Convert 0.72727272 to a rational number in simplest form
To convert 0.72727272 to a rational number in simplest form, let's call it x and express it as an equation:
x = 0.72727272
To get rid of the repeating decimal, we can multiply x by a power of 10 that eliminates the repeating part. In this case, 10 is enough to eliminate the repeating part. Multiply both sides of the equation by 10:
10x = 7.2727272
Next, subtract the original equation from the equation multiplied by 10:
10x - x = 7.2727272 - 0.72727272
9x = 6.54545448
Simplifying the equation:
x = 6.54545448 / 9
To further simplify, we can divide both the numerator and denominator by their greatest common divisor, which is 1:
x = 6.54545448 ÷ 9
x ≈ 0.72727272
Therefore, 0.72727272 as a rational number in simplest form is approximately 0.72727272.
To convert the repeating decimal 0.72727272 to a rational number in simplest form, we can use the concept of infinite geometric series.
Step 1: Let x = 0.72727272
Step 2: Multiply x by 100 to shift the decimal point two places to the right: 100x = 72.727272.
Step 3: Subtract x from 100x to eliminate the repeating part: 100x - x = 99x = 72.727272 - 0.727272 = 72.
Step 4: Divide both sides of the equation by 99 to solve for x: 99x/99 = 72/99.
Step 5: Simplify the fraction 72/99 by dividing both the numerator and denominator by their greatest common divisor, which is 9: 8/11.
So, the rational number equivalent to the repeating decimal 0.72727272 is 8/11 in simplest form.
To convert a decimal number to a rational number in simplest form, we need to find the fraction that is equivalent to the given decimal. In this case, the decimal is 0.72727272.
Let's denote the decimal as x:
x = 0.72727272
To convert this decimal to a fraction, we'll set up an equation where x is equal to the fraction p/q, where p is the numerator, and q is the denominator:
x = p/q
To find p and q, we'll multiply both sides of the equation by a power of 10 that eliminates the decimal:
10x = 10(p/q)
This equation can be simplified to:
10x = (10p)/q
Now, we need to observe that the repeating decimal portion 0.72727272 can be written as 0.72, repeating. To represent this, we can subtract the repeating portion from the original number:
x - 0.72 = 0.00727272
Multiplying the equation by 100 (to shift two decimal places to the right), we have:
100(x - 0.72) = 100(0.00727272)
100x - 72 = 0.727272
In this equation, 100x represents the non-repeating portion of the decimal, and 0.727272 represents the repeating portion.
Now, solve for 100x:
100x = 72 + 0.727272
100x = 72.727272
We can simplify the repeating decimal by multiplying both sides of the equation by 100 (to shift two decimal places to the right):
10000x = 7272.727272
Now, subtract the original equation from this new equation:
10000x - 100x = 7272.727272 - 72.727272
9900x = 7200
Divide both sides by 9900:
x = 7200/9900
To simplify this fraction, we can find the greatest common divisor (GCD) of 7200 and 9900.
The GCD of 7200 and 9900 is 900. Divide both the numerator and the denominator by the GCD:
7200/900 = 8/1
Therefore, the rational number that is equivalent to the decimal 0.72727272 in simplest form is 8/1 or 8.