Math: Basic Tutorials : Arithmetic Operations

Arithmetic Operations

A strong foundation in basic arithmetic—addition, subtraction, multiplication, and division—is essential for accurate medication dosing, IV calculations, and other critical nursing tasks. This tutorial will review these core operations, as well as keywords that imply each operation.

Understanding Arithmetic Operations

Keywords for each mathematical operation help identify which type of calculation is needed to solve a problem.

Operation	Keywords
Addition	sum, sum of, increased by, more than, combined, together, total of, plus, added to, comparatives (e.g., greater than)
Subtraction	difference, difference between/of, decreased by, minus, less, less than, fewer than, left, left over, smaller than
Multiplication	product, product of, times, multiplied by, factor of, twice, triple
Division	quotient, divide, groups of, split, average, ratio of, per

Note: addition and subtraction are inverse operations of each other meaning you can find an equivalent question using the reverse operation. Just as Multiplication and Division are inverses of each other.

$12 + 20 = 32$ and $32 - 20 = 12$

12 \times 8 = 96

and

96 \div 8 = 12

Click on the titles below to view each example.

Example 1

Example 1:
Find the sum of $321$ , $22$ , and $1345$
Solution:	Addition is implied, so	$321 + 22 + 1345 = 1688$

Example 2

Example 2:
How many groups of eight parts can be found in three hundred thirty-six parts?
Solution:	"Groups of" implies division, so	$336 \div 8 = 42$ groups

Example 3

Example 3:
If there are 96 aircraft maintenance technicians working at each terminal and there are 12 terminals, how many technicians are there?
Solution:	There are 96 technicians in each terminal, and we want to multiply by a factor of 12.	$96 \times 12 = 11522$ technicians

Mastering Addition & Subtraction

Being an expert at addition and subtraction means having a flexible approach to problem-solving. By mastering multiple strategies, you can confidently tackle problems even when a preferred method slips your mind or when tools like calculators aren’t available. Different techniques, such as breaking numbers apart, using number lines, or applying mental math shortcuts, allow for greater accuracy and efficiency. Understanding these methods also strengthens number sense, making complex calculations easier to manage. This section will explore various ways to approach addition and subtraction, helping you build confidence and adaptability in solving mathematical problems.

Moving Beyond the Standard Algorithm

Moving beyond the standard algorithm (see figure 1) opens the door to a deeper understanding of numbers and greater flexibility in problem-solving. While traditional column addition and subtraction are reliable methods, alternative strategies such as breaking numbers apart (decomposition), using number lines, and applying mental math techniques can make calculations more intuitive and efficient. These approaches help develop number sense, allowing for quicker estimations and creative solutions, especially when working with large numbers or solving real-world problems. By exploring different methods, learners gain a well-rounded mathematical foundation that extends beyond rote memorization and fosters true numerical fluency.

Figure 1

Addition & Subtraction Strategies

Click on the titles below to view each method.

Breaking Down Numbers by Place Value

Breaking numbers into parts can make addition and subtraction easier.

Example 1:
$3467 + 2859$
Break it into place values:	$= (3000 + 2000) + (400 + 800) + (60 + 50) + (7 + 9)$ $= 5000 + 1200 + 110 + 16$ $= (5000 + 1000) + (200 + 100) + (10 + 10) + 6$ $= 6000 + 300 + 20 + 6$
Solution:	$= 6326$
Example 2:
$7532 - 4268$
Break it down:	$= (7000 - 4000) + (500 - 200) + (30 - 60) + (2 - 8)$
We can use negative numbers.	$= 3000 + 300 - 30 - 6$ $= 3000 + 270 - 6$ $= 3000 + 264$
Solution:	$= 3264$
Try It: Solve $6743 + 3185$ and $8301 - 4527$

Number Line Strategy

Using a number line helps visualize additions and subtratctions

Example 1:
$2374 + 1648$

Start with 2375 Add 1000 Add 600 Add 40 Add 8	$2375 + 1000 = 3375$ $3375 + 600 = 3975$ $3975 + 40 = 4015$ $4015 + 8 = 4023$
Solution:	$= 4023$
Example 2:
$5623 - 2487$

Start with 5623 Subtract 2000 Subtract 400 Subtract 80 Subtract 7	$5628 - 2000 = 3623$ $3623 - 400 = 3223$ $3223 - 80 = 3143$ $3143 - 7 = 3136$
Solution:	$= 3136$
Try It: Solve $7235 + 4528$ and $9801 - 5643$

Making Numbers Simpler

This method makes numbers easier to work with by adjusting them slightly.

