GSEB Class 7 Maths Solutions Chapter 1 Integers Exercise 1.3

Question 1. Find each of the following products:
(a) \( 3 \times (-1) \)
(b) \( (-1) \times 225 \)
(c) \( (-21) \times (-30) \)
(d) \( (-316) \times (-1) \)
(e) \( (-15) \times 0 \times (-18) \)
(f) \( (-12) \times (-11) \times (10) \)
(g) \( 9 \times (-3) \times (-6) \)
(h) \( (-18) \times (-5) \times (-4) \)
(i) \( (-1) \times (-2) \times (-3) \times 4 \)
(j) \( (-3) \times (-6) \times (-2) \times (-1) \)
Answer:
(a) To find the product of 3 and -1, we multiply their absolute values (3 and 1) and then apply the negative sign, as one number is positive and the other is negative. This gives us \( 3 \times (-1) = -(3 \times 1) = -3 \).
(b) To find the product of -1 and 225, we multiply their absolute values (1 and 225) and then apply the negative sign, as one number is negative and the other is positive. This results in \( (-1) \times 225 = -(1 \times 225) = -225 \).
(c) When multiplying two negative numbers, the result is always positive. We can break down 21 into \( (20+1) \) for easier calculation. Then, multiply each part by 30 and sum the results to get 630.
\( (-21) \times (-30) = +[21 \times 30] \)
\( = [(20 + 1) \times 30] \)
\( = 20 \times 30 + 1 \times 30 = 600 + 30 = 630 \)
(d) Multiplying a negative number by another negative number always gives a positive result. So, -316 times -1 equals positive 316.
\( (-316) \times (-1) = + (316 \times 1) = 316 \)
(e) Any number multiplied by zero will always result in zero. Therefore, when -15, 0, and -18 are multiplied together, the final product is 0.
\( (-15) \times 0 \times (-18) = +[(-15) \times 0] \times (-18) \)
\( = 0 \times (-18) = 0 \)
(f) When an even number of negative integers are multiplied, the final product is positive. Multiplying -12 by -11 gives 132. Then, multiplying 132 by 10 results in 1320.
\( (-12) \times (-11) \times 10 = + [12 \times 11 \times 10] \)
\( = [132 \times 10] = 1320 \)
(g) Since there are two negative integers (an even number) being multiplied, the final outcome will be positive. We multiply 9 by 3 by 6 to achieve the final product of 162.
\( 9 \times (-3) \times (-6) = + [9 \times 3 \times 6] = 162 \)
(h) Because there are three negative integers (an odd number) in this product, the final result will be negative. Multiplying 18 by 5 by 4 gives 360, so the final answer is -360.
\( (-18) \times (-5) \times (-4) = - [18 \times 5 \times 4] \)
\( = - [18 \times 20] = -360 \)
(i) With three negative numbers (an odd count) in the multiplication, the final result will be negative. We multiply 1, 2, 3, and 4 together to get 24. Therefore, the product is -24.
\( (-1) \times (-2) \times (-3) \times 4 = - [1 \times 2 \times 3 \times 4] \)
\( = - [2 \times 12] = -24 \)
(j) There are four negative numbers (an even count) in this product, so the overall result will be positive. Multiplying 3, 6, 2, and 1 together yields a product of 36.
\( (-3) \times (-6) \times (-2) \times (-1) = + [3 \times 6 \times 2 \times 1] \)
\( = + [36 \times 1] = 36 \)
In simple words: When you multiply numbers, remember that an even count of negative signs gives a positive answer, while an odd count gives a negative answer. Also, multiplying by zero always results in zero.

Exam Tip: Remember the rules of multiplying integers: same signs (positive x positive or negative x negative) give a positive product, and different signs (positive x negative) give a negative product. Always count the negative signs.

