NCERT Class 7 Ganita Prakash 2 Chapter 02 Operations with Integers PDF Download

Chapter 2: Operations with Integers

2.1 A Quick Recap of Integers

Rakesh's Puzzle: A Number Game

Rakesh gives you a challenge.

"I have thought of two numbers", he says. "Their sum is 25, and their difference is 11."

Can you tell me the two numbers?

You don't need to use any formulas. Just try different pairs of numbers and then check:

1. Do the two numbers add up to 25?
2. Is the difference between them 11?

(Remember: the difference means first number - second number.)

Write your guesses like this:

Did you find the right pair?

Now that you've found the correct pair, Rakesh gives you a second challenge:

"Think of two numbers whose sum is 25, but their difference is -11."

Use the same method. Try different pairs of numbers and fill in the table again. You will notice that if you swap the numbers from the first puzzle, you get the answer to Rakesh's second puzzle. That is, the first number is 7 and the second is 18!

Figure it Out

Let us try to find a few more pairs of numbers from their sums and differences:

(a) Sum = 27, Difference = 9
(b) Sum = 4, Difference = 12
(c) Sum = 0, Difference = 10
(d) Sum = 0, Difference = -10
(e) Sum = -7, Difference = -1
(f) Sum = -7, Difference = -13

Choose a partner and take turns to play this game. In each turn, one of you can think of two integers, and give their sum and difference; the other person must then figure out the integers. After some practice, you can perform this magic trick for your family members and surprise them!

Teacher's Note

When you get a sum and difference, write down two equations: first number + second number = sum, and first number - second number = difference. Adding these equations cancels one variable, making it much faster than guessing. This is the algebraic shortcut behind Rakesh's puzzle.

Carrom Coin Integers

A carrom coin is struck to move it to the right. Each strike moves the coin a certain number of units of distance rightwards based on the force of the strike.

[Figure: A number line from 0 to 12 with a yellow coin, See in your textbook]

To begin with, the coin is at point 0. If the coin is struck twice, with the first strike moving it by 4 units and the second strike moving it by 3 units, what will be the final position of the coin?

It is clear that the coin will be \( 4 + 3 = 7 \) units from 0.

If the coin is struck twice, and if the two movements are known, can you give a formula for the final position of the coin?

If the first strike moves the coin \( a \) units to the right and if the second strike moves the coin \( b \) units to the right, then the final position is \( P = a + b \), where \( P \) is the distance of the coin from the starting point 0.

[Figure: A diagram showing the number line with the coin at position 0 and arrows indicating movements a and b to reach position P, See in your textbook]

Now, suppose the coin can be struck to move it in either direction - left or right.

[Figure: A number line centered at 0 with a yellow coin and arrows pointing left and right, See in your textbook]

The coin is at 0. If it is struck twice (the direction of the two strikes may be the same or different) can you give a formula for the final position of the coin?

One way to do this is to consider different cases of the directions of the strikes:

• both are rightward,
• both are leftward,
• the first one is rightward and the second one is leftward, and
• the first one is leftward and the second one is rightward.

An efficient way to model this situation is to use positive and negative integers. First, let us model the line on which the coin moves as a number line.

\[ \ldots -5 \text{ } -4 \text{ } -3 \text{ } -2 \text{ } -1 \text{ } 0 \text{ } 1 \text{ } 2 \text{ } 3 \text{ } 4 \text{ } 5 \text{ } 6 \text{ } \ldots \]

Let us consider the rightward movement positive and the leftward movement negative.

Suppose the first strike moves the coin rightward by 5 units from 0, and the second strike leftward by 7 units, then we take the

First Movement = 5 units.
Second Movement = -7 units.

What is the final position of the coin?

This can be found by simply adding the two movements: \( 5 + (-7) = -2 \).

So the coin is at -2, or it is 2 units to the left of 0.

In general, if the first strike moves the coin \( a \) units (which is positive if the strike is to the right and negative if the strike is to the left), and the second strike \( b \) units (which is positive if the strike is to the right and negative if the strike is to the left), then the final position \( P \), after the two strikes, is again \( P = a + b \).

Teacher's Note

Remember that a negative number represents a leftward movement on the number line. When you add \( 5 + (-7) \), you are starting at 5 and moving 7 units left, which lands you at -2. Always think of the sign as telling you the direction, not just as a minus sign.

Based on this new model, answer the following questions:

1. If the first movement is -4 and the final position is 5, what is the second movement?

Key Points

• Integers include both positive and negative whole numbers, plus zero. You can use them to represent movements in opposite directions on a number line.
• When adding integers, think of rightward movement as positive and leftward movement as negative. Adding a negative number is the same as moving left.
• For any sum of two integers, you can find the original integers if you also know their difference. The two numbers satisfy the equations: sum = first + second, and difference = first - second.
• The formula \( P = a + b \) works for the final position of the coin no matter which directions the strikes go, as long as you use positive for rightward and negative for leftward.

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