This skill teaches a human to understand and reliably use basic arithmetic.
After studying and practicing this skill, the learner should be able to:
- Understand numbers and place value.
- Add, subtract, multiply, and divide whole numbers and decimals.
- Work with negative numbers.
- Understand and calculate with fractions.
- Use percentages.
- Apply the correct order of operations.
- Estimate answers and recognize implausible results.
- Translate ordinary real-world problems into arithmetic.
- Check arithmetic independently.
The goal is not merely to memorize procedures. The learner should understand what the operations mean and know when to use them.
Numbers represent quantities.
In the decimal number system, the value of a digit depends on its position.
Example:
4,582.37
means:
4 thousands = 4,000
5 hundreds = 500
8 tens = 80
2 ones = 2
3 tenths = 0.3
7 hundredths= 0.07
Therefore:
4,582.37
= 4,000 + 500 + 80 + 2 + 0.3 + 0.07
Understand the comparison symbols:
7 > 3 7 is greater than 3
3 < 7 3 is less than 7
7 = 7 the quantities are equal
Do not confuse a digit with its value. In 500, the digit 5 represents 500, not 5.
Addition combines quantities.
8 + 5 = 13
If you have 8 objects and receive 5 more, you now have 13.
The numbers being added are called addends. The result is the sum.
Align numbers by place value.
347
+ 286
-----
633
Working from right to left:
7 + 6 = 13
write 3, carry 1
4 + 8 + 1 = 13
write 3, carry 1
3 + 2 + 1 = 6
Result:
633
For decimals, align decimal points:
12.50
+ 3.78
-------
16.28
Order does not affect the sum:
4 + 9 = 9 + 4
Grouping does not affect the sum:
(2 + 3) + 7 = 2 + (3 + 7)
Adding zero changes nothing:
18 + 0 = 18
Subtraction can represent taking away or finding a difference.
12 - 5 = 7
If you have 12 objects and remove 5, 7 remain.
It can also answer:
"How much larger is 12 than 5?"
Answer: 7.
Align place values.
532
- 178
-----
354
Regroup when the upper digit is too small.
Addition and subtraction are inverse operations.
If:
12 - 5 = 7
then:
7 + 5 = 12
Use this relationship to check subtraction.
Subtraction is not commutative:
10 - 3 = 7
3 - 10 = -7
Order matters.
Numbers below zero are negative.
A number line helps:
-3 -2 -1 0 1 2 3
Negative numbers occur naturally in situations such as temperatures, debts, elevations, and gains/losses.
Adding a positive number moves upward:
-4 + 7 = 3
Adding a negative number moves downward:
5 + (-8) = -3
Subtracting a number is equivalent to adding its opposite:
5 - 8
= 5 + (-8)
= -3
5 - (-8)
= 5 + 8
= 13
positive × positive = positive
negative × negative = positive
positive × negative = negative
negative × positive = negative
Examples:
4 × 3 = 12
(-4) × (-3) = 12
(-4) × 3 = -12
The same sign rules apply to division.
Multiplication represents equal groups or scaling.
4 × 6 = 24
can mean four groups containing six items each.
It can also mean scaling 6 by a factor of 4.
Learn the multiplication table through at least 10 × 10 fluently.
Examples:
6 × 7 = 42
8 × 9 = 72
10 × 7 = 70
Understanding is more important than memorization, but fluency with common facts makes later arithmetic much easier.
Example:
23 × 14
Break it apart:
23 × 14
= 23 × (10 + 4)
= 230 + 92
= 322
This illustrates the distributive property:
a(b + c) = ab + ac
a × 0 = 0
a × 1 = a
Order does not matter:
4 × 7 = 7 × 4
Division can represent sharing or determining how many groups fit into a quantity.
20 ÷ 5 = 4
This can mean:
"20 items divided equally among 5 groups gives 4 per group."
or:
"How many groups of 5 fit into 20?"
Answer: 4.
Multiplication and division are inverse operations.
