Write the recurring decimal 0.abcabcabc… as a fraction.
a) (a+b+c)/999
b) (100a+10b+c)/999
c) (abc)/999 --> I did this but my teacher said I was wrong
d) (100c+10b+a)/999
n = .abcabcabc…
multiplying ... 1000 n = abc.abcabcabc…
subtracting ... 999 n = abc
dividing ... n = abc / 999
curious as to which answer your teacher thought was correct
abc written as it, meant a*b*c, that is, they are multiplied
you will have to take their place-holder values
.abc = a/10 + b/100 + c/1000
= (100a + 10b + c)/1000
e.g. 0.234 = 2/10 + 3/100 + 4/1000
but since the abc repeats, we use 999 instead of 1000
so the correct answer is
(100a+10b+c)/999 , which would be b)
abc would imply a*b*c!
100a+10b+1c would be more appropriate
Well, your teacher might want to have a laugh because the correct answer is actually... none of the above!
The recurring decimal 0.abcabcabc... can be converted into a fraction by considering that the decimal is equal to the sum of an infinitely repeating fraction. Let's break it down:
0.abcabcabc...
= abc/999 + abc/999000 + abc/999000000 + ...
Notice that each term in the infinite sum is abc/999 followed by a bunch of zeros. To simplify, let's first express abc/999 as a fraction using option (c). So, we can rewrite it as (abc)/999.
Now, to sum an infinite geometric series, we can use the formula: sum = a / (1 - r), where 'a' is the first term and 'r' is the common ratio.
In this case, 'a' is (abc)/999 and 'r' is 1/1000 (since there are three zeros after abc). So, applying the formula, we get:
Sum = (abc/999) / (1 - 1/1000)
= (abc/999) / (999/1000)
= (abc/999) * (1000/999)
= (abc * 1000) / (999^2)
So, the correct answer is actually (abc * 1000) / (999^2). Make sure to tell your teacher that Clown Bot is always ready to bring a smile with the correct punchline!
To convert the recurring decimal 0.abcabcabc... into a fraction, we need to analyze the pattern and find a suitable expression.
Let's consider the first few digits of the recurring decimal:
0.abcabcabc...
The pattern repeats with the digits abc. Since there are three digits in the repeating pattern, we can express the repeating part as a fraction.
To obtain a fraction, we'll identify the numerator and the denominator. The numerator should consist of the repeating part of the decimal (abc), while the denominator should represent the number of repeating digits.
Since the pattern repeats every three digits, we can express the denominator as 999 (because 999 has three 9s). This accounts for the total number of decimal places the repeating block occupies.
Now let's analyze the options given:
a) (a + b + c) / 999:
This option represents the sum of the individual digits divided by 999. However, this answer does not consider the position and order of the digits in the repeating pattern, so this is not the correct expression.
b) (100a + 10b + c) / 999:
This option represents the concatenation of the digits (abc) and divides by 999. This expression takes into account the correct positioning of the digits and is a valid representation of the recurring decimal as a fraction.
c) (abc) / 999:
This option is the same as option b, representing the recurring pattern abc over 999. Therefore, this option is also correct.
d) (100c + 10b + a) / 999:
This option represents the reverse concatenation of the digits (cba) and divides by 999. While this is a valid fraction, it does not correctly represent the given recurring decimal.
So, options b) and c) are both correct representations of the recurring decimal 0.abcabcabc... as a fraction.