**URGENT** Which of these nonterminating decimals can be converted into a rational number?
a. 0.874387438743..
b. 0.0000100020003..
c. 5.891420975..
d. 10.07401295..**
am i correct? if not please explain how to do it correctly because i'm struggling and this is due in an hour.
only repeating decimals can be converted to a rational number.
A is the only choice here.
Suppose it has a value, say, x. Then
10000x = 8743.87438743...
x = 0.87438743...
If you subtract, then that gives you
9999x = 8743.0000000...
x = 8743/9999
Or, if you have studied geometric series, you can see that
a = 0.8743
r = 0.00001
and the infinite sum is .8743/.9999
In general, the repeating digits can be placed over that many 9's to form a rational number. Of course, leading zeroes can complicate things.
0.232323 = 23/99
0.00232323 = 23/9900
and so on
To determine which of these nonterminating decimals can be converted into a rational number, we can look for patterns in the decimals.
a. 0.874387438743..
This decimal does not appear to have a repeating pattern, so it is not a rational number.
b. 0.0000100020003..
This decimal is interesting because it has a repeating pattern of zeros followed by increasing sequence of digits. However, it is important to note that the repeating pattern does not continue indefinitely. Therefore, this is not a rational number.
c. 5.891420975..
This decimal does not have any repeating patterns, so it is not a rational number.
d. 10.07401295..
Similar to the previous decimals, this decimal does not have any repeating patterns and is not a rational number.
In conclusion, none of the given decimals can be converted into a rational number, as they all lack repeating patterns.
To determine which of these nonterminating decimals can be converted into a rational number, we need to understand what a rational number is. A rational number is a number that can be expressed as the quotient or fraction of two integers.
To convert a decimal into a rational number, we need to find a pattern or develop a rule for the decimal representation.
Let's examine each of the options:
a. 0.874387438743...
This decimal has a repeating pattern of 8743. To convert this decimal into a rational number, we can write it as:
0.874387438743... = 0.874387438743438743...
Let's call this number x:
x = 0.874387438743438743...
Now, let's multiply x by 10 to eliminate the repeating part:
10x = 8.743874387438438743...
Subtracting the original number x from both sides:
10x - x = 8.743874387438438743... - 0.874387438743438743...
9x = 7.869486948694694694...
Now, divide both sides by 9 to solve for x:
x = 7.869486948694694694... / 9
By simplifying the fraction, we can write:
x = 874/999
Since 874/999 is a fraction of two integers, the decimal 0.874387438743... is a rational number.
b. 0.0000100020003...
This decimal seems to have an increasing pattern, but it is not repeating. Hence, it cannot be converted into a rational number.
c. 5.891420975...
This decimal does not have a clear pattern or repetition. Therefore, it cannot be converted into a rational number.
d. 10.07401295...
Similar to option c, this decimal does not have a clear pattern or repetition. So, it cannot be converted into a rational number.
Therefore, the correct answer is:
a. 0.874387438743..
I hope this explanation helps you. Good luck with your assignment! If you have any more questions, feel free to ask.