6 ÷ (1/6) × 2 = ? · MathPrisma

Gentle

6 \div \frac{1}{6} \times 2

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0:00 / 1:28

Transcript

6 ÷ 1/6 × 2. One line of arithmetic — and it starts fights in comment sections. Get your answer ready… let's see if it survives.

$6 \div \frac{1}{6} \times 2$

Trap number one: that 'MD' in PEMDAS looks like multiply-before-divide. Do 1/6 × 2 first, divide, and you get 18. Nope. Multiplication and division are the SAME rank.

$6 \div (\frac{1}{6} \times 2) \to 18 ?$

And when it's a tie, there's no drama — you read it like words on a page, left to right. So the first thing that happens is 6 divided by 1/6.

Which is trap number two — because dividing by one sixth is NOT dividing by six. Keep, change, flip: keep the 6, change divide to times, flip 1/6 into 6. Six times six — 36.

$6 \times 6 = 36$

Weird that dividing made it bigger? Ask what the division means: how many sixths fit inside 6? Each whole holds six of them, and you have six wholes. Thirty-six. It HAS to grow.

Now finish the tie, left to right. 36 times 2 — 72. Done.

$36 \times 2 = 72$

Two reflexes to rewire. Dividing by a number smaller than one makes things grow — and a tie between times and divide is settled by position. Left to right, like reading.

One line, two traps · 1:28

Two reflexes fight the right answer here, and almost everyone trips on at least one.

First, dividing by $\tfrac{1}{6}$ is not dividing by six. Keep · change · flip: dividing by a fraction means multiplying by its reciprocal, so $6 \div \tfrac{1}{6} = 6 \times 6 = 36$ . Dividing by a number smaller than one makes the result bigger — you’re really asking “how many sixths fit inside 6?”, and the answer is thirty-six of them.

Second, now that only $\div$ and $\times$ are left, they don’t take turns by rank. They run strictly left to right:

$6 \div \tfrac{1}{6} \times 2 = 36 \times 2 = 72$

What this hides

The “MD” in PEMDAS looks like multiply-then-divide, so the eye wants to do $\tfrac{1}{6} \times 2$ first and divide by that — which gives $18$ . But multiplication and division are the same rank; a tie is read like words on a page, left to right, and the leftmost one goes first. Meanwhile “divide by a sixth” sounds like shrinking, because with whole numbers dividing always shrinks — a divisor less than one quietly flips that instinct.

The transferable idea: $\div \tfrac{a}{b}$ is always $\times \tfrac{b}{a}$ , and operations of equal rank are settled by position, not by which symbol you were taught to name first.

Now train the eye — it opens on another $\div$ / $\times$ tie. Click the operation that fires first:

Which operation goes first?