Example 1:
$4998 + 3276$
Instead of adding directly, round 4998 to 5000 (adding 2) and adjust at the end:	$5000 + 3276 = 8276$
Now subtract the extra 2. (The inverse operation of adding 2 from above):	$8276 - 2 = 8274$
Solution:	$= 8274$
Example 2:
$8012 - 4991$
Instead of subtracting directly, round 4991 up to 5000 (a difference of 9) and adjust at the end. (Note: Why did we adjust the second number, but not the first?)	$8012 - 5000 = 3012$
Now add back the extra 9 (Going from 5000 back to 4991):	$3012 + 9 = 3021$
Solution:	$= 3021$
Try It: Solve $7999 + 3482$ and $9004 - 4998$

Checking Your Work with the Inverse Operation

To check subtraction, use addition. To check addition, use subtraction.

Example 1:
$6315 - 2478 = 3837$
Check by adding:	$3837 + 2478 = 6315$ ✓
Example 2:
$4329 + 5672 = 10.001$
Check by subtracting:	$10,001 - 5672 = 4329$ ✓
Try It: Solve $8675 - 3492$ and check with addition.

Mastering Multiplication

Multiplication is more than just a mechanical process. Understanding different strategies enhances problem-solving skills in algebra, calculus, and real-world applications.

Why Not Just Memorize or Use the Standard Algorithm?

Memorizing times tables is useful, but insufficient for complex calculations and mental flexibility.
The standard multiplication algorithm works but isn’t always the most efficient.
Alternative strategies can simplify large numbers, reduce errors, and improve mental math skills.

Multiplication Strategies

Click on the titles below to view each method.

Repeated Addition & Iterative Thinking

Multiplication can be viewed as an iterative process: foundational loops in programming and recursive functions in computer science.

Example 1:
$4 \times 3$
Multiplication represents repeated addition. So 4 × 3 means adding 3 four times.	$= 3 + 3 + 3 + 3$
Solution:	$= 12$
Example 2:
You invest $1000 in a savings account that earns 5% simple interest per year. How much money will you have after 4 years using repeated addition?
Interest for 1 year:	Interest $= 1000 \times 0.05 = 50$
After 1 year:	$$ 1000 + 50 = $ 1050$
Use repeated addition for 4 years:	Year 2: $$ 1050 + 50 = $ 1100$ Year 3: $$ 1100 + 50 = $ 1150$ Year 4: $$ 1150 + 50 = $ 1200$
Solution:	Final amount after 4 years: $= $ 1200$

Scaling & Proportional Reasoning

Rescaling numbers makes computation easier and is used in economics and physics.

Example 1:
$49 \times 26$
We can round down and rewrite as:	$= (50 \times 26) - (1 \times 26)$ $= 1300 - 26$
Solution:	$= 1274$
Example 2:
$102 \times 17$
We can round up and rewrite as:	$= (100 \times 17) + (2 \times 17)$ $= 1700 + 34$
Solution:	$= 1734$

Doubling & Halving

Halving one factor and doubling another preserves the product but simplifies calculations.

Example 1:
$16 \times 25$
We divide 16 by 2 and multiply 25 by 2 and rewrite as:	$= (16 \div 2) \times (25 \times 2)$ $= 8 \times 50$
Solution:	$= 400$
Example 2:
$48 \times 15$
We divide 48 by 2 and multiply 15 by 2 and rewrite as:	$= (48 \div 2) \times (15 \times 2)$ $= 24 \times 30$
Solution:	$= 720$

Distributive Property & Algebraic Expansion

Breaking numbers into smaller, manageable parts can speed up calculations.

