Express the repeating decimal 0.513 (the 13 is repeating) as a fraction in lowest terms using the infinite geometric series method.
the 1 and the there are repeating so it's .513131313131313... and 489/900 isn't giving me that. could you explain it again?
.513= .510+ 3/1000 + 3.10000+ 3/1E5 +3/1E6+ ...
notice that this is ...
= .510+ 3/1000 (1+ (1/10)+ (1/10)^3+..)
sum of
= .510 + 3/1000(1/.9)=+510/1000+3/900
= (510*9+300)/9000 check that
= (4590+ 300) /9000 = 4890/9000=489/900
what 9,000=900x_
Why did the decimal go to the circus? To become a fraction! Let's turn this repeating decimal into a fraction with a little clown magic.
Let's represent the repeating decimal as x = 0.5131313...
Now, let's multiply both sides by 100 since we have two repeating digits:
100x = 51.313131...
Subtracting the original equation x = 0.5131313... from 100x = 51.313131... gives us:
99x = 50.8
Dividing both sides of the equation by 99:
x = 50.8/99
So, our repeating decimal 0.513 (with the 13 repeating) can be expressed as the fraction 50.8/99.
And just like a clown car, we can't have any extra baggage! Let's simplify this fraction:
x = 50.8/99 = 508/990 = 254/495
So, the repeating decimal 0.513 as a fraction in lowest terms using the infinite geometric series method is 254/495.
To express the repeating decimal 0.513 (where the 13 is repeating) as a fraction in lowest terms using the infinite geometric series method, we can follow these steps:
Step 1: Identify the repeating part of the decimal
In this case, the repeating part is 13.
Step 2: Let x represent the repeating decimal
Let x = 0.5131313...
Step 3: Multiply the repeating decimal by an appropriate power of 10 to eliminate the repeating part
To eliminate the repeating part, we need to multiply x by 100, which moves the decimal point two places to the right:
100x = 51.313131...
Step 4: Subtract the original decimal x from the result obtained in step 3
When we subtract x from 100x, the repeating part cancels out:
100x - x = 51.313131... - 0.5131313...
99x = 51.313131... - 0.5131313...
Step 5: Simplify the equation obtained in step 4
To simplify the equation, we want to shift the decimal point to eliminate the repeating part:
99x = 51.313131... - 0.5131313...
99x = 51.3 (since 13 is repeating)
99x = 5131
Step 6: Solve for x
Divide both sides of the equation by 99:
99x/99 = 5131/99
x = 5131/99
Step 7: Express the fraction in lowest terms
To express the fraction 5131/99 in lowest terms, we can find their greatest common divisor (GCD) and divide both the numerator and denominator by it:
GCD(5131, 99) = 11
Dividing both the numerator and denominator by 11, we get:
5131/99 = (5131 ÷ 11) / (99 ÷ 11)
5131/99 = 467/9
Therefore, the repeating decimal 0.513 (with the 13 repeating) can be expressed as the fraction 467/9 in lowest terms using the infinite geometric series method.