Type 2 + 3 × 4 into two different calculators and you can get two different answers: 14 on one, 20 on the other. Neither is broken. They follow different rules about what to work out first, and the percent key varies just as much. Once you know where the keys are, knowing those rules is most of what using a calculator well comes down to.
Two Ways Calculators Work
Left to right, as you press. Most simple four-function desk calculators work out each step the moment you press the next operator. Enter 2 + 3 × 4 and the display shows 5 when you press ×, then 20 at the end.
The whole expression, by the order of operations. Scientific calculators, and most calculators that show the whole expression on screen, follow the order taught in school as PEMDAS or BODMAS:
- Parentheses first
- Then powers
- Then multiplication and division, left to right
- Then addition and subtraction, left to right
By that order, 2 + 3 × 4 = 2 + 12 = 14, and 12 ÷ 3 × 2 = 8, not 2. The basic calculator on this site works this way. If you want a sum worked out first, put it in parentheses: (2 + 3) × 4 = 20.
The difference matters with real numbers. A balance of $245.50, plus four charges of $18.75, minus a $9.25 refund, is 245.50 + 18.75 × 4 − 9.25 = 311.25. A calculator that works left to right turns the same keys into 1,047.75.
To test a calculator you haven't used before, enter 2 + 3 × 4 =. If you get 14, it follows the order of operations. If you get 20, it works left to right, and you'll need to press = after each step yourself.
The Percent Key
On paper, 10% is just 0.10. Most desk calculators give the % key a shortcut meaning when it follows + or −, and the basic calculator here does the same:
| You enter | Result | What it means |
|---|---|---|
| 200 × 15% | 30 | 15% of 200 |
| 50 + 10% | 55 | 50 plus 10% of 50 |
| 50 − 10% | 45 | 50 minus 10% of 50 |
| 10% | 0.1 | Ten hundredths |
That makes everyday sums one line long. For a $64.80 restaurant bill and an 18% tip, 64.80 × 18% = 11.664, so the tip is $11.66, and 64.80 + 18% = 76.464, so the total is $76.46.
Chained percentages are where people slip. Raising a price by 20% and then cutting it by 20% doesn't get you back where you started: 100 + 20% − 20% = 96, because the second 20% is taken from 120. In the same way, a 25% markup is undone by a 20% discount: 80 + 25% − 20% = 80.
Taking a Percentage Back Out
A total that already includes tax can't be reversed by subtracting the same percentage. If $107.25 includes 7.25% sales tax, 107.25 − 7.25% gives 99.47, which is wrong. Divide by 1 plus the rate instead: 107.25 ÷ 1.0725 = 100, so the price before tax was $100.
For a percentage change, divide the difference by the starting value. A score that rises from 70 to 84 has gone up by 20%: work out 84 − 70 = 14, then press ÷ 70 × 100 =. The percentage change calculator does the same for any two values.
Using This Calculator
- Live answer. The result appears under the expression as you type, so a slip shows up before you press =.
- Carrying on from a result. After =, pressing +, −, × or ÷ continues from the answer; pressing a digit starts a new calculation. That is how the percentage change example above works in two steps.
- Corrections. ⌫ deletes the last character and C clears everything.
- Negative numbers. ± changes the sign of everything on the display. To use a negative number partway through, press − straight after the operator: 7 × −3 = −21.
- History. Your last 12 calculations are listed under History; click one to bring it back and edit it. The list is cleared when you leave or reload the page.
- Keyboard. Type numbers, + − * / and parentheses directly, press Enter or = for the answer and Esc to clear.
- Commas. Commas are read as thousands separators, so 1,250 + 750 = 2000. Use a point for decimals: 2,5 is read as 25, not 2.5.
Division by zero shows a message instead of a number: 12 ÷ 0 can't be done, and 0 ÷ 0 is undefined.
Why 0.1 + 0.2 Shows 0.3
Computers store most decimals in binary, where 0.1 has no exact form, so many programs report 0.1 + 0.2 as 0.30000000000000004 and 1.1 × 3 as 3.3000000000000003. This calculator rounds every result to 12 significant digits, which removes that noise: 0.1 + 0.2 = 0.3 and 1.1 × 3 = 3.3. Repeating decimals stop at 12 digits, so 2 ÷ 3 shows 0.666666666667, and very large or very small results switch to scientific notation: 15,000,000,000,000,000 shows as 1.5e+16.
Checking Your Answer
Calculators don't make arithmetic mistakes, but people make keying mistakes. Three habits catch most of them:
- Estimate first. Round the numbers before you start: 48.7 × 21 is roughly 50 × 20 = 1,000. The calculator should give 1,022.7. If it shows 102.27 or 10,227, a decimal point went in the wrong place.
- Reverse the operation. If 1,200 × 12 = 14,400, then 14,400 ÷ 12 should give 1,200 back.
- Look for a difference divisible by 9. Swapping two neighboring digits always changes a number by a multiple of 9. Typing 1,342 instead of 1,432 makes a total 90 too small, so a total that is off by 9, 90 or 270 often means two digits were swapped. It is a long-standing bookkeeping check.
When You Need More Than the Basics
The basic calculator covers arithmetic and percentages. For powers, roots, logarithms or trigonometry, the scientific calculator uses the same keypad and history, adds function keys and parentheses, and switches between degrees and radians.
Try the examples above in the basic calculator.