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Answer :
- Convert the mixed number $7 \frac{2}{3}$ to an improper fraction: $\frac{23}{3}$.
- Multiply the length of each bulb by $n$, the number of bulbs: $n \times \frac{23}{3}$.
- Test each possible total length to find an integer value for $n$.
- The total length is 23 inches when $n = 3$, so the answer is $\boxed{23 \text{ inches}}$.
### Explanation
1. Understanding the Problem
We are given that the length of each bulb is $7 \frac{2}{3}$ inches. We need to find the total length of all the bulbs. The possible answers are 21 inches, $21 \frac{2}{3}$ inches, $22 \frac{1}{3}$ inches, and 23 inches. However, the number of bulbs is not given in the problem statement. Therefore, we need to determine the number of bulbs from the possible answers.
2. Setting up the Equation
Let $n$ be the number of bulbs with length $7 \frac{2}{3}$ inches. Then the total length is $n \times 7 \frac{2}{3}$ inches. We need to find $n$ such that $n \times 7 \frac{2}{3}$ is equal to one of the possible answers.
3. Converting to Improper Fraction
Convert the mixed number to an improper fraction: $7 \frac{2}{3} = \frac{7 \times 3 + 2}{3} = \frac{21 + 2}{3} = \frac{23}{3}$ inches.
4. Total Length Equation
The total length is $n \times \frac{23}{3}$ inches. We need to find an integer $n$ such that $n \times \frac{23}{3}$ is equal to one of the possible answers.
5. Converting Possible Answers
The possible answers are 21, $21 \frac{2}{3}$, $22 \frac{1}{3}$, and 23. Convert them to improper fractions with denominator 3: $21 = \frac{63}{3}$, $21 \frac{2}{3} = \frac{65}{3}$, $22 \frac{1}{3} = \frac{67}{3}$, $23 = \frac{69}{3}$.
6. Finding n
We need to find $n$ such that $n \times \frac{23}{3}$ is equal to one of $\frac{63}{3}$, $\frac{65}{3}$, $\frac{67}{3}$, $\frac{69}{3}$. This means $n \times 23$ must be equal to 63, 65, 67, or 69.
7. Checking n for 21
If $n \times 23 = 63$, then $n = \frac{63}{23}$, which is not an integer.
8. Checking n for $21 \frac{2}{3}$
If $n \times 23 = 65$, then $n = \frac{65}{23}$, which is not an integer.
9. Checking n for $22 \frac{1}{3}$
If $n \times 23 = 67$, then $n = \frac{67}{23}$, which is not an integer.
10. Checking n for 23
If $n \times 23 = 69$, then $n = \frac{69}{23} = 3$, which is an integer.
11. Final Answer
Therefore, the number of bulbs is 3, and the total length is $3 \times 7 \frac{2}{3} = 3 \times \frac{23}{3} = 23$ inches.
### Examples
Imagine you are decorating a room with a string of identical decorative lights. Each light bulb is $7 \frac{2}{3}$ inches long. If you use 3 of these bulbs, the total length they cover is 23 inches. This kind of calculation is useful in many scenarios, such as planning the layout of lighting, determining the amount of material needed for a project, or even calculating distances in construction or design.
- Multiply the length of each bulb by $n$, the number of bulbs: $n \times \frac{23}{3}$.
- Test each possible total length to find an integer value for $n$.
- The total length is 23 inches when $n = 3$, so the answer is $\boxed{23 \text{ inches}}$.
### Explanation
1. Understanding the Problem
We are given that the length of each bulb is $7 \frac{2}{3}$ inches. We need to find the total length of all the bulbs. The possible answers are 21 inches, $21 \frac{2}{3}$ inches, $22 \frac{1}{3}$ inches, and 23 inches. However, the number of bulbs is not given in the problem statement. Therefore, we need to determine the number of bulbs from the possible answers.
2. Setting up the Equation
Let $n$ be the number of bulbs with length $7 \frac{2}{3}$ inches. Then the total length is $n \times 7 \frac{2}{3}$ inches. We need to find $n$ such that $n \times 7 \frac{2}{3}$ is equal to one of the possible answers.
3. Converting to Improper Fraction
Convert the mixed number to an improper fraction: $7 \frac{2}{3} = \frac{7 \times 3 + 2}{3} = \frac{21 + 2}{3} = \frac{23}{3}$ inches.
4. Total Length Equation
The total length is $n \times \frac{23}{3}$ inches. We need to find an integer $n$ such that $n \times \frac{23}{3}$ is equal to one of the possible answers.
5. Converting Possible Answers
The possible answers are 21, $21 \frac{2}{3}$, $22 \frac{1}{3}$, and 23. Convert them to improper fractions with denominator 3: $21 = \frac{63}{3}$, $21 \frac{2}{3} = \frac{65}{3}$, $22 \frac{1}{3} = \frac{67}{3}$, $23 = \frac{69}{3}$.
6. Finding n
We need to find $n$ such that $n \times \frac{23}{3}$ is equal to one of $\frac{63}{3}$, $\frac{65}{3}$, $\frac{67}{3}$, $\frac{69}{3}$. This means $n \times 23$ must be equal to 63, 65, 67, or 69.
7. Checking n for 21
If $n \times 23 = 63$, then $n = \frac{63}{23}$, which is not an integer.
8. Checking n for $21 \frac{2}{3}$
If $n \times 23 = 65$, then $n = \frac{65}{23}$, which is not an integer.
9. Checking n for $22 \frac{1}{3}$
If $n \times 23 = 67$, then $n = \frac{67}{23}$, which is not an integer.
10. Checking n for 23
If $n \times 23 = 69$, then $n = \frac{69}{23} = 3$, which is an integer.
11. Final Answer
Therefore, the number of bulbs is 3, and the total length is $3 \times 7 \frac{2}{3} = 3 \times \frac{23}{3} = 23$ inches.
### Examples
Imagine you are decorating a room with a string of identical decorative lights. Each light bulb is $7 \frac{2}{3}$ inches long. If you use 3 of these bulbs, the total length they cover is 23 inches. This kind of calculation is useful in many scenarios, such as planning the layout of lighting, determining the amount of material needed for a project, or even calculating distances in construction or design.
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