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: $\frac{34}{9} + \frac{193}{90} = \frac{533}{90} = 5 \frac{83}{90}$.
- Calculate the sum of the fractions: $\frac{49}{60} + \frac{1}{5} + \frac{11}{30} = \frac{49}{60} + \frac{12}{60} + \frac{22}{60} = \frac{83}{60}$.
### 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: $\frac{49}{60} + \frac{1}{5} + \frac{11}{30}$.
2. Converting 3.(7) to a fraction
Let's solve the first problem. We need to convert the repeating decimals to fractions. First, let's convert $3.(7)$ to a fraction. Let $x = 3.(7)$. Then $10x = 37.(7)$. Subtracting the two equations, $10x - x = 37.(7) - 3.(7)$, so $9x = 34$, which means $x = \frac{34}{9}$.
3. Converting 2.1(4) to a fraction
Next, let's convert $2.1(4)$ to a fraction. Let $y = 2.1(4)$. Then $10y = 21.(4)$. Let $z = 21.(4)$. Then $10z = 214.(4)$. Subtracting the two equations, $10z - z = 214.(4) - 21.(4)$, so $9z = 193$, which means $z = \frac{193}{9}$. Thus, $10y = \frac{193}{9}$, so $y = \frac{193}{90}$.
4. Adding the fractions
Now, we need to calculate the sum of the two fractions: $\frac{34}{9} + \frac{193}{90}$. To add these fractions, we need a common denominator, which is 90. So, we have $\frac{34}{9} = \frac{340}{90}$. Therefore, $\frac{340}{90} + \frac{193}{90} = \frac{340 + 193}{90} = \frac{533}{90}$.
5. Expressing as a mixed number
Now, let's express the fraction $\frac{533}{90}$ as a mixed number. We have $533 \div 90 = 5$ with a remainder of $83$. So, $\frac{533}{90} = 5 \frac{83}{90}$. Therefore, the answer to the first problem is $5 \frac{83}{90}$.
6. Finding a common denominator
Now, let's solve the second problem. We need to calculate the sum of three fractions: $\frac{49}{60} + \frac{1}{5} + \frac{11}{30}$. To add these fractions, we need a common denominator. The least common multiple of 60, 5, and 30 is 60. So, we rewrite the fractions with the common denominator: $\frac{49}{60}$, $\frac{1}{5} = \frac{12}{60}$, and $\frac{11}{30} = \frac{22}{60}$.
7. Adding the fractions
Now, we can calculate the sum: $\frac{49}{60} + \frac{12}{60} + \frac{22}{60} = \frac{49 + 12 + 22}{60} = \frac{83}{60}$.
8. Expressing as a mixed number
Finally, let's express the fraction $\frac{83}{60}$ as a mixed number. We have $83 \div 60 = 1$ with a remainder of $23$. So, $\frac{83}{60} = 1 \frac{23}{60}$. Therefore, the answer to the second problem is $\frac{83}{60}$.
9. Final Answer
The first problem's answer is $5 \frac{83}{90}$, which corresponds to option C. The second problem's answer is $\frac{83}{60}$, which corresponds to option B.
### Examples
Understanding fractions and decimals is crucial in everyday life. For instance, when you're cooking and need to adjust ingredient quantities, knowing how to add and convert fractions and decimals ensures your recipe turns out perfectly. Similarly, when calculating discounts at a store, being able to work with fractions and decimals helps you determine the final price accurately, ensuring you're getting the best deal.
- Convert the repeating decimal $2.1(4)$ to a fraction: $2.1(4) = \frac{193}{90}$.
- Add the two fractions: $\frac{34}{9} + \frac{193}{90} = \frac{533}{90} = 5 \frac{83}{90}$.
- Calculate the sum of the fractions: $\frac{49}{60} + \frac{1}{5} + \frac{11}{30} = \frac{49}{60} + \frac{12}{60} + \frac{22}{60} = \frac{83}{60}$.
### 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: $\frac{49}{60} + \frac{1}{5} + \frac{11}{30}$.
2. Converting 3.(7) to a fraction
Let's solve the first problem. We need to convert the repeating decimals to fractions. First, let's convert $3.(7)$ to a fraction. Let $x = 3.(7)$. Then $10x = 37.(7)$. Subtracting the two equations, $10x - x = 37.(7) - 3.(7)$, so $9x = 34$, which means $x = \frac{34}{9}$.
3. Converting 2.1(4) to a fraction
Next, let's convert $2.1(4)$ to a fraction. Let $y = 2.1(4)$. Then $10y = 21.(4)$. Let $z = 21.(4)$. Then $10z = 214.(4)$. Subtracting the two equations, $10z - z = 214.(4) - 21.(4)$, so $9z = 193$, which means $z = \frac{193}{9}$. Thus, $10y = \frac{193}{9}$, so $y = \frac{193}{90}$.
4. Adding the fractions
Now, we need to calculate the sum of the two fractions: $\frac{34}{9} + \frac{193}{90}$. To add these fractions, we need a common denominator, which is 90. So, we have $\frac{34}{9} = \frac{340}{90}$. Therefore, $\frac{340}{90} + \frac{193}{90} = \frac{340 + 193}{90} = \frac{533}{90}$.
5. Expressing as a mixed number
Now, let's express the fraction $\frac{533}{90}$ as a mixed number. We have $533 \div 90 = 5$ with a remainder of $83$. So, $\frac{533}{90} = 5 \frac{83}{90}$. Therefore, the answer to the first problem is $5 \frac{83}{90}$.
6. Finding a common denominator
Now, let's solve the second problem. We need to calculate the sum of three fractions: $\frac{49}{60} + \frac{1}{5} + \frac{11}{30}$. To add these fractions, we need a common denominator. The least common multiple of 60, 5, and 30 is 60. So, we rewrite the fractions with the common denominator: $\frac{49}{60}$, $\frac{1}{5} = \frac{12}{60}$, and $\frac{11}{30} = \frac{22}{60}$.
7. Adding the fractions
Now, we can calculate the sum: $\frac{49}{60} + \frac{12}{60} + \frac{22}{60} = \frac{49 + 12 + 22}{60} = \frac{83}{60}$.
8. Expressing as a mixed number
Finally, let's express the fraction $\frac{83}{60}$ as a mixed number. We have $83 \div 60 = 1$ with a remainder of $23$. So, $\frac{83}{60} = 1 \frac{23}{60}$. Therefore, the answer to the second problem is $\frac{83}{60}$.
9. Final Answer
The first problem's answer is $5 \frac{83}{90}$, which corresponds to option C. The second problem's answer is $\frac{83}{60}$, which corresponds to option B.
### Examples
Understanding fractions and decimals is crucial in everyday life. For instance, when you're cooking and need to adjust ingredient quantities, knowing how to add and convert fractions and decimals ensures your recipe turns out perfectly. Similarly, when calculating discounts at a store, being able to work with fractions and decimals helps you determine the final price accurately, ensuring you're getting the best deal.