If:
5 × 4 = 20
then:
20 ÷ 5 = 4
20 ÷ 4 = 5
Use multiplication to check division.
Not every division produces a whole number.
17 ÷ 5 = 3 remainder 2
because:
5 × 3 + 2 = 17
It can also be written:
17/5 = 3.4
Never divide by zero.
8 ÷ 0
is undefined.
This is not the same as:
0 ÷ 8 = 0
A fraction represents part of a whole or a ratio.
3/4
The top number, 3, is the numerator. The bottom number, 4, is the denominator.
It means three parts when a whole has been divided into four equal parts.
1/2 = 2/4 = 3/6 = 50/100
Multiplying or dividing the numerator and denominator by the same nonzero number does not change the fraction's value.
8/12
Divide numerator and denominator by 4:
8/12 = 2/3
Fractions need a common denominator.
1/4 + 2/4 = 3/4
For:
1/2 + 1/3
use denominator 6:
1/2 = 3/6
1/3 = 2/6
therefore:
3/6 + 2/6 = 5/6
Do NOT add denominators:
1/2 + 1/3 ≠ 2/5
Multiply numerators and denominators:
2/3 × 4/5 = 8/15
To divide by a fraction, multiply by its reciprocal:
2/3 ÷ 4/5
= 2/3 × 5/4
= 10/12
= 5/6
Decimals are another representation of fractional quantities.
0.5 = 5/10 = 1/2
0.25 = 25/100 = 1/4
Place value matters:
0.5 = five tenths
0.05 = five hundredths
Therefore:
0.5 > 0.05
Align decimal points:
14.25
- 3.70
-------
10.55
4.27 × 10 = 42.7
4.27 × 100 = 427
427 ÷ 10 = 42.7
427 ÷ 100 = 4.27
Understand this as a change in place value rather than merely memorizing "move the decimal point."
Percent means "per hundred."
25% = 25/100 = 0.25 = 1/4
To convert a percentage to a decimal:
18% = 0.18
To find a percentage of a quantity:
20% of 80
= 0.20 × 80
= 16
"What percentage of 80 is 20?"
20 / 80 = 0.25 = 25%
A $50 item increases by 20%.
Increase:
0.20 × 50 = 10
New value:
50 + 10 = 60
Equivalently:
50 × 1.20 = 60
A $50 item is discounted by 20%.
Discount:
0.20 × 50 = 10
New price:
50 - 10 = 40
Equivalently:
50 × 0.80 = 40
Percentage changes are relative to a starting quantity.
A 50% increase followed by a 50% decrease does NOT return to the original value.
Example:
100 increased 50% = 150
150 decreased 50% = 75
When an expression contains multiple operations, use this order:
- Parentheses or other grouping
- Exponents
- Multiplication and division, from left to right
- Addition and subtraction, from left to right
Example:
3 + 4 × 5
Multiply first:
3 + 20 = 23
not:
7 × 5 = 35
Example:
(3 + 4) × 5 = 35
Parentheses deliberately change the order.
Multiplication does not universally come "before division."
They have equal priority and are evaluated left to right.
24 ÷ 6 × 2
= 4 × 2
= 8
Likewise, addition and subtraction have equal priority.
Do not treat arithmetic as blind symbol manipulation.
Before or after calculating, estimate the expected answer.
Example:
49 × 21
Estimate:
50 × 20 = 1,000
Exact answer:
49 × 21 = 1,029
The exact answer is close to the estimate, so it is plausible.
If your calculation produced 10,290, the estimate would immediately suggest that a mistake had occurred.
Develop the habit:
Estimate → Calculate → Check plausibility
Arithmetic becomes useful when you can determine which operation represents a real situation.
"You have $25 and receive another $18."
Use addition:
25 + 18 = 43
"You have $60 and spend $17."
Use subtraction:
60 - 17 = 43
"Seven boxes contain 12 items each."
Use multiplication:
7 × 12 = 84
"84 items are shared among 7 people."
