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230 − 220 × 0.5 = 5! Explained: Why the Answer Is 120

Type
Math puzzle
Reading
3 min

Short answer: 230−220×0.5230-220\times 0.5 equals 120, not 5. The viral image writes “= 5!”, and the exclamation mark is not excitement: it is the factorial sign, and 5!=1205! = 120. So the equation is true, but only for people who read it correctly.

Contents7 sections

The calculation, step by step

Multiplication comes before subtraction (the order of operations), so the 220×0.5220\times 0.5 part is done first:

230−220×0.5  =  230−110  =  120.230-220\times 0.5 \;=\; 230-110 \;=\; 120.

And the right-hand side:

5!=5×4×3×2×1=120.5! = 5\times 4\times 3\times 2\times 1 = 120.

Both sides are 120, so 230−220×0.5=5!230-220\times 0.5=5! is a correct equation.

Why so many people get 5

The trap works in two layers.

  1. Left to right instead of multiplication first. Reading the expression like a sentence gives 230−220=10230-220=10, then 10×0.5=510\times 0.5=5. That matches the “5” on the right, so it feels confirmed.
  2. The exclamation mark looks like punctuation. In a meme, “= 5!” reads like an excited “five!” Most people never see the factorial sign outside a math class, so they skip it.

The two mistakes support each other: the wrong method lands exactly on the number that is written, so nothing seems off. People who do the order of operations correctly get 120, see “5”, and think the image is wrong. Both groups end up arguing, which is exactly why the post spreads.

The rule that decides it

The standard order of operations is taught in the US as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), and elsewhere as BODMAS or BIDMAS. The names differ, the rule is the same: multiplication and division are done before addition and subtraction.

If the author had wanted the answer 5, the expression would need parentheses: (230−220)×0.5=5(230-220)\times 0.5=5.

Any scientific calculator, phone calculator in scientific mode, spreadsheet or programming language gives 120 when you type 230-220*0.5.

What 5! means

For a positive whole number nn, the factorial n!n! is the product of all whole numbers from 1 to nn:

n!=n×(n−1)×⋯×2×1.n! = n\times(n-1)\times\dots\times 2\times 1.

It grows very fast: 3!=63!=6, 4!=244!=24, 5!=1205!=120, 6!=7206!=720, 10!=3,628,80010!=3{,}628{,}800. It counts the ways to put nn different things in a row: 5 books can be lined up on a shelf in 5!=1205!=120 different orders. By convention, 0!=10! = 1.

Try the same trick on these

Each one is built the same way. Work it out before opening the answer.

Puzzle 1

Is 25−19×0.5=3! 25-19\times 0.5 = 3!\, true?

Show Answer

No. 19×0.5=9.519\times 0.5=9.5, so the left side is 25−9.5=15.525-9.5=15.5, while 3!=63!=6. Reading left to right would give (25−19)×0.5=3(25-19)\times 0.5=3, which is the trap.

Puzzle 2

Fill in the blank so that __−6×0.5=3!\_\_ - 6\times 0.5 = 3! is true when read correctly.

Show Answer

3!=63!=6 and 6×0.5=36\times 0.5=3, so the blank is 6+3=96+3=\mathbf{9}. Check: 9−3=6=3!9-3=6=3!.

Puzzle 3

Why does 24−18×0.5=3!24-18\times 0.5 = 3! not work as a version of the meme?

Show Answer

Read left to right it gives (24−18)×0.5=3(24-18)\times 0.5=3, which matches the “3” and fools people, but the correct value is 24−9=1524-9=15, not 3!=63!=6. The original meme is clever because both readings line up: nn for the wrong method and n!n! for the right one. For a−b×0.5a-b\times 0.5 that needs a−b2=n\tfrac{a-b}{2}=n and a−b2=n!a-\tfrac{b}{2}=n!, which gives b=2 n!−4nb=2\,n!-4n and a=2 n!−2na=2\,n!-2n. With n=5n=5: b=220b=220, a=230a=230.

Frequently asked questions

What is 230 − 220 × 0.5?

It is 120. Multiply first: 220×0.5=110220\times 0.5=110, then subtract: 230−110=120230-110=120.

Is 230 − 220 × 0.5 = 5! true?

Yes. The exclamation mark is the factorial, and 5!=1205!=120, which is the correct value of the left side.

Why does my calculator say 5?

Some basic four-function calculators apply each operation as soon as you press the next key, so they compute 230−220=10230-220=10, then 10×0.5=510\times 0.5=5. A scientific calculator follows the order of operations and shows 120.