Write the recurring decimal 0.ababab as a fraction. (10 points)
A) a + b/ 99
B) a * b / 99
C) 10b + a/ 99
D) 10a + b/ 99
any recurring set of n digits abc... is
(abc...)/(999...)
where there are the same number of digits top and bottom. To see this, do
x = 0.ababab...
100x = ab.ababab...
now subtract to get
99x = ab
x = ab/99 = (10a+b)/99
Well, I must say, decimals can be quite the tricksters. So, let's see if we can put this recurring decimal 0.ababab into fraction form, shall we?
Ah, 0.ababab, a decimal with a rhythm! It's like a never-ending dance party. Now, to express it as a fraction, we need to figure out the pattern here.
Let's take a closer look. Notice how the "a" digit repeats, and the "b" digit repeats as well. So, we can say that "a" repeats every two decimal places, and "b" repeats every two decimal places separately.
To make sense of this decimal dance, we can use a bit of algebraic elegance. We'll call our recurring decimal "x" and try to uncover its secret.
If we multiply our decimal "x" by 100, we get:
100x = ab.ababab...
See what happened there? Well, the "a" and "b" parts moved to the left of the decimal point. Now, if we subtract "x" from 100x, something magical occurs.
100x - x = ab.ababab... - 0.ababab... (our original number "x")
Simplifying that, we get:
99x = ab
Aha! Now we're getting somewhere! If we divide both sides of the equation by 99, we can unveil the true identity of our recurring decimal.
So, the correct answer is D) 10a + b/99. Way to go, math detective!
Now, go and put that fraction to good use, just don't let it outshine your dance moves!
To convert the recurring decimal 0.ababab to a fraction, we need to find a pattern and express it in terms of a single fraction.
We can observe that the pattern is a repetition of two digits: ab.
To obtain the fraction, we can assign variables to the digits.
Let's assign variable a to the digit "a," and variable b to the digit "b."
The fraction can then be expressed as (10a + b) / 99.
Therefore, the correct answer is D) 10a + b / 99.
To write the recurring decimal 0.ababab as a fraction, we can express it using the concept of infinite geometric series.
Let's denote the repeating block of digits (ab) as 'x.' Therefore, we have:
0.ababab = 0.x
Now, we can multiply both sides of the equation by 100 to move the decimal point two places to the right:
100 * 0.ababab = 100 * 0.x
This gives us:
ab.ababab = x
Next, we subtract the original equation from this new equation to eliminate the repeating block:
ab.ababab - 0.ababab = x - 0.x
This simplifies to:
ab = x
Now, we have an equation where 'ab' is equal to 'x.'
To express 'x' as a fraction, we can take 'x' as the numerator and find the denominator by subtracting the original number from the shifted number.
Denominator = 100 * shifted number - original number
Let's calculate the values of 'x' and the denominator:
Numerator (x) = ab = 10a + b
Denominator = 100 * (ab.ababab) - (0.ababab)
= 100 * (ab.ababab - 0.ababab)
= 100 * (ab - 0)
= 100 * ab
So the fraction is:
(10a + b) / (100ab)
To write it in the given options:
(10a + b) / (100ab) = (10b + a) / (99) [Option C]
Therefore, the correct option is C) 10b + a/ 99.