Use division:
84 ÷ 7 = 12
"A car travels 180 miles in 3 hours at a constant rate."
180 ÷ 3 = 60 miles per hour
"A $120 item is discounted by 25%, then $8 shipping is added."
Discount:
120 × 0.25 = 30
Discounted price:
120 - 30 = 90
Add shipping:
90 + 8 = 98
Final cost:
$98
Do not simply calculate every number you see. First determine what the numbers represent and what question is being asked.
A competent arithmetic user does not assume the first calculation is correct.
Use one or more of these methods:
For:
83 - 29 = 54
check:
54 + 29 = 83
For:
198 × 51 = 10,098
estimate:
200 × 50 = 10,000
The result is plausible.
For:
17 × 24
Method 1:
17 × 24 = 408
Method 2:
17 × (20 + 4)
= 340 + 68
= 408
Agreement increases confidence.
If 5 notebooks cost $4 each:
5 × $4 = $20
An answer such as "$1.25" should be questioned even before checking the calculation.
Watch specifically for these errors:
- Misaligning place values or decimal points.
- Forgetting a carried or borrowed value.
- Reversing subtraction.
- Confusing multiplication with addition.
- Dividing in the wrong direction.
- Dividing by zero.
- Adding fraction denominators.
- Losing negative signs.
- Applying operations in the wrong order.
- Treating percentage points and percentage changes as identical.
- Using a calculator without checking whether the result is reasonable.
- Performing arithmetic before understanding what a word problem asks.
When an answer is wrong, identify the type of error rather than merely memorizing the corrected answer.
Develop convenient transformations.
Instead of:
99 + 47
think:
100 + 47 - 1 = 146
Instead of:
25 × 16
think:
100/4 × 16
= 400
or:
16 × 100 ÷ 4
= 400
For 15% of 80:
10% of 80 = 8
5% of 80 = 4
therefore:
15% of 80 = 12
Mental arithmetic is not about avoiding written calculation at all costs. It is about recognizing simple numerical relationships.
Calculators are tools, not substitutes for arithmetic understanding.
Use a calculator when:
- Numbers are cumbersome.
- Many calculations must be performed.
- High numerical precision is useful.
- The arithmetic itself is not what you are trying to practice.
Before entering a calculation:
- Determine what operation is required.
- Estimate the expected magnitude.
- Enter the calculation.
- Inspect the result.
- Check whether the result makes sense.
If a calculator says:
4,923 × 19 = 935.37
you should reject the result immediately because:
5,000 × 20 ≈ 100,000
The calculator was almost certainly given the wrong input.
Do not proceed solely because you have read a section. Practice until you can perform the skill accurately without referring to the instructions.
Practice:
38 + 47
903 - 278
26 × 14
144 ÷ 12
Answers:
85
625
364
12
Practice:
-8 + 13
7 - 12
-6 × 4
-35 ÷ -7
Answers:
5
-5
-24
5
Practice:
1/3 + 1/6
7/8 - 3/8
2/5 × 3/4
3/7 ÷ 9/14
Answers:
1/2
1/2
3/10
2/3
Practice:
4.75 + 2.8
12.4 - 7.65
15% of 200
45 is what percent of 180?
Answers:
7.55
4.75
30
25%
Practice:
8 + 3 × 5
(8 + 3) × 5
36 ÷ 6 + 4 × 2
20 - 12 ÷ 3
Answers:
23
55
14
16
Solve these without looking at the answers first.
A restaurant bill is $48. A 20% tip is added.
What is the total?
A package contains 24 bottles. You buy 7 packages.
How many bottles do you have?
You have $250. You spend $89.50 and then receive $40.
How much do you have?
A 300-mile trip takes 5 hours at a constant average speed.
What is the average speed?
A $160 jacket is discounted by 25%.
What is the sale price?
You have 3/4 of a pizza. You eat 1/3 of the amount you currently have.
How much pizza did you eat, and how much remains?
