RBSE Solutions Class 7 Maths Chapter 1 Integers Exercise 1.2

Question 1. Find the product of the following:
(i) (-3) x 4
(ii) (-1) x 24
(iii) (-30) x (-24)
(iv) (-214) x 0
(v) (-15) x (-7) x 6
(vi) (-5) x (-7) x (-4)
(vii) (-3) x (-2) x (-1) x (-5)
Answer:
(i) \( (-3) \times 4 = -12 \)
(ii) \( (-1) \times 24 = -24 \)
(iii) \( (-30) \times (-24) = 720 \) Multiplying two negative numbers always gives a positive result.
(iv) \( (-214) \times 0 = 0 \)
(v) \( (-15) \times (-7) \times 6 = 105 \times 6 = 630 \)
(vi) \( (-5) \times (-7) \times (-4) = 35 \times (-4) = -140 \)
(vii) \( (-3) \times (-2) \times (-1) \times (-5) = [(-3) \times (-2)] \times [(-1) \times (-5)] = 6 \times 5 = 30 \)
In simple words: When you multiply numbers, remember that two negative signs make a positive, but an odd number of negative signs makes a negative. Also, anything multiplied by zero is always zero.

🎯 Exam Tip: Pay close attention to the number of negative signs in a multiplication. An even count of negative signs results in a positive product, while an odd count results in a negative product.

Question 2. Start with (- 1) x 5 and make a pattern to show that (- 1) x (- 1) = + 1
Answer:
The pattern shows how multiplying by -1 changes a positive number to negative, and then a negative number back to positive.
\( (-1) \times 5 = -5 \)
\( (-1) \times 4 = -4 \)
\( (-1) \times 3 = -3 \)
\( (-1) \times 2 = -2 \)
\( (-1) \times 0 = 0 \)
\( (-1) \times (-1) = +1 \)
In simple words: When you multiply by -1, the number changes its sign. A positive number becomes negative, and a negative number becomes positive. This pattern helps understand why multiplying two negative numbers gives a positive answer.

🎯 Exam Tip: To show patterns in math, start with a known fact and gradually change one part of the problem to reveal the rule, like how multiplying by -1 affects the sign of numbers.

Question 3. The rate of decreasing the temperature in a refrigerator is 3°C per minute. A thing whose temperature is 25°C, is kept in refrigerator. After how much time, the temperature of thing will be - 2°C?
Answer:
Starting temperature \( = 25^\circ C \)
Final desired temperature \( = -2^\circ C \)
Total temperature drop needed \( = 25^\circ C - (-2^\circ C) = 25^\circ C + 2^\circ C = 27^\circ C \)
Rate of temperature decrease \( = 3^\circ C \) per minute.
Time taken to reach \( -2^\circ C \)
\( = \frac{\text{Total temperature drop}}{\text{Rate of decrease}} \)
\( = \frac{27^\circ C}{3^\circ C \text{ per minute}} = 9 \text{ minutes} \)
So, it will take 9 minutes for the temperature to drop to \( -2^\circ C \). This calculation assumes a constant cooling rate within the refrigerator.
In simple words: The refrigerator cools by 3 degrees every minute. The temperature needs to go from 25 degrees down to -2 degrees. This is a total drop of 27 degrees. So, 27 divided by 3 means it will take 9 minutes.

🎯 Exam Tip: When calculating temperature changes, remember to subtract the final temperature from the initial temperature to find the total change, being careful with negative signs.

Question 4. In a game, two balls are given on selecting a blue card and three balls are gained on selecting a red card. Sheetal has 27 balls with her. During game she gets 9 blue cards continuously. How much balls, she have remaining?
Answer:
Initial number of balls with Sheetal \( = 27 \)
Balls given for selecting one blue card \( = 2 \)
Number of blue cards Sheetal gets \( = 9 \)
Total balls Sheetal has to give \( = 9 \times 2 = 18 \) balls.
Remaining balls with Sheetal \( = \text{Initial balls} - \text{Balls given away} \)
\( = 27 - 18 = 9 \) balls.
So, Sheetal has 9 balls remaining. The game involves losing balls with blue cards, but gaining with red cards, creating a dynamic score.
In simple words: Sheetal started with 27 balls. She got 9 blue cards, and each blue card means giving away 2 balls. So, she gave away 18 balls in total. She had 27 balls, gave away 18, which leaves her with 9 balls.

🎯 Exam Tip: Read word problems carefully to identify whether items are being added or subtracted, especially when different actions (like getting blue vs. red cards) have different outcomes.

