Write the repeating decimal first as a geometric series and then as a fraction (a ratio of two integers) 3.8549 = 3.85494949_ 0A 3.8549 = 0.0049(0.01)k k=0 B. 3.8549 = 3.85 + 2 0.00490.01)k k=0 0 C 3.8549 = 0.0049 + 2 3.85(0.001)k k=0 0 D: 3.8549 = 3.85(0.1)k k=0 Write the repeating decimal as a fraction (a ratio of two integers) 3.8549 (Type an integer or a simplified fraction:)
The Correct Answer and Explanation is:
To solve this problem, we first need to represent the repeating decimal 3.85494949… as a geometric series. The repeating part is “49”, so this part repeats every two decimal places. Here’s how we can break it down:
Step 1: Express the repeating decimal as a geometric series
Let’s first separate the non-repeating part from the repeating part.
We have:3.85494949…=3.85+0.00494949…3.85494949… = 3.85 + 0.00494949…3.85494949…=3.85+0.00494949…
Now, focus on the repeating part 0.00494949…0.00494949…0.00494949…. This is a repeating decimal, which can be written as a geometric series:0.00494949…=0.0049⋅(1+0.01+0.012+0.013+… )0.00494949… = 0.0049 \cdot (1 + 0.01 + 0.01^2 + 0.01^3 + \dots)0.00494949…=0.0049⋅(1+0.01+0.012+0.013+…)
This is a geometric series with:
- First term a=0.0049a = 0.0049a=0.0049
- Common ratio r=0.01r = 0.01r=0.01
So, we can express it as:0.00494949…=0.0049∑k=0∞(0.01)k0.00494949… = 0.0049 \sum_{k=0}^{\infty} (0.01)^k0.00494949…=0.0049k=0∑∞(0.01)k
This series sums up to:S=a1−r=0.00491−0.01=0.00490.99=499900S = \frac{a}{1 – r} = \frac{0.0049}{1 – 0.01} = \frac{0.0049}{0.99} = \frac{49}{9900}S=1−ra=1−0.010.0049=0.990.0049=990049
Step 2: Combine the non-repeating part and the sum of the geometric series
Now, we add the non-repeating part 3.853.853.85 and the sum we just calculated:3.85494949…=3.85+4999003.85494949… = 3.85 + \frac{49}{9900}3.85494949…=3.85+990049
To combine these, express 3.853.853.85 as a fraction:3.85=3851003.85 = \frac{385}{100}3.85=100385
Thus, the total expression becomes:3.85494949…=385100+4999003.85494949… = \frac{385}{100} + \frac{49}{9900}3.85494949…=100385+990049
Now, we need a common denominator to combine the fractions. The least common denominator (LCD) is 9900:385100=385×99100×99=381159900\frac{385}{100} = \frac{385 \times 99}{100 \times 99} = \frac{38115}{9900}100385=100×99385×99=990038115
Now, add the fractions:381159900+499900=38115+499900=381649900\frac{38115}{9900} + \frac{49}{9900} = \frac{38115 + 49}{9900} = \frac{38164}{9900}990038115+990049=990038115+49=990038164
Thus, the repeating decimal 3.85494949… as a fraction is:381649900\frac{38164}{9900}990038164
Final Answer:
The correct geometric series representation is:3.85494949…=3.85+∑k=0∞0.0049⋅(0.01)k3.85494949… = 3.85 + \sum_{k=0}^{\infty} 0.0049 \cdot (0.01)^k3.85494949…=3.85+k=0∑∞0.0049⋅(0.01)k
And as a fraction, the repeating decimal 3.8549 is:381649900\frac{38164}{9900}990038164