Question 2. Verify the following:
(a) \( 18 \times [7 + (-3)] = [18 \times 7] + [18 \times (-3)] \)
(b) \( (-21) \times [(-4) + (-6)] = [(-21) \times (-4)] + [(-21) \times (-6)] \)
Answer:
(a) To check this, we first calculate the left side (L.H.S.) by adding the numbers inside the bracket and then multiplying by 18. For the right side (R.H.S.), we multiply 18 by each number in the bracket separately and then add the products. Both sides should simplify to the same value, which is 72, confirming the statement.
L.H.S. \( = 18 \times [7 + (-3)] \)
\( = 18 \times [7 - 3] = 18 \times [4] = 72 \)
R.H.S. \( = (18 \times 7) + [18 \times (-3)] \)
\( = 126 + [-54] = 126 - 54 = 72 \)
\( \therefore \) L.H.S. \( = \) R.H.S.
\( \implies 18 \times [7 + (-3)] = [18 \times 7] + [18 \times (-3)] \)
(b) To verify this statement, we first work out the left-hand side (L.H.S.) by adding the negative numbers in the bracket and then multiplying by -21. For the right-hand side (R.H.S.), we multiply -21 by each negative number separately and then add those products. Both calculations give 210, showing the statement is true.
L.H.S. \( = (-21) \times [(-4) + (-6)] \)
\( = (-21) \times [-10] = + (21 \times 10) = 210 \)
R.H.S. \( = [(-21) \times (-4)] + [(-21) \times (-6)] = [(+84)] + [(+126)] = 84 + 126 = 210 \)
\( \therefore \) L.H.S. \( = \) R.H.S.
\( \implies (-21) \times [(-4) + (-6)] = [(-21) \times (-4)] + [(-21) \times (-6)] \)
In simple words: This question checks the distributive property, which means you can either add numbers first and then multiply, or multiply each number separately and then add. Both methods should always give the same answer.

Exam Tip: The distributive property \( a \times (b + c) = (a \times b) + (a \times c) \) is a fundamental concept. Practice applying it with both positive and negative integers to build confidence.

Question 3.
(i) For any integer a, what is \( (-1) \times a \) equal to?
(ii) Determine the integer whose product with \( (-1) \) is
(a) -22
(b) 37
(c) 0
Answer:
(i) When you multiply any whole number 'a' by -1, the result is simply the opposite sign of 'a', which is \( -a \).
\( (-1) \times a = -a \)
(ii)
(a) To get -22 when multiplying by -1, the number must be 22, because -1 multiplied by a positive number gives its negative counterpart.
\( \therefore (-1) \times 22 = -22 \)
(b) To obtain 37 when multiplying by -1, the original number must be -37, as multiplying two negative numbers results in a positive number.
\( (-1) \times (-37) = 37 \)
(c) If the product with -1 is 0, then the number itself must be 0, because any number multiplied by zero is always zero.
\( (-1) \times (0) = 0 \) [\( \therefore \) Product of a negative integer and zero is zero.]
In simple words: Multiplying any number by -1 simply flips its sign. If the number was positive, it becomes negative; if it was negative, it becomes positive. If it was zero, it stays zero.

Exam Tip: Understanding the effect of multiplying by -1 is crucial for working with integers. It's the same as finding the additive inverse of a number.

Question 4. Starting from \( (-1) \times 5 \), write various products showing some pattern to show \( (-1) \times (-1) = 1 \).
Answer: We can show how multiplying -1 by -1 equals 1 by following a pattern. Starting with -1 times 5, we get -5. As we decrease the number multiplied by -1 (from 5 to 4, then to 3, and so on), the product increases by 1 each time. This pattern continues until we reach -1 times 0, which is 0. Following this trend, when we multiply -1 by -1, the product increases by 1 from the previous result of 0, giving us 1.
\( (-1) \times 5 = -5 \)
\( (-1) \times 4 = -4 = (-5) + 1 \)
\( (-1) \times 3 = -3 = (-4) + 1 \)
\( (-1) \times 2 = -2 = (-3) + 1 \)
\( (-1) \times 1 = -1 = (-2) + 1 \)
\( (-1) \times 0 = 0 = (-1) + 1 \)
\( (-1) \times (-1) = 1 = 0 + 1 \)
In simple words: By observing the pattern, as the number multiplied by -1 decreases by one, the answer increases by one. This leads us to conclude that -1 times -1 equals 1.

