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