A developer has land that has x feet of lake frontage. The land is to be subdivided into lots, each of which is to have either 80 feet or 100 feet of lake frontage. If \(\frac{1}{9}\) of the lots are to have 80 feet of frontage each and the remaining 40 lots are to have 100 feet of frontage each, what is the value of x?
Show Hint
- 400
- 3,200
- 3,700
- 4,400
- 4,760
The Correct Option is D
Solution and Explanation
This is a multi-step word problem. The key is to first determine the total number of lots using the information about the 'remaining' lots, then calculate the number of lots of each type, and finally sum their frontages to find the total frontage, x.
Step 2: Detailed Explanation:
1. Find the total number of lots. - Let \(N\) be the total number of lots. - We are told that \(\frac{1}{9}\) of the lots have 80 feet of frontage. - This means the fraction of lots that have 100 feet of frontage is the remainder: \(1 - \frac{1}{9} = \frac{8}{9}\). - The problem states that these "remaining 40 lots" all have 100 feet of frontage. - Therefore, we can set up the equation: \(\frac{8}{9} N = 40\). - To solve for \(N\), multiply both sides by \(\frac{9}{8}\): \(N = 40 \times \frac{9}{8} = 5 \times 9 = 45\). - So, there are 45 lots in total.
2. Find the number of each type of lot. - Number of 80-foot lots = \(\frac{1}{9} \times N = \frac{1}{9} \times 45 = 5\) lots. - Number of 100-foot lots = 40 lots (this was given in the problem).
3. Calculate the total frontage, x. - The total frontage is the sum of the frontages from both types of lots. - Total frontage = (Number of 80-foot lots \(\times\) 80 feet) + (Number of 100-foot lots \(\times\) 100 feet) \[ x = (5 \times 80) + (40 \times 100) \] \[ x = 400 + 4000 \] \[ x = 4400 \] Step 3: Final Answer:
The value of x is 4,400 feet.
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