it is required to find a rational number which is equal to the recurring decimal number 0.327327327...,(a)find the value of 1000x-x (b)hence, express 0.327327327... in the form a/b, where a and b are integers.
.327... = 327/999 = 109/333
1000x = 327.327327...
x = 0.327327...
999x = 327
To find a rational number equal to the recurring decimal number 0.327327327..., we can use algebraic manipulations.
First, let's find the value of 1000x - x:
Let x = 0.327327327...
Multiply both sides by 1000 to remove the decimals:
1000x = 327.327327...
Subtract x from both sides:
1000x - x = 327.327327... - 0.327327327...
Simplifying the right side:
1000x - x = 327
We have 999x = 327.
To solve for x, divide both sides by 999:
x = 327/999
Therefore, the value of 1000x - x is 327/999.
Now, to express 0.327327327... in the form a/b, where a and b are integers, we can use the value of x we found.
Recall that x = 327/999.
To write x in the form a/b, we can simplify the fraction 327/999:
Both 327 and 999 are divisible by 3:
327 ÷ 3 = 109
999 ÷ 3 = 333
So, x can be expressed as
x = 109/333
Therefore, 0.327327327... is equal to the rational number 109/333.