Lucy wants to find out the number of frogs in a pond near her house. She catches 54 of them, puts a leg band on each frog, and puts them back in the pond. The next week, she catches 70 frogs. Of the 70 frogs, 13 have a leg band. Based on this information, what is a reasonable estimate of the number of frogs in the pond?
A) 208 B) 632 C) 291 D) 111
Let's use a proportion to estimate the total number of frogs in the pond:
$\frac{\text{number of banded frogs}}{\text{total number of frogs in first sample}}=\frac{\text{number of banded frogs in second sample}}{\text{total number of frogs in second sample}}$
Plugging in the given values, we get:
$\frac{54}{x}=\frac{13}{70}$
Cross-multiplying and solving for $x$, we get:
$x=\frac{54\times 70}{13}\approx 291$
So an estimate of 291 frogs in the pond is reasonable. Therefore, the answer is $\boxed{\textbf{(C) }291}$.
i do not understand can you put it in simple words bot
Sure! We can use a formula to help estimate the total number of frogs in the pond. We know that Lucy caught 54 frogs and put leg bands on all of them. Later, when she caught 70 more frogs, 13 of them already had leg bands. By comparing these numbers, we can figure out how many frogs must be in the pond overall.
We write out a formula using these numbers and solve it to get our estimate. Using this method, we estimate that there are around 291 frogs in the pond.
To estimate the number of frogs in the pond, we can use the information given.
First, we know that Lucy caught 54 frogs and put a leg band on each of them. So, we can assume that these 54 frogs are the ones with leg bands.
Next, Lucy caught 70 frogs the following week, and 13 of them already had leg bands. This means that the remaining 70 - 13 = 57 frogs are new frogs.
Therefore, the total number of frogs in the pond would be the sum of the frogs with leg bands and the new frogs, which is 54 + 57 = 111.
So, a reasonable estimate of the number of frogs in the pond is 111.
The answer is (D) 111.