Math worksheet featuring problems on division, integer operations, properties of integers, inequalities, and real-world applications.

Let's solve each question step by step.

---

1. Fill in the blanks:

(a) $ 30 - \_\_\_\_\_\_ = 3 $

Let the missing number be $ x $.
So, $ 30 - x = 3 $
$ x = 30 - 3 = 27 $

✔ Answer: $ 27 $

---

(b) $ -89 + \_\_\_\_\_\_ = 89 $

Let the missing number be $ x $.
$ -89 + x = 89 $
$ x = 89 + 89 = 178 $

✔ Answer: $ 178 $

---

(c) $ \_\_\_\_\_\_ + 1 = -205 $

Let the missing number be $ x $.
$ x + 1 = -205 $
$ x = -205 - 1 = -206 $

✔ Answer: $ -206 $

---

(d) $ \_\_\_\_\_\_ \times (-12) = 120 $

Let the missing number be $ x $.
$ x \times (-12) = 120 $
$ x = \frac{120}{-12} = -10 $

✔ Answer: $ -10 $

---

(e) $ 24 + 12 \div 4 - 5 \times 3 = ? $

Apply BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction):

- $ 12 \div 4 = 3 $
- $ 5 \times 3 = 15 $

Now:
$ 24 + 3 - 15 = 27 - 15 = 12 $

✔ Answer: $ 12 $

---

2. Find the value of $ P $, if $ p \times (-9) = 135 $

We have:
$ p \times (-9) = 135 $
$ p = \frac{135}{-9} = -15 $

✔ Answer: $ -15 $

---

3. The additive inverse of zero (0) is ________

Additive inverse of a number $ a $ is the number that, when added to $ a $, gives 0.

So, $ 0 + x = 0 \Rightarrow x = 0 $

✔ Answer: $ 0 $

---

4. Match the properties of Integers:

| Property | Match |
|--------|-------|
| (a) Distributive law over addition | (v) $ a \times (b + c) = a \times b + a \times c $ |
| (b) Associative law for multiplication | (iii) $ (a \times b) \times c = a \times (b \times c) $ |
| (c) Additive Identity | (i) $ a + 0 = a = 0 + a $ |
| (d) Commutative law over addition | (ii) $ a + b = b + a $ |
| (e) Multiplicative Identity | (iv) $ a \times 1 = 1 \times a = a $ |

✔ Matching:
- (a) → (v)
- (b) → (iii)
- (c) → (i)
- (d) → (ii)
- (e) → (iv)

---

5. Use the sign of >, < or = in the box to make the statement true

(a) $ 25 - 40 + 10 \_\_\_\_\_\_ 25 - 40 - 10 $

Left: $ 25 - 40 + 10 = -15 + 10 = -5 $
Right: $ 25 - 40 - 10 = -15 - 10 = -25 $

Since $ -5 > -25 $, use >

✔ Answer: $ > $

---

(b) $ (-9) + (-6) \_\_\_\_\_\_ (-9) - (-6) $

Left: $ -9 + (-6) = -15 $
Right: $ -9 - (-6) = -9 + 6 = -3 $

Since $ -15 < -3 $, use <

✔ Answer: $ < $

---

(c) $ 35 + (-70) - (-35) \_\_\_\_\_\_ (-24) - (15) + 39 $

Left: $ 35 - 70 + 35 = (35 + 35) - 70 = 70 - 70 = 0 $

Right: $ -24 - 15 + 39 = -39 + 39 = 0 $

So, $ 0 = 0 $

✔ Answer: $ = $

---

6. Evaluate:

(a) $ (-1) \times (-2) \times (-3) \times (-4) \times (-5) $

Count the negative signs: 5 negatives → odd number ⇒ result is negative

Magnitude: $ 1 \times 2 \times 3 \times 4 \times 5 = 120 $

So, answer is $ -120 $

✔ Answer: $ -120 $

---

(b) $ 795 \times (-25) + (-795) \times 75 $

Factor out $ 795 $:

= $ 795 \times (-25) + (-795) \times 75 $
= $ 795 \times (-25) - 795 \times 75 $
= $ 795 \times [(-25) - 75] $
= $ 795 \times (-100) = -79500 $

✔ Answer: $ -79500 $

---

(c) $ (-59) + (-19) + 59 $

Group: $ (-59 + 59) + (-19) = 0 - 19 = -19 $

✔ Answer: $ -19 $

---

7. An elevator descends at 7 m/min. Starts from 5m above ground. How long to reach –20m?

Current position: $ +5 $ m
Target: $ -20 $ m

Total distance to descend: $ 5 - (-20) = 5 + 20 = 25 $ meters

Speed: 7 m/min

Time = $ \frac{25}{7} \approx 3.57 $ minutes

But let's keep it as a fraction: $ \frac{25}{7} $ min

Or convert to minutes and seconds:
$ \frac{25}{7} = 3 \frac{4}{7} $ min ≈ 3 min $ \frac{4}{7} \times 60 \approx 34.28 $ sec

But usually we leave as $ \frac{25}{7} $ minutes or $ 3\frac{4}{7} $ minutes

✔ Answer: $ \boxed{\frac{25}{7}} $ minutes

---

8. With the help of number line, find how much greater is 3 than –3

On number line:
- $ 3 $ is to the right of $ -3 $
- Distance between them: $ 3 - (-3) = 3 + 3 = 6 $

So, 3 is 6 units greater than –3.

✔ Answer: $ 6 $

---

9. The sum of two integers is –1500. One of the numbers is 599. Find the other number.

Let the unknown number be $ x $.
Then:
$ x + 599 = -1500 $
$ x = -1500 - 599 = -2099 $

✔ Answer: $ -2099 $

---

✔ Final Answers Summary:

1.
(a) 27
(b) 178
(c) -206
(d) -10
(e) 12

2. $ P = -15 $

3. Additive inverse of 0 is 0

4. Matching:
(a)-(v), (b)-(iii), (c)-(i), (d)-(ii), (e)-(iv)

5.
(a) >
(b) <
(c) =

6.
(a) -120
(b) -79500
(c) -19

7. $ \frac{25}{7} $ minutes (≈ 3.57 min)

8. 6

9. $ -2099 $

Let me know if you'd like a visual explanation for any part!