What learning arithmetic really means
The short version:
- Arithmetic is the math of counting, sharing, and comparing amounts using the four operations, fractions, decimals, and percents.
- Fractions, decimals, and percents are three ways of writing the same number: 3/4 is 0.75 is 75%.
- Estimating first is the fastest way to catch mistakes, because a wrong answer usually looks wrong before you check the arithmetic.
Arithmetic is the part of math you use without a worksheet: splitting a bill, doubling a recipe, checking a sale price, or reading a phone plan. It is also the foundation everything else stands on, because a mistake in fractions becomes a mistake in algebra later. The four lessons below cover the skills that show up most often, and the four-function calculator is there for checking totals once you have made your own estimate.
Arithmetic is the math of counting, sharing, and comparing amounts. It covers the four operations plus fractions, decimals, percents, and ratios. Number sense, fractions, decimals, percentages, ratios, and estimation.
The arithmetic learning path, in order
Follow these lessons in order. Fractions come first because percents and ratios are both special kinds of fractions, and arithmetic practice after each one keeps the skill from fading.
A Practical Guide to Fractions
Why it comes here: Fractions are the language of parts, and percents, ratios, probability, and algebraic fractions all reuse the same rules you learn here.
What you need first: You need multiplication and division facts up to 12, and the idea that a whole can be cut into equal pieces.
Understanding Percentages
Why it comes here: A percent is just a fraction with a denominator of 100, so this lesson turns fraction skill into money, tax, tips, and discount skill.
What you need first: You need fractions, decimals, and the ability to multiply by numbers like 0.15 without panic.
Ratios and Proportions
Why it comes here: Ratios compare two amounts, and proportions scale that comparison up or down. Recipes, maps, unit prices, and mixtures all work this way.
What you need first: You need fractions and equivalent fractions, plus cross-multiplication, which is one small step past division.
Order of Operations
Why it comes here: Without a shared order, 2 + 3 × 4 could mean 20 or 14. This lesson makes long expressions give one answer that everyone agrees on.
What you need first: You need the four operations, exponents, and the habit of finishing what is inside parentheses first.
Arithmetic formulas, with real numbers
| Task | Rule | Worked example | Answer |
|---|---|---|---|
| Add fractions | Rewrite with a common denominator, then add the tops | 2/3 + 1/4 = 8/12 + 3/12 | 11/12 |
| Multiply fractions | Multiply the tops, multiply the bottoms, then simplify | 3/5 × 10/9 = 30/45 | 2/3 |
| Divide fractions | Flip the second fraction and multiply | 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 | 1 7/8 |
| Find a percent of a number | Change the percent to a decimal and multiply | 18% of 250 = 0.18 × 250 | 45 |
| Percent change | (new − old) ÷ old, then multiply by 100 | (63 − 45) ÷ 45 = 0.4 | 40% increase |
| Solve a proportion | Cross-multiply, then divide | 3/8 = x/40 → 8x = 120 | x = 15 |
The percent-change row catches more people than any other. The bottom of the fraction is always the amount you started with, so a price rising from 45 to 63 is a 40% increase, while the same price falling from 63 to 45 is only a 28.6% decrease. Same two numbers, different starting point, different answer. Try both directions in the arithmetic calculator to see why the base matters.
Arithmetic words you need to know
These words appear in word problems constantly, and misreading one of them is a more common cause of wrong answers than bad arithmetic.
- Numerator
- The numerator is the top number of a fraction, and it counts how many equal parts you have.
- Denominator
- The denominator is the bottom number of a fraction, and it names the size of each equal part.
- Equivalent fractions
- Equivalent fractions look different but have the same value, such as 2/4, 3/6, and 1/2.
- Improper fraction
- An improper fraction has a numerator larger than its denominator, such as 11/4, which equals 2 3/4.
- Mixed number
- A mixed number combines a whole number and a fraction, such as 2 3/4, and is often easier to picture than 11/4.
- Decimal
- A decimal writes a fraction using place value after a point, so 0.75 means seventy-five hundredths.
- Percent
- A percent is a number out of one hundred, so 40% means 40 out of every 100, or the decimal 0.4.
- Ratio
- A ratio compares two amounts, written 3:2 or 3 to 2, without saying how large the total is.
- Proportion
- A proportion is a statement that two ratios are equal, such as 3/8 = 15/40.
- Factor
- A factor is a whole number that divides another number evenly, so the factors of 12 are 1, 2, 3, 4, 6, and 12.
- Multiple
- A multiple is what you get when you multiply a number by a whole number, so 15 and 20 are multiples of 5.
- Estimate
- An estimate is a quick rough answer used to check whether the exact answer is sensible.
When each part of arithmetic is taught
Arithmetic is taught over about six school years, and each year adds one new kind of number. Find your row to see what comes next.
| School level | What you learn at that stage |
|---|---|
| Grades 2-3 | Addition and subtraction with regrouping, multiplication facts, and the meaning of division as fair sharing. |
| Grades 4-5 | Long division, factors and multiples, equivalent fractions, adding fractions with unlike denominators, and decimals. |
| Grade 6 | Dividing fractions, ratios and unit rates, percents, negative numbers, and the order of operations with exponents. |
| Grade 7 and adult refreshers | Percent increase and decrease, tax and tip, proportional reasoning, and multi-step word problems with units. |
Common questions about arithmetic
Why do I need a common denominator to add fractions but not to multiply them?
Adding means counting pieces, and you can only count pieces that are the same size. Thirds and quarters are different sizes, so 2/3 + 1/4 has to become 8/12 + 3/12 before the tops can be added, giving 11/12. Multiplying means taking a part of a part, which does not require matching sizes: 1/2 of 1/3 is 1/6 no matter what. That is why 3/5 × 10/9 can be done straight across.
What is the fastest way to find a percent in my head?
Break it into 10% and 5% chunks. Ten percent of 250 is 25, so 20% is 50 and 5% is 12.5. That means 18% is 25 + 25 minus 5, which is 45. This works because 10% just moves the decimal point one place left. For a tip, 15% is 10% plus half of that 10%, so on a $42 bill it is 4.20 + 2.10 = $6.30.
Is PEMDAS a strict left-to-right list?
Not quite. Multiplication and division have equal rank, and so do addition and subtraction. Within each rank you work left to right. So 24 ÷ 6 × 2 is 4 × 2 = 8, not 24 ÷ 12 = 2. Reading PEMDAS as four steps instead of six is the single most common cause of wrong answers on long expressions, and the order of operations lesson works through several of them.
When should I use a fraction and when should I use a decimal?
Use fractions when the numbers divide cleanly or when you need an exact value, because 1/3 is exact and 0.333 is not. Use decimals for money, measurement, and anything you will compare or sort, since 0.62 is obviously larger than 0.6 while 5/8 versus 3/5 takes a moment. Many problems are easiest if you convert everything to one form first and stay there.
How do I check an answer without redoing the whole problem?
Estimate with friendly numbers, then compare. If you calculated 18% of 250 and got 450, the estimate that 20% is about 50 tells you immediately that the decimal point slipped. For division, multiply your answer back: 15/8 should give 3/4 when multiplied by 2/5. Two seconds of estimating catches most errors that ten minutes of rechecking would find.
What to do next
The mistake to watch for
Most arithmetic errors come from losing place value, applying a percent to the wrong base, or treating a fraction bar as if it did not group the numerator and denominator.
Your next study session
Choose the lesson that matches what tripped you up most recently, work five problems on paper, then verify the totals in the four-function calculator. When your estimates start matching your exact answers, move on to arithmetic practice for mixed questions.