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...