Exam Tip: When asked to demonstrate a property, showing a clear, consistent pattern with a sequence of calculations is an effective method. Ensure each step follows the pattern logically.

Question 5. Find the product, using suitable properties:
(a) \( 26 \times (-48) + (-48) \times (-36) \)
(b) \( 8 \times 53 \times (-125) \)
(c) \( 15 \times (-25) \times (-4) \times (-10) \)
(d) \( (-41) \times 102 \)
(e) \( 625 \times (-35) + (-625) \times 65 \)
(f) \( 7 \times (50 - 2) \)
(g) \( (-17) \times (-29) \)
(h) \( (-57) \times (-19) + 57 \)
Answer:
(a) We use the distributive property here. Since \( (-48) \) is common, we can take it out. Then, we add 26 and -36 inside the bracket, which equals -10. Finally, multiplying -48 by -10 gives a positive product of 480.
\( 26 \times (-48) + (-48) \times (-36) \)
[Using distributivity of multiplication over addition]
\( = (-48)[26 + (-36)] \)
\( = (-48)[-10] \)
\( = (-48) \times (-10) = 480 \)
(b) Using the associative property, we can rearrange the multiplication to make it simpler. First, multiply 8 by -125 to get -1000. Then, multiply -1000 by 53, which results in -53000.
\( 8 \times 53 \times (-125) \)
\( = [8 \times (-125)] \times 53 \) [Using associative property for multiplication]
\( = [-1000] \times 53 \)
\( = -[1000 \times 53] \)
\( = -53000 \)
(c) We apply the associative property of multiplication to group numbers smartly. Multiplying -25 by -4 gives 100. Multiplying -10 by 15 gives -150. Finally, multiplying 100 by -150 gives the product -15000.
Using associative property for multiplication, we have
\( 15 \times (-25) \times (-4) \times (-10) \)
\( = [(-25) \times (-4)] \times [15 \times (-10)] \)
\( = [100] \times [-150] \)
\( = -[100 \times 150] = -15000 \)
(d) We can use the distributive property by breaking 102 into \( (100 + 2) \). Then, multiply -41 by 100 and -41 by 2 separately. Adding these two products (-4100 and -82) gives the final result of -4182.
\( (-41) \times 102 \)
\( = (-41) \times [100 + 2] \)
\( = (-41) \times 100 + (-41) \times 2 \) [\( \implies a(b + c) = a \times b + a \times c \)]
\( = -4100 + (-82) = -4182 \)
(e) We use the distributive property here. First, rewrite \( (-625) \times 65 \) as \( 625 \times (-65) \) to make 625 a common factor. Then, add -35 and -65 inside the bracket, which gives -100. Finally, multiply 625 by -100 to get -62500.
\( 625 \times (-35) + (-625) \times 65 \)
\( = 625 \times (-35) + 625 \times (-65) \)
\( = 625 [(-35) + (-65)] \)
\( = 625 \times [-100] \)
\( = -[625 \times 100] = -62500 \)
(f) Using the distributive property, we multiply 7 by 50 and then 7 by 2. Subtracting the second product from the first one \( (350 - 14) \) gives us the final result of 336.
\( 7 \times (50 - 2) \)
\( = 7 \times 50 - 7 \times 2 \)
\( = 350 - 14 = 336 \)
(g) Multiplying two negative numbers gives a positive result. We can rewrite 29 as \( (30 - 1) \) and use the distributive property. This means multiplying 17 by 30 and then subtracting 17 multiplied by 1. The calculation simplifies to 510 minus 17, which equals 493.
\( (-17) \times (-29) \)
\( = + [17 \times 29] \)
\( = 17 \times (30 - 1) \) [\( \implies a(b - c) = a \times b - a \times c \)]
\( = 17 \times 30 - 17 \times 1 \)
\( = 510 - 17 = 493 \)
(h) To simplify this expression, we first rewrite 57 as \( (-1) \times (-57) \). This lets us take out (-57) as a common factor. Then, we add -19 and -1 inside the bracket, which equals -20. Finally, multiplying -57 by -20 gives a positive product of 1140.
\( (-57) \times (-19) + 57 \)
\( = (-57) \times (-19) + [(-1) \times (-57)] \) [\( \implies 57 = (-1) \times (-57) \)]
\( = (-57) \times [(-19) + (-1)] \)
\( = (-57) \times [-20] \)
\( = + [57 \times 20] = 1140 \)
In simple words: Look for opportunities to use mathematical rules like the distributive property or the associative property. These properties allow you to rearrange or break down calculations to make them easier to solve, especially with positive and negative numbers.

