How you would convert the repeating, nonterminating decimal to a fraction? Explain the process as you solve the problem. 0.1515.....
let x = .151515...
multiply by 100, (note there were 2 digits repeating, and there are 2 zeros in 100)
100x = 15.151515...
subtract the two equations:
99x = 15
x = 15/99 = 5/33 , check with a calculator.
Suppose you have some initial decimals that don't repeat,
e.g.
.34123123123...
let x = .34123123123...
three digits repeat, so I multiply by 1000
1000x = 341.23123123...
subtract them, make sure to line up the decimals and notice the 123123... match up
999x = 340.89
x = 340.89/999 , but we don't want decimals in our fraction
= 34089/99900
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To convert a repeating, nonterminating decimal to a fraction, you can use the method of infinite geometric series. Let's take the repeating decimal 0.1515... and convert it to a fraction.
Step 1: Assign a variable to the repeating block.
Let's say x = 0.1515...
Step 2: Identify how many digits repeat.
In this case, the repeating block is two digits long: 15.
Step 3: Create an equation.
To eliminate the repeating part, we can subtract it from the original number:
100x = 15.1515...
Step 4: Shift decimals to align the repeating block.
To shift the decimal places, multiply both sides of the equation by 100, which is 10 to the power of the number of digits in the repeating block:
10000x = 1515.1515...
Step 5: Subtract the original equation from the shifted equation to eliminate repeating digits.
Subtract the original equation (100x) from the shifted equation (10000x):
(10000x - 100x) = (1515.1515... - 15.1515...)
Simplifying the equation:
9900x = 1500
Step 6: Solve for x.
Divide both sides by 9900:
x = 1500/9900
Step 7: Reduce the fraction if possible.
To simplify the fraction further, we can divide both the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 1500 and 9900 is 300:
x = (1500/300) / (9900/300)
x = 5/33
Therefore, the fraction equivalent of 0.1515... is 5/33.