Example 1:
$38 \times 7$
Break the numbers into more manageable parts and rewrite as:	$= (30 \times 7) + (8 \times 7)$ $= 210 + 56$
Solution:	$= 266$
Example 2:
$232 \times 46$
Break the numbers down as much as you need to make the equation manageable. For example this equation can be broken down even further:	$= (200 \times 46) + (30 \times 46) + (2 \times 46)$ $= (200 \times 40) + (200 \times 6) + (30 \times 40 + (30 \times 6) + (2 \times 46)$ $= 8000 + 1200 + 1200 + 180 + 92$
Solution:	$= 10,672$

Mastering Division

Division is a fundamental mathematical operation that helps distribute quantities into equal parts. It is closely related to multiplication, as division is essentially the process of finding out how many times one number is contained within another. This lesson explores different strategies for division and demonstrates its connection to multiplication.

Division as the Inverse of Multiplication

Division and multiplcation are inverse operations. If you know that: $a \times b = c$ then the related division facts are $c \div b = a$ .

Example:

If $6 \times 4 = 24$ then, $24 \div 4 = 6$ and $24 \div 6 = 4$ .

Division Strategies

Click on the titles below to view each method.

Long Division Method

Recognize how many multiples of the divisor go into parts of the dividend.

Steps:

Divide the dividend by the divisor
Multiply the quotient by the divisor
Subtract the product from the dividend
Bring down the next digit (if any) and repeat

Example 1:
$125 \div 5$
5 does not go into 1, so move to next number 5 goes 2 times into 12 (remainder 2) Bring down 5, making it 25 5 goes 5 times into 25 The quotient is 25
Solution:	$= 25$

Example 1:

$125 \div 5$

5 does not go into 1, so move to next number
5 goes 2 times into 12 (remainder 2)
Bring down 5, making it 25
5 goes 5 times into 25
The quotient is 25

Solution:

$= 25$

When multiples are recognized quickly, then this method can be reduced to short division by using mental math.

Partial Quotients Method

You can break the dividend into smaller, more manageable parts.

Example 1:
$196 \div 7$
Recognize that 7 goes into 196 multiple times. A large, easy multiple of 7 is 140 (since 7 × 20 = 140) Subtract 140 from the dividend (196) Find how many times 7 goes into 56 Add up the partial quotients	$196 \div 7$ $140 = 7 \times 20$ $196 - 140 = 56$ $56 = 7 \times 8$ $20 + 8$
Solution:	$= 28$

Example 1:

$196 \div 7$

Recognize that 7 goes into 196 multiple times.
A large, easy multiple of 7 is 140 (since 7 × 20 = 140)
Subtract 140 from the dividend (196)
Find how many times 7 goes into 56
Add up the partial quotients

$196 \div 7$
$140 = 7 \times 20$
$196 - 140 = 56$
$56 = 7 \times 8$
$20 + 8$

Solution:

$= 28$

Estimation & Rounding in Division

For quick approximations, round numbers before dividing.

Example 1:
Solve $198 \div 4$ by estimation.
Round 198 to 200 to get an aproximate answer.	$200 \div 4$
Solution:	$= 50$ (aproximate answer) $= 49.5$ (exact answer)

Example 1:

Solve $198 \div 4$ by estimation.

Round 198 to 200 to get an aproximate answer.

$200 \div 4$

Solution:

$= 50$ (aproximate answer)

$= 49.5$ (exact answer)

Repeated Subtraction Method

Division can be understood as repeated subtraction of the divisor from the dividend.

Example 1:
Solve $20 \div 5$ using repeated subtraction.
Keep subtracting 5 until you reach 0. The number of times you subtract 5 is the quotient.	$20 - 5 = 15$ $15 - 5 = 10$ $10 - 5 = 5$ $5 - 5 = 0$
Solution:	We subtracted 5 four times, therefore the quotient is: $= 4$

Example 1:

Solve $20 \div 5$ using repeated subtraction.

Keep subtracting 5 until you reach 0. The number of times you subtract 5 is the quotient.

$20 - 5 = 15$
$15 - 5 = 10$
$10 - 5 = 5$
$5 - 5 = 0$

Solution:

We subtracted 5 four times, therefore the quotient is:

$= 4$

Arithmetic Operations Practice Quiz Activity PDF

Practice

Last Updated: Jun 5, 2025 4:16 PM
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Math: Basic Tutorials : Arithmetic Operations

Arithmetic Operations

Understanding Arithmetic Operations

Mastering Addition & Subtraction

Addition & Subtraction Strategies

Mastering Multiplication

Multiplication Strategies

Mastering Division

Division as the Inverse of Multiplication

Division Strategies

Practice