Use algebra for 5.235235235... and 3.9999999... Also write these numbers as a fraction not a mixed number. How do I do this?
Use algebra for 5.235235235... and 3.9999999... ???? , trying to interpret what that means
taking a stab:
let x = 5.235235235... , I see 3 digits repeating, so I will multiply by 1000
1000x = 5235.235235..
subtract them:
999x = 5230
x = 5230/999 , check with your calculator, it will be correct
do the other decimal in the same way, note that only ONE digit repeats, so what would you multiply by ?
Don't be surprised by the unexpected result.
To represent recurring decimals such as 5.235235235... and 3.9999999... as fractions, we need to use algebraic techniques. Let's start with 5.235235235...:
Step 1: Assign a variable to the repeating part. In this case, let's use "x" to represent the repeating decimal.
Step 2: Multiply both sides of the equation by a power of 10 to shift the decimal point to the right of the repeating part. Since there are three digits in the repeating part (235), we will multiply by 1000.
1000x = 5235.235235...
Step 3: Subtract the original equation from the equation in step 2 to eliminate the repeating part.
1000x - x = 5235.235235... - 5.235235235...
999x = 5230
Step 4: Solve for x by dividing both sides by 999.
x = 5230/999
To write this as a fraction, we simplify the fraction 5230/999:
5230/999 = 1046/199
Therefore, 5.235235235... can be written as the fraction 1046/199.
Similarly, let's express 3.9999999... as a fraction:
Step 1: Assign a variable to the repeating part. Let's use "y" in this case.
Step 2: Multiply both sides of the equation by a power of 10 to shift the decimal point to the right of the repeating part. Since there are seven nines in the repeating part, we will multiply by 10^7.
10^7y = 39999999.9999999...
Step 3: Subtract the original equation from the equation in step 2 to eliminate the repeating part.
10^7y - y = 39999999.9999999... - 3.9999999...
9999999y = 39999996
Step 4: Solve for y by dividing both sides by 9999999.
y = 39999996/9999999
To write this as a fraction, we simplify the fraction 39999996/9999999:
39999996/9999999 = 4
Therefore, 3.9999999... can be written as the fraction 4.
In summary, 5.235235235... is equivalent to the fraction 1046/199, and 3.9999999... is equivalent to the fraction 4.