The exercise shows the following questions:

1. The sum of the squares of two consecutive natural numbers is 113. Find the numbers.
2. Tinu is younger than Pinky by three years. The product of their ages is 180. Find their ages.
3. The length of the rectangle is greater than twice its breadth by 3 cm. If the area of the rectangle is 24 sq. cm, find its length and breadth.

Let me solve these problems one by one for you.

Problem 1: The sum of the squares of two consecutive natural numbers is 113. Find the numbers.

Let's denote the two consecutive natural numbers as $x$ and $x+1$.

The equation for the sum of their squares is: $x^2 + (x+1)^2 = 113$

Expanding and simplifying: $x^2 + x^2 + 2x + 1 = 113$ $2x^2 + 2x + 1 = 113$ $2x^2 + 2x - 112 = 0$ $x^2 + x - 56 = 0$

Now, solving this quadratic equation: $x = \frac{-1 \pm \sqrt{1 + 4 \cdot 56}}{2}$ $x = \frac{-1 \pm \sqrt{225}}{2}$ $x = \frac{-1 \pm 15}{2}$

This gives two values for $x$:

1. $x = \frac{14}{2} = 7$
2. $x = \frac{-16}{2} = -8$ (which we discard as it is not a natural number)

So, $x = 7$, and the two consecutive natural numbers are 7 and 8.

Answer for Problem 1: The numbers are 7 and 8.

Would you like detailed solutions for the other questions?

Here are five related questions for practice:

1. What is the sum of squares of three consecutive numbers if the middle number is 10?
2. Find two consecutive numbers whose cubes sum up to 189.
3. If the product of two consecutive numbers is 182, what are those numbers?
4. The difference of squares of two consecutive natural numbers is 25. Find the numbers.
5. Find the sum of the squares of two consecutive odd numbers that add up to 24.

Tip:

When dealing with consecutive numbers, defining the numbers in terms of variables like $x$ and $x+1$ can simplify setting up equations and solving them.