In this problem, we shall see how to write numbers whose decimal is repeating as fraction_ Suppose we have the number 0.123 0.123123123 We can write this number as 123 103 123 123 106 109 Express this sum using series notation_ Calculate the sum of the series and express the answer as simplified fraction:
The Correct Answer and Explanation is:
To express a repeating decimal like 0.123‾0.\overline{123}0.123 (where the “123” repeats) as a fraction, we can use series notation. Here’s the step-by-step breakdown:
Step 1: Express the repeating decimal as a series
We can rewrite the decimal 0.123‾0.\overline{123}0.123 as: 0.123123123…=0.123+0.000123+0.000000123+…0.123123123\ldots = 0.123 + 0.000123 + 0.000000123 + \ldots0.123123123…=0.123+0.000123+0.000000123+…
This is an infinite geometric series with the first term a=0.123a = 0.123a=0.123 and the common ratio r=10−3r = 10^{-3}r=10−3 (since each new term is 100010001000 times smaller than the previous term).
So, the sum of this infinite series is given by the formula for the sum of a geometric series: S=a1−rS = \frac{a}{1 – r}S=1−ra
where a=0.123a = 0.123a=0.123 and r=10−3r = 10^{-3}r=10−3.
Step 2: Calculate the sum of the series
Substitute a=0.123a = 0.123a=0.123 and r=10−3r = 10^{-3}r=10−3 into the formula: S=0.1231−10−3=0.1230.999S = \frac{0.123}{1 – 10^{-3}} = \frac{0.123}{0.999}S=1−10−30.123=0.9990.123
Now simplify: S=123999S = \frac{123}{999}S=999123
Step 3: Simplify the fraction
To simplify 123999\frac{123}{999}999123, we can find the greatest common divisor (GCD) of 123 and 999. Using the Euclidean algorithm:
- 999÷123=8999 \div 123 = 8999÷123=8 with a remainder of 999−8×123=999−984=15999 – 8 \times 123 = 999 – 984 = 15999−8×123=999−984=15
- 123÷15=8123 \div 15 = 8123÷15=8 with a remainder of 123−8×15=123−120=3123 – 8 \times 15 = 123 – 120 = 3123−8×15=123−120=3
- 15÷3=515 \div 3 = 515÷3=5 with a remainder of 000
So, the GCD of 123 and 999 is 3. Now, divide both the numerator and the denominator of 123999\frac{123}{999}999123 by 3: 123999=123÷3999÷3=41333\frac{123}{999} = \frac{123 \div 3}{999 \div 3} = \frac{41}{333}999123=999÷3123÷3=33341
Thus, the simplified fraction for 0.123‾0.\overline{123}0.123 is 41333\frac{41}{333}33341.
Final Answer:
The repeating decimal 0.123‾0.\overline{123}0.123 can be expressed as the simplified fraction: 41333\frac{41}{333}33341
