Write 2.445353535353... as a fraction
I'm learning integrals right now and this is part of the exercises under it. Is there a trick to this? Using integrals?
Thank you very much!
2.44 + .0053535353 .......
2.44 + (1/100)(.53535353 .... )
.53535353... = .53 + .53/100 + .53/100^2...
= a + a r + a r^2 + a r^3.....
where
a = .53 and r = .01
sum of that infinite series
= a/(1-r)
so
.53 /(.99) = 53/99
so
we have
2.44 + (1/100)(53/99)
(1/100)( 244 + 53/99)
I think you can take it from there
or
let x = 2.44535353..
times 100
100x = 244.535353..
subtract them
99x = 242.09
x = 242.09/99
= 24209/9900 or 2 4409/9900
Do it Reiny's way (but mine is more fun)
And Math should be and is fun.
To write the recurring decimal 2.445353535353... as a fraction, we can use the method of converting recurring decimals to fractions.
Step 1: Let x = 2.445353535353...
Step 2: Multiply both sides of the equation by a power of 10 to eliminate the decimal places in x. Since there is one digit after the decimal point, multiply by 10:
10x = 24.453535353535...
Step 3: Subtract the original equation from the equation obtained in Step 2:
10x - x = 24.453535353535... - 2.445353535353...
Simplifying, we get:
9x = 22.008181818181...
Step 4: Now, the recurring part of x repeats after every two decimal places. So, to eliminate the recurring part, multiply both sides of the equation by 100:
900x = 2200.8181818181...
Step 5: Subtract the equation obtained in Step 3 from the equation obtained in Step 4:
900x - 9x = 2200.8181818181... - 22.008181818181...
Simplifying, we get:
891x = 2178.81
Step 6: Divide both sides of the equation by 891:
x = 2178.81 / 891
Now, to simplify the fraction further:
x = 36 / 11
Therefore, 2.445353535353... is equal to the fraction 36/11.
Regarding your question on using integrals to solve this, the method mentioned above is a more straightforward approach for converting recurring decimals to fractions. Integrals are not directly applicable in this specific case. However, integrals can be powerful tools for evaluating functions and solving other types of mathematical problems.