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Answer :
- Convert the repeating decimal $3,(7)$ to a fraction: $3,(7) = \frac{34}{9}$.
- Convert the repeating decimal $2,1(4)$ to a fraction: $2,1(4) = \frac{193}{90}$.
- Add the two fractions: $\frac{34}{9} + \frac{193}{90} = \frac{533}{90} = 5 \frac{83}{90}$.
- Add the fractions $\frac{49}{60} + \frac{1}{5} + \frac{11}{30} = \frac{83}{60}$.
- The answers are $5 \frac{83}{90}$ and $\frac{83}{60}$.
### Explanation
1. Problem Analysis
We are given two problems. The first problem asks us to express the sum of two repeating decimals, $3,(7)$ and $2,1(4)$, as a fraction. The second problem asks us to calculate the sum of three fractions: $\frac{49}{60}$, $\frac{1}{5}$, and $\frac{11}{30}$.
2. Converting 3.(7) to a fraction
First, let's convert the repeating decimal $3,(7)$ to a fraction. Let $x = 3.(7)$. Then $10x = 37.(7)$. Subtracting $x$ from $10x$, we get $9x = 37.(7) - 3.(7) = 34$. Thus, $x = \frac{34}{9}$.
3. Converting 2.1(4) to a fraction
Next, let's convert the repeating decimal $2,1(4)$ to a fraction. Let $y = 2.1(4)$. Then $10y = 21.(4)$. Let $z = 21.(4)$. Then $10z = 214.(4)$. Subtracting $z$ from $10z$, we get $9z = 214.(4) - 21.(4) = 193$. Thus, $z = \frac{193}{9}$, and $y = \frac{z}{10} = \frac{193}{90}$.
4. Adding the fractions
Now, let's add the two fractions: $\frac{34}{9} + \frac{193}{90} = \frac{340}{90} + \frac{193}{90} = \frac{533}{90}$.
5. Simplifying the fraction
Let's simplify the fraction $\frac{533}{90}$. $533 = 5 \times 90 + 83$, so $\frac{533}{90} = 5 \frac{83}{90}$. Therefore, $3,(7) + 2,1(4) = 5 \frac{83}{90}$.
6. Adding the fractions
Now, let's add the fractions $\frac{49}{60} + \frac{1}{5} + \frac{11}{30}$. Find a common denominator, which is 60. So, $\frac{49}{60} + \frac{1}{5} + \frac{11}{30} = \frac{49}{60} + \frac{12}{60} + \frac{22}{60} = \frac{49+12+22}{60} = \frac{83}{60}$.
7. Final Answer
Therefore, the first answer is $5 \frac{83}{90}$, which corresponds to option C. The second answer is $\frac{83}{60}$, which corresponds to option B.
### Examples
Understanding how to convert repeating decimals to fractions is useful in various real-life scenarios, such as calculating precise measurements in construction or engineering, where even small errors can accumulate and cause significant problems. Similarly, adding fractions is a fundamental skill in cooking, where recipes often require combining different proportions of ingredients. Mastering these mathematical concepts ensures accuracy and efficiency in practical tasks.
- Convert the repeating decimal $2,1(4)$ to a fraction: $2,1(4) = \frac{193}{90}$.
- Add the two fractions: $\frac{34}{9} + \frac{193}{90} = \frac{533}{90} = 5 \frac{83}{90}$.
- Add the fractions $\frac{49}{60} + \frac{1}{5} + \frac{11}{30} = \frac{83}{60}$.
- The answers are $5 \frac{83}{90}$ and $\frac{83}{60}$.
### Explanation
1. Problem Analysis
We are given two problems. The first problem asks us to express the sum of two repeating decimals, $3,(7)$ and $2,1(4)$, as a fraction. The second problem asks us to calculate the sum of three fractions: $\frac{49}{60}$, $\frac{1}{5}$, and $\frac{11}{30}$.
2. Converting 3.(7) to a fraction
First, let's convert the repeating decimal $3,(7)$ to a fraction. Let $x = 3.(7)$. Then $10x = 37.(7)$. Subtracting $x$ from $10x$, we get $9x = 37.(7) - 3.(7) = 34$. Thus, $x = \frac{34}{9}$.
3. Converting 2.1(4) to a fraction
Next, let's convert the repeating decimal $2,1(4)$ to a fraction. Let $y = 2.1(4)$. Then $10y = 21.(4)$. Let $z = 21.(4)$. Then $10z = 214.(4)$. Subtracting $z$ from $10z$, we get $9z = 214.(4) - 21.(4) = 193$. Thus, $z = \frac{193}{9}$, and $y = \frac{z}{10} = \frac{193}{90}$.
4. Adding the fractions
Now, let's add the two fractions: $\frac{34}{9} + \frac{193}{90} = \frac{340}{90} + \frac{193}{90} = \frac{533}{90}$.
5. Simplifying the fraction
Let's simplify the fraction $\frac{533}{90}$. $533 = 5 \times 90 + 83$, so $\frac{533}{90} = 5 \frac{83}{90}$. Therefore, $3,(7) + 2,1(4) = 5 \frac{83}{90}$.
6. Adding the fractions
Now, let's add the fractions $\frac{49}{60} + \frac{1}{5} + \frac{11}{30}$. Find a common denominator, which is 60. So, $\frac{49}{60} + \frac{1}{5} + \frac{11}{30} = \frac{49}{60} + \frac{12}{60} + \frac{22}{60} = \frac{49+12+22}{60} = \frac{83}{60}$.
7. Final Answer
Therefore, the first answer is $5 \frac{83}{90}$, which corresponds to option C. The second answer is $\frac{83}{60}$, which corresponds to option B.
### Examples
Understanding how to convert repeating decimals to fractions is useful in various real-life scenarios, such as calculating precise measurements in construction or engineering, where even small errors can accumulate and cause significant problems. Similarly, adding fractions is a fundamental skill in cooking, where recipes often require combining different proportions of ingredients. Mastering these mathematical concepts ensures accuracy and efficiency in practical tasks.
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