Answer :
- Calculate the length of each bulb as an improper fraction: $7 \frac{2}{3} = \frac{23}{3}$.
- Test each possible total length to see if the number of bulbs is an integer.
- For 21 inches: $n = \frac{63}{23}$ (not an integer).
- For $21 \frac{2}{3}$ inches: $n = \frac{65}{23}$ (not an integer).
- For $22 \frac{1}{3}$ inches: $n = \frac{67}{23}$ (not an integer).
- For 23 inches: $n = 3$ (an integer).
- The total length is \boxed{23} inches.
### Explanation
1. Understanding the Problem
We are given that the length of each hyacinth bulb is $7
\frac{2}{3}$ inches. We need to find the total length of all the bulbs, given the possible answers: 21 inches, $21
\frac{2}{3}$ inches, $22
\frac{1}{3}$ inches, and 23 inches. To find the total length, we need to determine the number of bulbs, which must be a whole number.
2. Calculating Length per Bulb
Let $n$ be the number of bulbs. The length of each bulb is $7
\frac{2}{3} =
\frac{7
\times 3 + 2}{3} =
\frac{21 + 2}{3} =
\frac{23}{3}$ inches. The total length is $n
\times
\frac{23}{3}$ inches.
3. Checking Possible Answers
We will check each of the possible answers to see if $n$ is an integer. If the total length is 21 inches, then $n
\times
\frac{23}{3} = 21$, so $n =
\frac{21
\times 3}{23} =
\frac{63}{23}
\approx 2.74$, which is not an integer.
If the total length is $21
\frac{2}{3} =
\frac{65}{3}$ inches, then $n
\times
\frac{23}{3} =
\frac{65}{3}$, so $n =
\frac{65/3}{23/3} =
\frac{65}{23}
\approx 2.83$, which is not an integer.
If the total length is $22
\frac{1}{3} =
\frac{67}{3}$ inches, then $n
\times
\frac{23}{3} =
\frac{67}{3}$, so $n =
\frac{67/3}{23/3} =
\frac{67}{23}
\approx 2.91$, which is not an integer.
If the total length is 23 inches, then $n
\times
\frac{23}{3} = 23$, so $n =
\frac{23
\times 3}{23} = 3$, which is an integer.
4. Finding the Solution
Since $n$ must be an integer, the total length is 23 inches, and the number of bulbs is 3.
### Examples
Understanding how to calculate total lengths based on individual items is useful in many real-life situations. For instance, if you are buying multiple pieces of lumber for a construction project, knowing the length of each piece and the total desired length helps you determine how many pieces you need. Similarly, if you're calculating the total fabric needed for sewing multiple items, this concept applies directly.
- Test each possible total length to see if the number of bulbs is an integer.
- For 21 inches: $n = \frac{63}{23}$ (not an integer).
- For $21 \frac{2}{3}$ inches: $n = \frac{65}{23}$ (not an integer).
- For $22 \frac{1}{3}$ inches: $n = \frac{67}{23}$ (not an integer).
- For 23 inches: $n = 3$ (an integer).
- The total length is \boxed{23} inches.
### Explanation
1. Understanding the Problem
We are given that the length of each hyacinth bulb is $7
\frac{2}{3}$ inches. We need to find the total length of all the bulbs, given the possible answers: 21 inches, $21
\frac{2}{3}$ inches, $22
\frac{1}{3}$ inches, and 23 inches. To find the total length, we need to determine the number of bulbs, which must be a whole number.
2. Calculating Length per Bulb
Let $n$ be the number of bulbs. The length of each bulb is $7
\frac{2}{3} =
\frac{7
\times 3 + 2}{3} =
\frac{21 + 2}{3} =
\frac{23}{3}$ inches. The total length is $n
\times
\frac{23}{3}$ inches.
3. Checking Possible Answers
We will check each of the possible answers to see if $n$ is an integer. If the total length is 21 inches, then $n
\times
\frac{23}{3} = 21$, so $n =
\frac{21
\times 3}{23} =
\frac{63}{23}
\approx 2.74$, which is not an integer.
If the total length is $21
\frac{2}{3} =
\frac{65}{3}$ inches, then $n
\times
\frac{23}{3} =
\frac{65}{3}$, so $n =
\frac{65/3}{23/3} =
\frac{65}{23}
\approx 2.83$, which is not an integer.
If the total length is $22
\frac{1}{3} =
\frac{67}{3}$ inches, then $n
\times
\frac{23}{3} =
\frac{67}{3}$, so $n =
\frac{67/3}{23/3} =
\frac{67}{23}
\approx 2.91$, which is not an integer.
If the total length is 23 inches, then $n
\times
\frac{23}{3} = 23$, so $n =
\frac{23
\times 3}{23} = 3$, which is an integer.
4. Finding the Solution
Since $n$ must be an integer, the total length is 23 inches, and the number of bulbs is 3.
### Examples
Understanding how to calculate total lengths based on individual items is useful in many real-life situations. For instance, if you are buying multiple pieces of lumber for a construction project, knowing the length of each piece and the total desired length helps you determine how many pieces you need. Similarly, if you're calculating the total fabric needed for sewing multiple items, this concept applies directly.
