Question

A gardener has 1000 plants. He wants to plant these in such a way that the number of rows and the number of columns remain the same. Find the minimum number of plants he needs more for this.
a)12
b)24
c)11
d)23

Answer 1

Hint: First find the square root of a given number. When you get the value in decimals, just find the range of integers which has this number in between them. Square root will lie between two integers with difference. So, now, take the minimum integers and then square it. Find the difference between this square and the number given which is 1000. This difference will be the minimum number added to 1000 to make it the same number of rows and columns. This is done because rows, columns and products give the total number. So, it is the same then the product must be a perfect square. Now, we need to find the nearest perfect square to 1000.

Complete step-by-step answer:
Let for example we have ‘x’ rows of plants and ‘y’ columns as shown.

In the given figure we can say, we have ‘x’ plants like that ‘y’ column. So, total plants are their product. Pants $=x.y$ .

Given the equation condition is $x=y$ , So, now we can multiply, we get pants $={{x}^{2}}$ .
The number of plants in the question is a value of 1000. So, we need a square root of 1000. It can be written as, $\sqrt{1000}$ .

Now take 1000 as a multiplied product of 100, 10 $\sqrt{100\times 10}=10\sqrt{=10}$ .

By using the range of $\sqrt{10}$ we can say that $31<10\sqrt{10}<32$ .

So the minimum integers greater than $\sqrt{1000}$ is 32. So, difference between squares of the above 2 numbers is: ${{32}^{2}}-{{\left( \sqrt{1000} \right)}^{2}}=1024-1000$ .

By simplifying the above number we get the difference as : =24.

So, we need 24 more plants to make it a perfect square.

Therefore, we need 24 plantations to make it in a way that the number of rows and columns are equal.

Therefore Option (B) is the correct answer.


Note: The $\sqrt{1000}$ can be found by division method instead of writing it to $10\sqrt{10}$ directly applying the division method to 1000.