Express the recurring decimal into fraction.
a)0.02469999999...
b)123.76111111...
c)542.8888888...
d)789.456456456...
I work only on the decimal part, then add on the whole number at the end
I will use b) , then you follow my steps to do the others.
For the numerator:
write down all the digits to the end of the first repeat : 761
subract the digits that did NOT repeat:
761 - 76 = 685
For the denominator:
write down a 9 for each repeating digits.
There is only 1 digit repeating, so I will use one 9
Follow this with a 0 for each digits that is not repeating. I count two digits before the repeat, so two zeros. My denominator is 900
the repeating decimal becomes 685/900
or 137/180 in lowest terms
check with a calculator:
137/180 =.76111...
to finish off, add the original 123, so
123.76111... = 123 137/180 or 22277/180
btw, you can always check your answer by division on your calculator
I have worked on a) but on checking answer does not satisfied a).
my answer is 2223/90000=0.0247
Please check this.
Should have mentioned that
.9999....
.099999...
.00999...
etc, are special cases
e.g.
show that .99999.... = 1
.099999.... = .1
.009999... = .01
etc
1/3 = .3333...
2/3 = .6666...
add them:
3/3 = .99999...
1 = .9999...
1/30 = .03333...
2/30 = .06666..
add them:
3/30 = .0999...
1/10 = .09999....
.1 = .09999...
etc
so 0.02469999999...
= 0.0246 + 0.0000999999
= .0246 + .0001
= .0247
or 247/10000 ,
The others should have worked for you
OK this i understand but how will I find reaped digit when i will change the fraction into recurring decimal that reaped digit is 9,8,7,6,5 in this case.
To express recurring decimals into fractions, we can follow a few steps:
a) 0.02469999999...
Step 1: Let's assign a variable to the recurring part. Here, the recurring part is 0246.
Step 2: Multiply both sides of the equation by 10 raised to the power of the number of digits in the recurring part. In this case, we have 4 digits in the recurring part, so we'll multiply by 10^4.
10^4 * x = 246.9999...
Step 3: Subtract the original equation from the one obtained in Step 2 to eliminate the recurring part.
10^4 * x - x = 246.9999... - 0.02469999999...
9976 * x = 246.975
Step 4: Solve for x by dividing both sides of the equation by the non-recurring part coefficient.
x = 246.975 / 9976
This fraction is already simplified, so the final result is:
x = 0.024721927904464
b) 123.76111111...
Step 1: The recurring part here is 76.
Step 2: Multiply both sides of the equation by 10 raised to the power of the number of digits in the recurring part. In this case, we have 2 digits in the recurring part, so we'll multiply by 10^2.
10^2 * x = 12376.1111...
Step 3: Subtract the original equation from the one obtained in Step 2 to eliminate the recurring part.
10^2 * x - x = 12376.1111... - 123.76
99 * x = 12252.3511
Step 4: Solve for x by dividing both sides of the equation by the non-recurring part coefficient.
x = 12252.3511 / 99
Simplify the fraction if needed.
x = 123.75565656565656...
c) 542.8888888...
Step 1: The recurring part here is 888.
Step 2: Multiply both sides of the equation by 10 raised to the power of the number of digits in the recurring part. In this case, we have 3 digits in the recurring part, so we'll multiply by 10^3.
10^3 * x = 542888.888...
Step 3: Subtract the original equation from the one obtained in Step 2 to eliminate the recurring part.
10^3 * x - x = 542888.888... - 542.888
999 * x = 542346
Step 4: Solve for x by dividing both sides of the equation by the non-recurring part coefficient.
x = 542346 / 999
Simplify the fraction if needed.
x = 542.8888888888889...
d) 789.456456456...
Step 1: The recurring part here is 456.
Step 2: Multiply both sides of the equation by 10 raised to the power of the number of digits in the recurring part. In this case, we have 3 digits in the recurring part, so we'll multiply by 10^3.
10^3 * x = 789456.456...
Step 3: Subtract the original equation from the one obtained in Step 2 to eliminate the recurring part.
10^3 * x - x = 789456.456... - 789.456
999 * x = 788667
Step 4: Solve for x by dividing both sides of the equation by the non-recurring part coefficient.
x = 788667 / 999
Simplify the fraction if needed.
x = 789.456456456456...