Question 5. Solve the following division:
(i) (-35) \( \div \) 7
(ii) 15 \( \div \) (-3)
(iii) - 25 \( \div \) (-25)
(iv) 25 \( \div \) (-1)
(v) 0 \( \div \) (-3)
(vi) 15 \( \div \) [1(-2) +1]
(vii) (-6) + 3 [(-2) + 1]
Answer:
(i) \( (-35) \div 7 = -5 \)
(ii) \( 15 \div (-3) = -5 \)
(iii) \( -25 \div (-25) = 1 \) Dividing two negative numbers always results in a positive number.
(iv) \( 25 \div (-1) = -25 \)
(v) \( 0 \div (3) = 0 \)
(vi) \( 15 \div [1(-2) + 1] = 15 \div [-2 + 1] = 15 \div [-1] = -15 \)
(vii) \( (-6) + 3 \times [(-2) + 1] = (-6) + 3 \times [-1] = -6 + (-3) = -6 - 3 = -9 \)
In simple words: When you divide numbers, if one number is negative and the other is positive, the answer is negative. If both numbers are negative, the answer is positive. Zero divided by any non-zero number is always zero.

🎯 Exam Tip: Remember the rules for signs in division: A positive divided by a negative (or vice versa) is negative, and a negative divided by a negative is positive. Always solve operations inside brackets first.

Question 6. A shopkeeper gains Rs.1 on selling a pen and loses 50 paise on selling a pencil. Represent the gain and loss in terms of integers.
(i) There is a loss of Rs.5 in a month. If he had sold 45 pens then find the number of pencils, sold by him in a month.
(ii) There is no profit and no loss in the second month. If he sold 70 pens and the number of pencils sold.
Answer:
Gain on selling 1 pen \( = +1 \) Rs. \( = +100 \) paise.
Loss on selling 1 pencil \( = -50 \) paise.

(i) Total loss in a month \( = -5 \) Rs. \( = -500 \) paise.
Gain from selling 45 pens \( = 45 \times 100 \) paise \( = 4500 \) paise.
Let \( x \) be the number of pencils sold.
Loss from selling \( x \) pencils \( = x \times (-50) \) paise \( = -50x \) paise.
Total profit/loss \( = \text{Gain from pens} + \text{Loss from pencils} \)
\( -500 = 4500 + (-50x) \)
\( -500 = 4500 - 50x \)
\( 50x = 4500 + 500 \)
\( 50x = 5000 \)
\( x = \frac{5000}{50} \)
\( \implies x = 100 \)
So, the shopkeeper sold 100 pencils. Understanding how different transactions combine to affect overall profit or loss is key here.

(ii) In the second month, there is no profit and no loss. This means the total gain equals the total loss.
Gain from selling 70 pens \( = 70 \times 100 \) paise \( = 7000 \) paise.
Let \( y \) be the number of pencils sold.
Loss from selling \( y \) pencils \( = y \times (-50) \) paise \( = -50y \) paise.
Since there is no profit/loss, total gain \( + \) total loss \( = 0 \)
\( 7000 + (-50y) = 0 \)
\( 7000 - 50y = 0 \)
\( 50y = 7000 \)
\( y = \frac{7000}{50} \)
\( \implies y = 140 \)
So, the shopkeeper sold 140 pencils.
In simple words: For a pen, the shopkeeper gets 100 paise (1 Rs.). For a pencil, he loses 50 paise.
(i) If he lost 500 paise in total and earned 4500 paise from pens, he must have lost 5000 paise from pencils. Since each pencil loses 50 paise, he sold 100 pencils.
(ii) If there was no profit or loss and he earned 7000 paise from pens, he must have lost 7000 paise from pencils. So, he sold 140 pencils (7000 divided by 50).

🎯 Exam Tip: Always convert all monetary values to a single unit (like paise) to avoid errors in calculations, especially when dealing with mixed units like Rupees and paise. Remember that "no profit, no loss" means total gains equal total losses.

Question 8. If going up a 60 feet multi storied by a lift is represented by positive integers then:
(i) How will we represented the height of flat at 60 feet above?
(ii) Represent the parking 15 feet below integer.
(iii) If lift goes upwards at the rate of 5 feet/ sec, represented as + 5 and if travels in an opposite direction, then what is the integer representing downward direction?
Answer:
(i) The height of the flat at 60 feet above ground level will be represented as \( +60 \). This uses positive integers to show upward movement.
(ii) The parking 15 feet below ground level will be represented as \( -15 \). Negative integers help us show positions below a reference point, like ground level.
(iii) If upward movement is \( +5 \) feet/sec, then the opposite direction, which is downward movement, will be represented as \( -5 \) feet/sec. Integers with opposite signs are perfect for showing opposite directions or changes.
In simple words: When going up, we use plus numbers. So, 60 feet up is \( +60 \). When going down, we use minus numbers. So, 15 feet below is \( -15 \). If going up fast is \( +5 \), then going down fast at the same speed is \( -5 \).

🎯 Exam Tip: Remember that positive and negative integers are used to represent opposite quantities or directions, such as above/below sea level, profit/loss, or moving up/down.