CBSE Class 7 Mathematics Integers Assignment Set 05

VII - Mathematics Assignments No. - 04 - Integers

Question 1. Which of the following statements are true?
(i) The sum of two integers is always an integer
(ii) The difference of two integers is always an integer
(iii) In integers, commutative property for subtraction is always true
(iv) \(-14 > -8 - (-7)\)
(v) \(|18 - (-7)| = |18| - |-7|\)
(vi) Sum of four integers can never be zero
Answer: (i) True: The set of integers is closed under addition.
(ii) True: The set of integers is closed under subtraction.
(iii) False: Subtraction is not commutative for integers (e.g., \(5 - 3 \neq 3 - 5\)).
(iv) False: RHS = \(-8 + 7 = -1\). Comparing \(-14\) and \(-1\), \(-14 < -1\).
(v) False: LHS = \(|18 + 7| = |25| = 25\). RHS = \(18 - 7 = 11\). Since \(25 \neq 11\), the statement is false.
(vi) False: The sum of four integers can be zero (e.g., \(2 + 3 + (-2) + (-3) = 0\)).
In simple words: Statements (i) and (ii) are true due to closure property, while the rest are false based on counterexamples.

Exam Tip: Always test properties with both positive and negative integers before concluding true or false.

Question 2. Subtract \(-8\) from \(11\). Also subtract \(11\) from \((-8)\). Are the two results equal?
Answer: First part: \(11 - (-8) = 11 + 8 = 19\)
Second part: \(-8 - 11 = -19\)
Since \(19 \neq -19\), the two results are not equal.
In simple words: Subtraction order matters for integers, so the results are opposite in sign.

Exam Tip: "Subtract A from B" means \(B - A\), not \(A - B\).

Question 3. Subtract the sum of \(-99\) and \(104\) from \(576\).
Answer: Sum of \(-99\) and \(104\) = \(-99 + 104 = 5\)
Subtracting this sum from \(576\):
\(576 - 5 = 571\)
In simple words: First find the sum inside the bracket, then subtract it from the given number.

Exam Tip: Always compute brackets or combined sums first before performing final subtraction.

Question 4. Subtract the sum of \((-123)\) and \((-154)\) from the sum of \(1023\) and \((-96)\).
Answer: First sum: \((-123) + (-154) = -277\)
Second sum: \(1023 + (-96) = 1023 - 96 = 927\)
Now, subtract the first sum from the second sum:
\(927 - (-277) = 927 + 277 = 1204\)
In simple words: Find both sums separately and then subtract the first total from the second total.

Exam Tip: Be careful with double negative signs: subtracting a negative number becomes addition (\(-(-a) = +a\)).

Question 5. Find the value of \(1 + 2 - 3 + 4 - 5 + 6 - 7 + 8 - 9 + \dots + 16 - 17 + 18 - 19\)
Answer: Group the terms starting from \(2\):
\(1 + (2 - 3) + (4 - 5) + (6 - 7) + (8 - 9) + \dots + (16 - 17) + (18 - 19)\)
Each pair inside the brackets evaluates to \(-1\):
There are 9 pairs from 2 to 19: \((2,3), (4,5), (6,7), (8,9), (10,11), (12,13), (14,15), (16,17), (18,19)\).
Total sum = \(1 + 9 \times (-1)\)
\(= 1 - 9 = -8\)
In simple words: Group consecutive numbers into pairs that result in -1 and add them to 1.

Exam Tip: Look for repeating patterns in long series to simplify calculation.

Question 6. Find the value of:
(i) \(410 + (-56) + (-96) + 17 + (-50)\)
(ii) \(-81 + (-24) + 75 + (-25) + 230\)
Answer:
(i) \(410 - 56 - 96 + 17 - 50\)
Sum of positive numbers = \(410 + 17 = 427\)
Sum of negative numbers = \(-56 - 96 - 50 = -202\)
Total = \(427 - 202 = 225\)

(ii) \(-81 - 24 + 75 - 25 + 230\)
Sum of positive numbers = \(75 + 230 = 305\)
Sum of negative numbers = \(-81 - 24 - 25 = -130\)
Total = \(305 - 130 = 175\) (Note: printing typo in original answer key states 178, correct arithmetic yields 175)
In simple words: Add all positive integers together and all negative integers together, then find their difference.

Exam Tip: Grouping all positive terms together and all negative terms together minimizes calculation errors.

Question 7. Find the value of the following products.
(i) \((-3) \times (-12) \times (-20) \times (15)\)
(ii) \((-15) \times (-20) \times (17) \times (5)\)
Answer:
(i) \((-3) \times (-12) = 36\)
\(36 \times (-20) = -720\)
\(-720 \times 15 = -10800\)

(ii) \((-15) \times (-20) = 300\)
\(300 \times 17 = 5100\)
\(5100 \times 5 = 25500\)
In simple words: Multiply step by step and count the total number of negative signs to determine the sign of the product.

Exam Tip: An odd number of negative factors gives a negative product; an even number of negative factors gives a positive product.

Question 8. If \(x \times (-1) = -50\), is the integer \(x\) positive or negative?
Answer:
\(x \times (-1) = -50\)
\(x = \frac{-50}{-1} = 50\)
Since 50 is greater than 0, the integer \(x\) is positive.
In simple words: Dividing a negative product by a negative integer results in a positive integer.

Exam Tip: The product of two integers with the same sign is always positive.

Question 9. State which is greater:
(i) \((7 + 8) \times 11\) and \(8 + 7 \times 11\)
(ii) \((9 - 8) \times 11\) and \(9 - 8 \times 11\)
Answer:
(i) First expression: \((7 + 8) \times 11 = 15 \times 11 = 165\)
Second expression: \(8 + 7 \times 11 = 8 + 77 = 85\)
Since \(165 > 85\), \((7 + 8) \times 11\) is greater.

(ii) First expression: \((9 - 8) \times 11 = 1 \times 11 = 11\)
Second expression: \(9 - 8 \times 11 = 9 - 88 = -79\)
Since \(11 > -79\), \((9 - 8) \times 11\) is greater.
In simple words: Evaluate each expression using the correct order of operations (BODMAS) and compare the final values.

Exam Tip: Always perform multiplication before addition or subtraction unless parentheses indicate otherwise.

Question 10. Write True or False for the following statements:
(i) Product of a positive and negative integer is positive
(ii) The product of three negative integers is a negative integer
(iii) For all non-zero integers \(a\) and \(b\), \(a \times b\) is always greater than either \(a\) or \(b\)
(iv) Every integer has both a predecessor and a successor
(v) The product of a negative and a positive integer may be zero
Answer:
(i) False (The product of a positive and a negative integer is always negative)
(ii) True (Three negative signs result in a negative product)
(iii) False (For example, if \(a = -2\) and \(b = 3\), \(a \times b = -6\), which is less than 3)
(iv) True (Every integer \(n\) has a predecessor \(n - 1\) and a successor \(n + 1\))
(v) False (The product of a non-zero negative and positive integer is always non-zero)
In simple words: Apply sign rules for multiplication and basic properties of integers to determine truth value.

Exam Tip: A single counterexample is enough to prove a universal statement false.