To solve this problem, we need to determine the number of toads in the pond. Let's follow the steps to find the solution:
1. Establish the initial ratio:
- We are given that the initial ratio of toads to frogs is [tex]\( 3:10 \)[/tex].
- Let the initial number of toads be [tex]\( 3x \)[/tex] and the number of frogs be [tex]\( 10x \)[/tex].
2. Account for the change:
- We are informed that 9 more frogs enter the pond.
- Therefore, the number of frogs becomes [tex]\( 10x + 9 \)[/tex].
3. Set up the new ratio:
- The new ratio of toads to frogs becomes [tex]\( 3:11 \)[/tex].
- Using our variables, the number of toads remains [tex]\( 3x \)[/tex] and the number of frogs is now [tex]\( 10x + 9 \)[/tex].
4. Set up the ratio equation:
- The new ratio can be formed into an equation:
[tex]\[
\frac{3x}{10x + 9} = \frac{3}{11}
\][/tex]
5. Solve the equation for [tex]\( x \)[/tex]:
- Cross multiply to solve for [tex]\( x \)[/tex]:
[tex]\[
11 \cdot 3x = 3 \cdot (10x + 9)
\][/tex]
- Simplify:
[tex]\[
33x = 30x + 27
\][/tex]
- Isolate [tex]\( x \)[/tex]:
[tex]\[
33x - 30x = 27
\][/tex]
[tex]\[
3x = 27
\][/tex]
[tex]\[
x = 9
\][/tex]
6. Calculate the number of toads:
- Recall that the number of toads is [tex]\( 3x \)[/tex].
- Substitute [tex]\( x = 9 \)[/tex]:
[tex]\[
3 \times 9 = 27
\][/tex]
Therefore, the number of toads in the pond is [tex]\( 27 \)[/tex].