Exam Tip: Identifying the appropriate property (distributive, associative, commutative) is key to solving these problems efficiently. Always check your work for sign errors, especially with multiple negative integers.

Question 6. A certain freezing process requires that room temperature be lowered from \( 40^\circ C \) at the rate of \( 5^\circ C \) every hour. What will be the room temperature 10 hours after the process begins?
Answer: The starting room temperature is \( 40^\circ C \). Each hour, the temperature drops by \( 5^\circ C \). Over 10 hours, the total temperature change will be a decrease of \( 50^\circ C \). To find the final temperature, we add this change to the initial temperature: \( 40^\circ C \) plus \( -50^\circ C \) equals \( -10^\circ C \). So, after 10 hours, the room temperature will be \( -10^\circ C \).
Room temperature \( = 40^\circ C \)
Change in temperature per hour \( = -5^\circ C \)
Change in temperature in 10 hours \( = 10 \times (-5^\circ C) = -50^\circ C \)
Room temperature after 10 hours \( = 40^\circ C + (-50^\circ C) = -10^\circ C \).
In simple words: The temperature starts at 40°C and goes down by 5°C each hour. After 10 hours, it will have dropped by 50°C. So, the new temperature will be 40°C minus 50°C, which is -10°C.

Exam Tip: Clearly define positive and negative changes. A "lowering" or "decrease" should be represented by a negative integer, and an "increase" or "rise" by a positive integer.

Question 7. In a class test containing 10 question, 5 marks are awarded for every correct answer and (- 2) marks are awarded for every incorrect answer and 0 for questions not attempted.
(i) Mohan gets four correct and six incorrect answers. What is his score?
(ii) Reshma gets five correct answers and five incorrect answers. What is her score?
(iii) Heena gets two correct and five incorrect answers out of seven questions she attempts. What is her score?
Answer:
Total number of questions \( = 10 \)
Marks for a correct answer \( = 5 \)
Marks for an incorrect answer \( = (-2) \)
Marks for unanswered question \( = 0 \)
(i) Mohan answered 4 questions correctly, earning \( 4 \times 5 = 20 \) marks. He answered 6 questions incorrectly, leading to \( 6 \times (-2) = -12 \) marks. Adding these, Mohan's total score is \( 20 + (-12) = 8 \).
(ii) Reshma gave 5 correct answers, getting \( 5 \times 5 = 25 \) marks. She also gave 5 incorrect answers, resulting in \( 5 \times (-2) = -10 \) marks. Her total score is \( 25 + (-10) = 15 \).
(iii) Heena answered 2 questions correctly, earning \( 2 \times 5 = 10 \) marks. She also answered 5 questions incorrectly, resulting in \( 5 \times (-2) = -10 \) marks. Since 3 questions were not attempted, they contributed \( 3 \times 0 = 0 \) marks. Adding all these up, Heena's total score is \( 10 + (-10) + 0 = 0 \).
In simple words: To calculate each person's score, multiply the number of correct answers by 5, and the number of wrong answers by -2. For Heena, also add 0 for unattempted questions, then sum up all the scores.

Exam Tip: Always account for all types of answers (correct, incorrect, not attempted) and their respective marks. Pay close attention to negative marks for incorrect answers.

