VerifiedHint: Both the rabbits and pigeons have only one head, so we can take a hint that the sum of rabbits and pigeons is $90$. Also we can make an equation from taking a hint from the total number of their legs given because a pigeon has only two legs while a rabbit has four legs.
Step by step solution:
Given: The total number of heads is given, $90$ of the rabbits and pigeons. While the sum of their legs that is $224$ are also given.
Let ‘x’ be the total number of rabbits in the zoo and ‘y’ be the total number of pigeons in the zoo.
So according to assumption we have to find the value of ‘y’.
As both rabbits and pigeons have only one head therefore we can make an equation here.
Total heads given $ = 90$, therefore ${\text{x + y = 90}}$-equation $\left( 1 \right)$
As pigeons have only $2$ legs and rabbits have four legs therefore we can make another equation here.
Total legs given $ = 224$
$4{\text{x + 2y}} = 224$ -equation $\left( 2 \right)$
Now we will multiply the equation $\left( 1 \right)$ by 2
We get $2{\text{x + 2y}} = 180$ -equation $\left( 3 \right)$
Now we will subtract equation $\left( 3 \right)$from equation $\left( 2 \right)$
$4{\text{x + 2y}} - {\text{2x}} - {\text{2y}} = 224 - 180$
$2{\text{x}} = 44$
${\text{x}} = 22$ -equation $\left( 4 \right)$
Put the value of equation $\left( 4 \right)$ in equation $\left( 1 \right)$
22 + y = 90
${\text{y}} = 68$
And we assume pigeons as ‘y’.
Hence the total number of pigeons in the zoo is $68$.
Note: Keep in mind while making the equation that we assume ‘x’ as rabbits and ‘y’ as pigeons. We multiplied ‘x’ by $4$ and ‘y’ by $2$ because rabbits have $4$ legs while pigeons have $2$ legs. We have multiplied the equation $\left( 1 \right)$ by 2 because we can get either ‘x’ or ‘y’ equal to solve the equation. If we have asked about the rabbit then we must have given the answer there are $22$ rabbits but here we have asked about pigeons only.