A store has 240 items. It sells 15% of them.
How many items were sold, and how many remain?
Four people split a $126 bill equally.
How much does each person pay?
A bank balance is -$35. You deposit $80 and then spend $27.
What is the new balance?
Evaluate:
18 + 24 ÷ 6 × 3 - 5
A $48 bill with a 20% tip:
20% of 48
= 0.20 × 48
= 9.60
Total:
48 + 9.60 = 57.60
Answer:
$57.60
24 bottles per package and 7 packages:
24 × 7 = 168
Answer:
168 bottles
Start with:
$250
Spend:
$89.50
Remaining:
250 - 89.50 = 160.50
Receive:
$40
New amount:
160.50 + 40 = 200.50
Answer:
$200.50
Average speed is distance divided by time:
300 ÷ 5 = 60
Answer:
60 miles per hour
Find 25% of $160:
0.25 × 160 = 40
Subtract the discount:
160 - 40 = 120
Answer:
$120
You have:
3/4
You eat one-third of that amount:
1/3 × 3/4
= 3/12
= 1/4
So you ate:
1/4 of a pizza
Amount remaining:
3/4 - 1/4
= 2/4
= 1/2
Answer:
1/4 of a pizza was eaten.
1/2 of a pizza remains.
Notice the distinction between:
eating 1/3 of a pizza
and:
eating 1/3 of the 3/4 of a pizza available.
The wording determines the calculation.
Find 15% of 240:
0.15 × 240 = 36
Items remaining:
240 - 36 = 204
Answer:
36 items were sold.
204 items remain.
Divide the total equally among four people:
126 ÷ 4 = 31.5
Answer:
$31.50 per person
Initial balance:
-35
Deposit $80:
-35 + 80 = 45
Spend $27:
45 - 27 = 18
Answer:
$18
Evaluate:
18 + 24 ÷ 6 × 3 - 5
Division and multiplication have equal priority, so work from left to right:
24 ÷ 6 = 4
then:
4 × 3 = 12
Now:
18 + 12 - 5
Addition and subtraction also have equal priority and are evaluated from left to right:
18 + 12 = 30
30 - 5 = 25
Answer:
25
When faced with an unfamiliar arithmetic problem, use this procedure.
Determine exactly what quantity you need to find.
Do not begin calculating merely because numbers appear in the problem.
Write down the relevant numbers and what each represents.
Example:
6 boxes
18 pencils per box
question: total pencils
Ask what operation describes the situation.
For equal groups:
number of groups × amount per group
Therefore:
6 × 18
Before calculating exactly:
6 × 20 ≈ 120
So the answer should be somewhat below 120.
6 × 18 = 108
The answer is not merely:
108
It is:
108 pencils
108 is somewhat below the estimate of 120, so the result is plausible.
Use this general pattern:
Understand → Represent → Estimate → Calculate → Check
Words can suggest an operation, but do not rely blindly on keywords. Understand the relationship between the quantities.
Typical relationships include:
Use addition when quantities are being combined.
existing amount + additional amount
Example:
15 books + 8 books = 23 books
Use subtraction when something is removed or when finding a difference.
starting amount - removed amount
or:
larger quantity - smaller quantity
Use multiplication for repeated equal quantities or scaling.
number of groups × amount per group
Example:
8 tables × 6 chairs per table = 48 chairs
Use division when sharing equally, determining a rate, or finding how many groups fit into another quantity.
total ÷ number of groups
or:
total ÷ amount per group
Translate percentages into fractions or decimals before calculating.
35% = 35/100 = 0.35
Then determine whether the problem requires finding a percentage, increasing by a percentage, or decreasing by a percentage.
Numbers in real problems usually represent something.
Examples:
12 meters
$35
8 kilograms
60 miles per hour
4 hours
Track units during calculations.
Example:
180 miles ÷ 3 hours
= 60 miles/hour
Units can help reveal mistakes.