Question 8. A cement company earns a profit of Rs 8 per bag of white cement sold and a loss of Rs 5 per bag of grey cement sold.
(a) The company sells 3,000 bags of white cement and 5,000 bags of grey cement in a month. What is its profit or loss?
(b) What is the number of white cement bags it must sell to have neither profit nor loss, if the number of grey bags sold is 6,400 bags?
Answer:
(a) For part (a), we first calculate the total profit from white cement by multiplying 3,000 bags by Rs 8 profit per bag, which is Rs 24,000. Next, we calculate the total loss from grey cement by multiplying 5,000 bags by Rs 5 loss per bag, which is Rs 25,000. Since the total loss (Rs 25,000) is greater than the total profit (Rs 24,000), the company has a net loss. The net loss is Rs \( (25,000 - 24,000) = \text{Rs } 1000 \).
Profit of Rs 8 is earned on a bag of white cement and a loss of Rs 5 on a bag of grey cement.
Number of white cement bags sold \( = 3,000 \)
Number of grey cement bags sold \( = 5,000 \)
\( \therefore \) Profit \( = 3000 \times \text{Rs } 8 = \text{Rs } 24,000 \)
Loss \( = 5000 \times \text{Rs } 5 = \text{Rs } 25,000 \)
Here Loss \( > \) Profit
\( \therefore \) Loss \( = \text{Rs } (25,000 - 24,000) = \text{Rs } 1000 \)
(b) For part (b), if 6,400 bags of grey cement are sold, the total loss will be \( 5 \times 6400 = \text{Rs } 32,000 \). To achieve neither profit nor loss, the company needs to make an equal profit from white cement sales, meaning a profit of Rs 32,000. Since each white cement bag gives a profit of Rs 8, the company must sell \( 32,000 \div 8 = 4,000 \) white cement bags.
Number of grey cement bags sold \( = 6400 \)
\( \therefore \) Total loss \( = 5 \times 6400 = 32000 \)
For no profit and no loss, there should be a profit of Rs 32000.
\( \therefore \) Number of white cement bags sold to earn a profit of Rs \( 32000 = 32000 \div 8 = 4000 \) bags
In simple words: Calculate total profit from white cement and total loss from grey cement, then subtract to find the net profit or loss. For no overall profit or loss, the total profit must exactly match the total loss.

Exam Tip: Clearly distinguish between profit (positive value) and loss (negative value) in your calculations. For break-even scenarios, ensure total profit equals total loss.

Question 9. Replace the blank with an integer to make it a true statement.
(a) \( (-3) \times \_ = 27 \)
(b) \( 5 \times \_ = -35 \)
(c) \( \_ \times (-8) = -56 \)
(d) \( \_ \times (-12) = 132 \)
Answer:
(a) To make the statement true, we need a number that, when multiplied by -3, gives 27. Since a negative number multiplied by a negative number gives a positive result, and \( 3 \times 9 = 27 \), the blank should be filled with -9.
\( (-3) \times (-9) = 27 \) [\( \implies 3 \times 9 = 27 \)]
(b) For the product to be -35 when multiplied by 5, the missing number must be negative. Since \( 5 \times 7 = 35 \), the blank should be filled with -7.
\( 5 \times (-7) = -35 \) [\( \implies 5 \times 7 = 35 \)]
(c) To get -56 when multiplied by -8, the unknown number must be positive. We know \( 7 \times 8 = 56 \), so the blank should be filled with 7.
\( 7 \times (-8) = -56 \) [\( \implies 7 \times 8 = 56 \)]
(d) For the product to be positive 132 when multiplied by -12, the missing number must also be negative. Since \( 11 \times 12 = 132 \), the blank should be filled with -11.
\( (-11) \times (-12) = 132 \) [\( \implies 11 \times 12 = 132 \)]
In simple words: To find the missing number, use the rules of multiplication with positive and negative integers. If the answer is positive and one number is negative, the other must be negative. If the answer is negative and one number is positive, the other must be negative.

Exam Tip: Always remember that the product of two numbers with the same sign is positive, and the product of two numbers with different signs is negative. This rule helps determine the sign of the missing integer.