Final answer:
To express repeating decimals as quotients of integers, set up an equation with the decimal as a variable, move the repeating part to the left of the decimal using multiplication, then solve for your variable to find the fraction form.
Step-by-step explanation:
To convert repeating decimals into fractions, you can set the decimal as a variable (x), multiply it by a power of 10 to move the repeating decimal block (the numerator) to the left of the decimal point, and then create an equation by subtracting this from the original variable. The result can then be solved for x, giving you the fraction form of the repeating decimal.
Converting 0.151515...
Let x = 0.151515...
100x = 15.151515... (Here, we multiply by 100 to get two repeating digits to the left of the decimal)
Subtract the first equation from the second equation (100x - x):
99x = 15
Now solve for x: x = 15/99, which simplifies to 5/33.
Converting 0.181818...
Let x = 0.181818...
100x = 18.181818...
Again, subtract the first equation from the second equation (100x - x):
99x = 18
Solve for x: x = 18/99, which simplifies to 2/11.
Converting 1.785785...
Let x = 1.785785...
1000x = 1785.785785...
Subtract the first equation from the second equation (1000x - x):
999x = 1784
Solve for x: x = 1784/999.
Converting 2.439543954395...
Let x = 2.439543954395...
100000x = 243954.395439543954...
Subtract the first equation from the second equation (100000x - x):
99999x = 243952
Solve for x: x = 243952/99999.