Change to fractional notation in simplest form using limit or sum of an infinite series 4.234 (34 repeating)
times 1000 n ... 4234.3434...
times 10 n ... 42.3434...
subtract to cancel decimals
990 n = 4192 ... n = 4192/990
reduce the fraction
To express the number 4.234 with the repeating decimal 34 in fractional notation using the concepts of limits or the sum of an infinite series, we can follow these steps:
Step 1: Assign variables
Let x = 4.234(34).
Step 2: Multiply by a power of 10 to align the repeating part
Multiply both sides of the equation by 100 to remove the repetition:
100x = 423.434343...
Step 3: Subtract the original equation from the multiplied equation
Subtracting 100x - x eliminates the repeating part:
100x - x = 423.434343... - 4.234343...
Simplifying, we get:
99x = 419.2
Step 4: Solve for x
Divide both sides of the equation by 99 to isolate x:
x = 419.2 / 99
Step 5: Simplify the expression
To express the fraction 419.2 / 99 in simplest form, we need to find the greatest common divider (GCD) between the numerator and the denominator. In this case, they have a GCD of 1.
So, the final fractional notation of 4.234(34) is:
4.234(34) = 419.2 / 99