Mixed Operations with integers worksheet - Free Printable

Problem: Mixed Operations with Directed Numbers (Integers)

We are tasked with solving a series of arithmetic expressions involving integers. The operations include addition, subtraction, multiplication, division, and exponentiation. We must follow the order of operations (PEMDAS/BODMAS):

1. Parentheses/Brackets
2. Exponents/Orders
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

Let's solve each problem step by step.

---

Problem 1:

$$
144 \div (-9) - 4 + 20
$$

#### Step-by-Step Solution:
1. Perform the division first:
$$
144 \div (-9) = -16
$$
2. Substitute back into the expression:
$$
-16 - 4 + 20
$$
3. Perform the subtraction and addition from left to right:
$$
-16 - 4 = -20
$$
$$
-20 + 20 = 0
$$

#### Final Answer:
$$
\boxed{0}
$$

---

Problem 2:

$$
-11 \cdot (-23) + (-6) - (+2) - 5 + 1
$$

#### Step-by-Step Solution:
1. Perform the multiplication first:
$$
-11 \cdot (-23) = 253
$$
2. Substitute back into the expression:
$$
253 + (-6) - (+2) - 5 + 1
$$
3. Simplify the signs:
$$
253 - 6 - 2 - 5 + 1
$$
4. Perform the addition and subtraction from left to right:
$$
253 - 6 = 247
$$
$$
247 - 2 = 245
$$
$$
245 - 5 = 240
$$
$$
240 + 1 = 241
$$

#### Final Answer:
$$
\boxed{241}
$$

---

Problem 3:

$$
4 \times 5 \div (-10) + 1
$$

#### Step-by-Step Solution:
1. Perform the multiplication:
$$
4 \times 5 = 20
$$
2. Perform the division:
$$
20 \div (-10) = -2
$$
3. Substitute back into the expression:
$$
-2 + 1
$$
4. Perform the addition:
$$
-2 + 1 = -1
$$

#### Final Answer:
$$
\boxed{-1}
$$

---

Problem 4:

$$
15 - 37 + 2 \times (-4 + 10) - 1
$$

#### Step-by-Step Solution:
1. Solve the expression inside the parentheses:
$$
-4 + 10 = 6
$$
2. Substitute back into the expression:
$$
15 - 37 + 2 \times 6 - 1
$$
3. Perform the multiplication:
$$
2 \times 6 = 12
$$
4. Substitute back into the expression:
$$
15 - 37 + 12 - 1
$$
5. Perform the addition and subtraction from left to right:
$$
15 - 37 = -22
$$
$$
-22 + 12 = -10
$$
$$
-10 - 1 = -11
$$

#### Final Answer:
$$
\boxed{-11}
$$

---

Problem 5:

$$
-18 \times 6 + (-7) + (-5)^3
$$

#### Step-by-Step Solution:
1. Perform the multiplication:
$$
-18 \times 6 = -108
$$
2. Solve the exponentiation:
$$
(-5)^3 = -125
$$
3. Substitute back into the expression:
$$
-108 + (-7) + (-125)
$$
4. Simplify the signs:
$$
-108 - 7 - 125
$$
5. Perform the addition and subtraction from left to right:
$$
-108 - 7 = -115
$$
$$
-115 - 125 = -240
$$

#### Final Answer:
$$
\boxed{-240}
$$

---

Problem 6:

$$
6 - (-13 + 20) - 3 \times (-5)
$$

#### Step-by-Step Solution:
1. Solve the expression inside the parentheses:
$$
-13 + 20 = 7
$$
2. Substitute back into the expression:
$$
6 - 7 - 3 \times (-5)
$$
3. Perform the multiplication:
$$
3 \times (-5) = -15
$$
4. Substitute back into the expression:
$$
6 - 7 - (-15)
$$
5. Simplify the signs:
$$
6 - 7 + 15
$$
6. Perform the addition and subtraction from left to right:
$$
6 - 7 = -1
$$
$$
-1 + 15 = 14
$$

#### Final Answer:
$$
\boxed{14}
$$

---

Problem 7:

$$
-78 \div \left[ -98 - (-132) - 21 \right] + 14
$$

#### Step-by-Step Solution:
1. Solve the expression inside the brackets:
$$
-98 - (-132) - 21 = -98 + 132 - 21
$$
$$
-98 + 132 = 34
$$
$$
34 - 21 = 13
$$
2. Substitute back into the expression:
$$
-78 \div 13 + 14
$$
3. Perform the division:
$$
-78 \div 13 = -6
$$
4. Substitute back into the expression:
$$
-6 + 14
$$
5. Perform the addition:
$$
-6 + 14 = 8
$$

#### Final Answer:
$$
\boxed{8}
$$

---

Problem 8:

$$
\left[ 12 - (1 - 7) + 1 \right] \times \left[ 42 - 53 - (-6) \right]
$$

#### Step-by-Step Solution:
1. Solve the expression inside the first set of brackets:
$$
12 - (1 - 7) + 1 = 12 - (-6) + 1
$$
$$
12 - (-6) = 12 + 6 = 18
$$
$$
18 + 1 = 19
$$
2. Solve the expression inside the second set of brackets:
$$
42 - 53 - (-6) = 42 - 53 + 6
$$
$$
42 - 53 = -11
$$
$$
-11 + 6 = -5
$$
3. Substitute back into the expression:
$$
19 \times (-5)
$$
4. Perform the multiplication:
$$
19 \times (-5) = -95
$$

#### Final Answer:
$$
\boxed{-95}
$$

---

Problem 9:

$$
\left[ 7 - (1 - 32) + 18 \right] \div \left( 1 - 16 + 13 \right) \times \left[ -48 - (-53) + 2 \right]
$$

#### Step-by-Step Solution:
1. Solve the expression inside the first set of brackets:
$$
7 - (1 - 32) + 18 = 7 - (-31) + 18
$$
$$
7 - (-31) = 7 + 31 = 38
$$
$$
38 + 18 = 56
$$
2. Solve the expression inside the second set of parentheses:
$$
1 - 16 + 13 = -15 + 13 = -2
$$
3. Solve the expression inside the third set of brackets:
$$
-48 - (-53) + 2 = -48 + 53 + 2
$$
$$
-48 + 53 = 5
$$
$$
5 + 2 = 7
$$
4. Substitute back into the expression:
$$
56 \div (-2) \times 7
$$
5. Perform the division:
$$
56 \div (-2) = -28
$$
6. Perform the multiplication:
$$
-28 \times 7 = -196
$$

#### Final Answer:
$$
\boxed{-196}
$$

---

Final Answers:

1. $\boxed{0}$
2. $\boxed{241}$
3. $\boxed{-1}$
4. $\boxed{-11}$
5. $\boxed{-240}$
6. $\boxed{14}$
7. $\boxed{8}$
8. $\boxed{-95}$
9. $\boxed{-196}$