Basic Rules for Mixed Operations

When solving problems with both addition and subtraction:

1. Work from left to right – Operations should be performed in the order they appear
2. No operator precedence – Unlike multiplication/division, addition and subtraction have equal priority
3. Maintain accuracy – Carefully perform each operation step-by-step
4. Check your work – Verify results by reversing the operations

Examples for Simple Mixed Operations

Problem 1: 32 + 15 – 8

Solution:

1. First step is to do the addition: 32 + 15 = 47
2. The second step is to do the subtraction: 47 – 8 = 39

Answer = 39

Verification: 39 + 8 = 47 ; 47 – 15 = 32 (✓ Correct number)

Problem 2: 54 – 17 + 23

Solution:

1. First step is to do the subtraction: 54 – 17 = 37
2. The second step is to do the addition: 37 + 23 = 60

Answer = 60

Verification: 60 – 23 = 37 ; 37 + 17 = 54 (✓ Correct number)

Examples for Multi-Step Mixed Operations

Problem 1: 125 + 236 – 78 + 45

Step-by-Step solution:

1. First, you have to do the addition of 125 + 236:
   a. I will group digits as follows:
        * For hundreds (100s): 100 + 200 = 300
        * For tens (10s): 20 + 30 = 50
        * For ones (1s): 5 + 6 = 11
   b. Now you can add the grouped digits together:
         * 300 + 50 = 350
         * 350 + 11 = 361
2. How to verify: 361 – 236 = 125 ✓

1. First Subtraction (361 – 78):
   • 361 – 70 = 291
   • 291 – 8 = 283
   • Verification: 283 + 78 = 361 ✓
2. Final Addition (283 + 45):
   • 283 + 40 = 323
   • 323 + 5 = 328
   • Verification: 328 – 45 = 283 ✓

Final Answer: 328

Alternative Approach (Grouping):
(125 + 236 + 45) – 78
= 406 – 78
= 328

Problem 2: 87 – 29 + 56 – 14

The solution can be carried out in a number of ways, but the most efficient way to get to the answer is to follow the following steps:

Step 1: The first subtraction (87 – 29) will be completed as follows;

a. In order to make the math easier, you could round 29 up to 30 and then subtract.

b. So, 87 – 30 = 57, then you would need to add 1 back into the solution (because you rounded up), which would give you 58;

c. Finally, you can check your subtraction: 58 + 29 = 87.

Solution to Step 1:

58

Step 2: The first addition (58 + 56), you can add:

a. 50 + 50 = 100

b. 8 + 6 = 14

c. Combine your two answers from (a) and (b) = 114

d. You can check your addition: 114 – 56 = 58

Solution to Step 2:

114

Step 3: You would now perform a subtraction to complete the problem:

a. 114 – 14 = 100

b. You can check your subtraction for this step: 100 + 14 = 114

Final Answer = 100

An alternate method of working is to do the Addition first and then do the Subtraction: (87 + 56) – (29 + 14) = 143 – 43 = 100

Examples for Word Problems with Mixed Operations

Example 1

Emily had 78 apples. She gave her friend 23 apples, then bought 35 more. How many apples does Emily have now?

Step 1: In this example, we have an initial quantity of apples:

Starting Amount = 78

Step 2: The first operation is a Subtraction operation of giving away apples:

a. This operation can be done.

i. By taking 78 – 23, in 3 steps;

(1) 78 – 20 = 58

(2) 58 – 3 = 55

ii. To verify your answer, you can check that 55 + 23 = 78

(The answer to this step is 55).

Step 3: The second operation is an Addition operation for buying more apples:

a. In this operation, you would take the answer from Step 2, which is (55) and add (35), using a similar method of working as the previous operation.

i. 55 + 35, in 3 steps,

(1) 50 + 30 = 80

(2) 5 + 5 = 10

(3) 80 + 10 = 90

ii. Again, to verify your answer, you can check that 90 – 35 = 55

(So the answer to this step is 90)

Another way of solving this is to use the following:

(78 + 35) – 23 = 113 – 23 = 90

Example 2

A bookstore had a sale of 142 books on Monday and 98 books on Tuesday. On Wednesday, the store also received a shipment of 125 books; however, it only sold a total of 80 books from that shipment. How many books does the bookstore have left now?

Solution Steps:

1. The first operation will be to find out how many total books were sold by adding together the books sold on Monday and Tuesday
** The calculation will be as follows
– (142 + 98) = 240
Step 1: 142 + 100 = 242

Step 2: 242 – 2 = 240 (Compensation method)
To verify, 240 – 98 = 142 ✓

2. The second operation will be to find out how many books were received in the shipment of 125 by adding that number to the total number of books sold (240)
** The calculation will be as follows
– (240 + 125) = 365
Step 1: 200 + 100 = 300

Step 2: 40 + 25 = 65

Step 3: 300 + 65 = 365
To verify, 365 – 125 = 240 ✓

3. The last operation will be to find out how many books were sold (80) by subtracting this number from the total number of books that the bookstore currently has
** The calculation will be as follows
– (365 – 80) = 285
Step 1: 365 – 60 = 305

Step 2: 305 – 20 = 285
To verify, 285 + 80 = 365 ✓

Another way to solve the problem is as follows: (142 + 98) – 80 + 125 = (240 – 80) + 125 = 160 + 125 = 285

Give yourself a challenge every day by solving mixed problems containing different types of operations that get progressively more difficult over time. Like any skill, consistent practice will improve your confidence and mastery. Use a math journal to document your progress and begin recognizing your potential to handle these types of problems.