Suppose the question asks for a distance but your calculation produces a quantity measured in hours. That is evidence that the setup may be wrong.
Be especially careful when units differ.
Example:
2 hours + 30 minutes
Do not calculate:
2 + 30 = 32
Convert to compatible units first.
Either:
2 hours = 120 minutes
then:
120 + 30 = 150 minutes
or recognize directly:
2 hours + 30 minutes = 2.5 hours
A ratio compares two quantities.
Example:
2 red objects for every 3 blue objects
can be written:
2:3
or:
2/3
A rate compares quantities with different units.
Example:
180 miles in 3 hours
Unit rate:
180 miles / 3 hours
= 60 miles/hour
If a recipe uses:
2 cups flour for 8 servings
then for 16 servings, the number of servings doubles, so the flour also doubles:
4 cups flour
The essential principle is that equivalent ratios preserve the same relationship.
The arithmetic mean, commonly called the average, is:
sum of values ÷ number of values
Example:
7, 9, 10, 14
Sum:
7 + 9 + 10 + 14 = 40
There are four values:
40 ÷ 4 = 10
Average:
10
Do not divide by the wrong number of values.
For:
5, 8, 12
there are three values, so:
(5 + 8 + 12) ÷ 3
= 25 ÷ 3
≈ 8.33
Rounding produces a simpler approximation.
To round to the nearest whole number:
4.2 → 4
4.7 → 5
Under the usual round-half-up convention taught in elementary arithmetic:
4.5 → 5
To round 6.784 to the nearest hundredth, inspect the thousandths digit:
6.784
Hundredths digit:
8
Thousandths digit:
4
Because 4 is below 5:
6.784 → 6.78
For:
6.786
the thousandths digit is 6, so:
6.786 → 6.79
Do not round unnecessarily during intermediate calculations.
Premature rounding can accumulate error.
Keep sufficient precision during the calculation and round the final result when appropriate.
Also be aware that calculators, software, finance, and scientific work may use different tie-breaking conventions for values exactly halfway between two choices.
Some arithmetic results are exact:
7 + 8 = 15
Others may be represented approximately:
1 ÷ 3 = 0.333333...
The exact value is:
1/3
A decimal approximation might be:
0.33
or:
0.333
depending on the required precision.
Use the symbol:
≈
to mean "approximately equal to."
Example:
1/3 ≈ 0.333
Do not write:
1/3 = 0.333
if you intend mathematical exactness, because 0.333 is only an approximation of 1/3.
Develop these habits:
- Read the entire problem before calculating.
- Identify what each number represents.
- Determine the operation before using a calculator.
- Estimate the expected magnitude.
- Keep place values aligned.
- Track negative signs carefully.
- Track units.
- Use parentheses when they make an expression clearer.
- Simplify fractions when appropriate.
- Check answers using an independent method.
- Question answers that contradict common sense.
- Prefer understanding a relationship over memorizing an isolated rule.
Accuracy is more important than speed while learning.
Speed should develop from repeated correct practice.
Complete this section without referring to earlier sections and, initially, without a calculator.
- 468 + 759
- 1,004 - 687
- 37 × 24
- 936 ÷ 18
- -17 + 29
- 14 - 31
- -8 × -7
- 72 ÷ -9
- 2/3 + 5/6
- 7/8 - 1/4
- 3/5 × 10/9
- 4/7 ÷ 2/3
- 17.85 + 6.7
- 30 - 12.46
- 4.8 × 2.5
- 18.9 ÷ 3
- Find 30% of 250.
- 24 is what percentage of 80?
- Increase 120 by 15%.
- Decrease 500 by 12%.
- 7 + 4 × 6
- (7 + 4) × 6
- 48 ÷ 8 × 3 + 2
- 50 - 18 ÷ 3 + 4
-
Nine boxes contain 16 items each. How many items are there?
-
A $72 meal receives a 20% tip. What is the total bill?
-
You have $500, spend $137.50, and then receive $85. How much money do you have?
