Class 7 Integers worksheet with problems on integer expressions, remainders, and addition properties.

Here are the step-by-step solutions for each question on the worksheet.

(1) Find the value of the following expression:

A) $(-126) - (-14) - 76 - (-43) - 97 - (-109)$

* Step 1: Simplify the signs. Remember that subtracting a negative number is the same as adding a positive number ($-- = +$).
* $-126 + 14 - 76 + 43 - 97 + 109$
* Step 2: Group the positive numbers and the negative numbers together to make it easier.
* Positives: $14 + 43 + 109 = 166$
* Negatives: $-126 - 76 - 97 = -299$
* Step 3: Combine the totals.
* $166 - 299 = -133$

B) $34 - 142 - 21 - (-181) - (-88) - (-41)$

* Step 1: Simplify the signs.
* $34 - 142 - 21 + 181 + 88 + 41$
* Step 2: Group positives and negatives.
* Positives: $34 + 181 + 88 + 41 = 344$
* Negatives: $-142 - 21 = -163$
* Step 3: Combine the totals.
* $344 - 163 = 181$

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(2) Remainder Problem

Original Order:
1. Divided by 8, remainder 3. Let the quotient be $Q_1$.
* Number $= 8 \times Q_1 + 3$
2. $Q_1$ divided by 7, remainder 2. Let the quotient be $Q_2$.
* $Q_1 = 7 \times Q_2 + 2$
3. $Q_2$ divided by 7, quotient 1, remainder 6.
* $Q_2 = 7 \times 1 + 6 = 13$

Now, work backward to find the original number:
* $Q_1 = 7(13) + 2 = 91 + 2 = 93$
* Original Number $= 8(93) + 3 = 744 + 3 = 747$

Reversed Order (Divisors: 7, then 7, then 8):
We start with the same number: 747.
1. Divide 747 by 7:
* $747 \div 7 = 106$ with a remainder of 5.
* Quotient is 106.
2. Divide the new quotient (106) by 7:
* $106 \div 7 = 15$ with a remainder of 1.
* Quotient is 15.
3. Divide the new quotient (15) by 8:
* $15 \div 8 = 1$ with a remainder of 7.

The final remainder is 7.

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(3) How many integers are there between...?

To find the number of integers strictly *between* two numbers $a$ and $b$, use the formula: $(b - a) - 1$.

* A) -5 and 6: $(6 - (-5)) - 1 = 11 - 1 = 10$
* B) -1 and 6: $(6 - (-1)) - 1 = 7 - 1 = 6$
* C) -2 and 2: $(2 - (-2)) - 1 = 4 - 1 = 3$ (Integers are -1, 0, 1)
* D) -4 and 7: $(7 - (-4)) - 1 = 11 - 1 = 10$

*(Note: The question asks "How many", but the options look like pairs of numbers. Based on standard math problems of this type, it usually asks to identify the count. However, looking at the layout, it seems to be asking which pair has a specific property or perhaps just listing ranges. Without a specific target number in the question stem, I will assume the question meant to ask for the count for option C as an example, or simply identifying the range. Given the ambiguity, let's look at Option C specifically: Between -2 and 2 are -1, 0, 1. That is 3 integers.)*

*Correction:* Looking closely at typical worksheets, this question often asks "Which of the following ranges contains X integers?" or similar. Since the prompt is cut off or vague ("Find how many integers are there between:" followed by options), I will provide the counts for all options so you can match it to your specific classroom instruction.
* Count for A: 10
* Count for B: 6
* Count for C: 3
* Count for D: 10

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(4) Railway Company Profit/Loss

* Business Class:
* Profit per ticket: Rs. 1671
* Tickets sold: 3865
* Total Profit: $1671 \times 3865 = 6,458,415$
* Economy Class:
* Loss per ticket: Rs. 165
* Tickets sold: 45723
* Total Loss: $165 \times 45723 = 7,544,295$
* Net Result:
* Profit - Loss
* $6,458,415 - 7,544,295 = -1,085,880$

Since the result is negative, it is a loss of Rs. 1,085,880. This matches option c.

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(5) Choose the correct operator

Compare $-5$ and $-5$.
Any number is equal to itself.
Therefore, $-5 = -5$.

Correct option: c. =

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(6) Fill in the blanks

$A + B = B + A$ represents the commutative property of addition.
(This property means you can swap the order of numbers when adding, and the answer stays the same).

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(7) Sum of the series

Series: $6 + (-6) + 6 + (-6) + 6 + (-6) + \dots$
Number of terms: 236

* Notice that every pair of terms $(6 + (-6))$ equals $0$.
* Since there are 236 terms, we can divide them into pairs: $236 / 2 = 118$ pairs.
* Each pair sums to 0.
* $118 \times 0 = 0$.

The sum is 0.

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Final Answer:

(1)
A) -133
B) 181

(2) 7

(3)
*(Counts for each option)*
A) 10 integers
B) 6 integers
C) 3 integers
D) 10 integers

(4) c. -1085880

(5) c. =

(6) commutative

(7) 0