Question

Steve saw $313$ birds during a bird-watching trip to Yellowstone National Park. He observed $15$ more trumpeter swans than sand-hill cranes. He saw twice as many mountain chickadees as dusky grouses. How many birds of each type did Steve observe if he saw half as many trumpeter swans as dusky grouses?

Answer 1

Find how many birds of each type did Steve observe:

Step-1: Make the equations:

Total number of birds $=313$

Trumpeter swans $=$$s$

Sandhill cranes $=$ $c$

Mountain chickadees $=$ $m$

Ducky grouses $=$ $g$

We have equations below based on given:

There are $15$ more trumpeter swans than sand-hill cranes.

$s=c+15$

There are half as many trumpeter swans as dusky grouses.

$$\begin{gathered} g=2s \\ \therefore g=2\left(c+15\right) \end{gathered}$$

There are twice as many mountain chickadees as dusky grouses.

$m=2g$

$$\begin{gathered} \Rightarrow m=2\left(2s\right) \\ \Rightarrow m=4s \\ \therefore m=4\left(c+15\right)\left(\because s=c+15\right) \end{gathered}$$

Total number of birds. he visited is given as $313$

$s+c+m+g=313$.

Step-2: Solve for $c$:

Substituting the value of $m,g$ and $s$ as above in $s+c+m+g=313$ and finding the value of $c$:

$c+15+c+2\left(c+15\right)+4\left(c+15\right)=313$

Now solve for $c$

$$\begin{gathered} 8c+15+30+60=313 \\ \Rightarrow 8c+105=313 \\ \Rightarrow 8c=313-105\left[\mathrm{subtracting}105\mathrm{from}\mathrm{both}\mathrm{sides}\right] \\ \Rightarrow 8c=208 \end{gathered}$$

$\Rightarrow$ $c=\frac{208}{8}\left[\mathrm{Dividing}8\mathrm{from}\mathrm{both}\mathrm{sides}\right]$

$\therefore$ $c=26$

Step-3: Calculate the other values:

Substitute value of $c$ in $s=c+15$

$$\begin{gathered} s=26+15 \\ \therefore s=41 \end{gathered}$$

Substitute value of $s$ in $g=2s$

$$\begin{gathered} g=2\times 41 \\ \therefore g=82 \end{gathered}$$

Substitute value of $g$ in $m=2g$

$$\begin{gathered} m=2\times 82 \\ \therefore m=164 \end{gathered}$$

Hence, the answer is

Trumpeter swans $=41$

Sandhill cranes $=26$

Mountain chickadees $=164$

Ducky grouses $=82$