-
A vehicle travels 420 miles in 7 hours at a constant average speed. What is its average speed?
-
A $240 item is discounted by 35%. What is the sale price?
-
A recipe for 4 people requires 3 cups of an ingredient. Assuming the recipe scales proportionally, how many cups are required for 10 people?
- 1,227
- 317
- 888
- 52
- 12
- -17
- 56
- -8
- 3/2, or 1 1/2
- 5/8
- 2/3
- 6/7
- 24.55
- 17.54
- 12
- 6.3
- 75
- 30%
- 138
- 440
- 31
- 66
- 20
- 48
- 144 items
- $86.40
- $447.50
- 60 miles per hour
- $156
- 7.5 cups
Do not consider the skill mastered merely because the material feels familiar.
A learner has basic arithmetic competence when they can reliably:
- Explain what addition, subtraction, multiplication, and division mean.
- Perform those operations with whole numbers and decimals.
- Correctly handle negative numbers.
- Add, subtract, multiply, divide, and simplify basic fractions.
- Convert among common fractions, decimals, and percentages.
- Calculate percentage amounts, increases, and decreases.
- Apply the order of operations correctly.
- Calculate simple ratios, rates, and averages.
- Estimate the approximate size of an answer.
- Translate ordinary word problems into arithmetic expressions.
- Track units correctly.
- Recognize obviously implausible results.
- Check a result using an inverse operation, estimation, or a second method.
- Use a calculator without surrendering judgment to it.
As a practical target, aim for at least 27 correct answers out of the 30 mastery-test questions, with no consistent weakness in an entire category.
If several errors come from the same concept, return to that section and practice that concept even if the total score is high.
Arithmetic is learned through retrieval and application, not reading alone.
Use this training cycle:
Read one section and work through its examples.
Do not merely read equations. Reproduce the reasoning yourself.
Solve several similar problems without looking at the examples.
Start slowly and prioritize correctness.
For every incorrect answer, determine why it was wrong.
Classify the error:
Concept error
Operation-selection error
Procedure error
Sign error
Place-value error
Fraction error
Order-of-operations error
Transcription/input error
Careless arithmetic error
Redo the problem correctly without copying the answer.
Then solve another problem of the same type.
Once individual operations are comfortable, mix problem types so that you must decide which operation to use.
This is important because real situations do not announce:
"This is a multiplication problem."
You must recognize the mathematical relationship yourself.
After a delay, solve similar problems again without reviewing first.
Successful delayed recall is stronger evidence of learning than immediate recognition.
A learner should be able to solve a problem such as the following from beginning to end:
"A store sells notebooks for $3.20 each. You buy 7 notebooks. A 25% discount is applied to the notebook total, and then $4.50 shipping is added. What is the final cost?"
Reasoning:
Cost before discount:
7 × 3.20 = 22.40
Discount:
25% of 22.40
= 0.25 × 22.40
= 5.60
After discount:
22.40 - 5.60 = 16.80
Add shipping:
16.80 + 4.50 = 21.30
Answer:
$21.30
Independent check:
A 25% discount means paying 75% of the original notebook cost:
22.40 × 0.75 = 16.80
Then:
16.80 + 4.50 = 21.30
The two methods agree.
The learner should also be able to estimate:
7 × about $3 = about $21
After a 25% discount, the merchandise should cost roughly $16, and adding about $5 shipping should produce a total around $21.
Therefore $21.30 is plausible.
Basic arithmetic is not the ability to execute memorized procedures in isolation.
Practical arithmetic competence consists of four connected abilities:
1. Understand the quantities.
2. Choose the correct mathematical operation.
3. Calculate accurately.
4. Determine whether the answer makes sense.
Whenever solving an arithmetic problem, use the complete loop:
Understand
↓
Represent
↓
Estimate
↓
Calculate
↓
Check
↓
Interpret
If the final answer cannot be interpreted in the context of the original problem, the work is not finished.
END OF SKILL