions. This helps combat climate change and reduces air pollution.
Which of the following is an example of a pull factor?
better jobs
harsh climate
poverty
war
The example of a pull factor is "better jobs."
Pull factors are factors that attract individuals to move to a particular location. In this case, the availability of better jobs is a positive factor that can pull people to a specific area in search of improved employment opportunities.
How do languages within a language group relate to each other?
They all developed on the same continent.
They are all spoken by people of a particular religious faith.
They all developed from a common ancestral language.
They are all spoken by people of a particular ethnic group.
The correct answer is: They all developed from a common ancestral language.
Languages within a language group relate to each other because they all developed from a common ancestral language. Over time, as populations spread out and diverged, languages evolved and changed, resulting in different languages within the same language group. These languages share similarities in terms of vocabulary, grammar, and pronunciation, tracing back to their common origin.