Posted April 9, 2015 By Presh Talwalkar. Read about me , or email me .

I came across an amazing way that ancient Egyptians divided numbers. It struck me as more intuitive than the long division method taught in school.

Video – Incredibly Easy Way To Divide Numbers – Ancient Math Secret

Watch the video from my YouTube channel.

Or keep reading for a text explanation, history lesson, and a mathematical proof.

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"All will be well if you use your mind for your decisions, and mind only your decisions." Since 2007, I have devoted my life to sharing the joy of game theory and mathematics. MindYourDecisions now has over 1,000 free articles with no ads thanks to community support! Help out and get early access to posts with a pledge on Patreon. .

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Divide 30 by 2.5

In school you would do this problem by shifting the decimal point on 2.5 to make 25, and doing the same on 30 to make 300. But the whole process seems a bit strange, and at times I would forget which way to shift the decimal. The long division steps also do not make any intuitive sense after that either.

The ancient Egyptians considered the problem in reverse. How many multiples of 2.5 can you count up until you get to 30?

We start by counting 1 part is 2.5.

1 part = 2.5

Then we double to get 2 parts is 5.

1 part = 2.5

2 parts = 5

We can multiply that by 5 to see that 10 parts are 25.

1 part = 2.5

2 parts = 5

10 part = 25

And now, we’ve pretty much solved the problem! We want to count up until we get to 30. So note that 5 + 25 = 30. The Egyptians would mark those lines with slash marks.

1 part = 2.5

2 parts = 5 \

10 part = 25 \

Now we find the answer by adding up the number of parts. So we have 2 + 10 = 12 parts. And that’s the answer!

We have 30 divided by 2.5 is equal to 12.

Wasn’t that easy?

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Learn How Egyptians Multiplied Numbers!

If you liked this method, you’ll also like to learn how ancient Egyptians multiplied numbers. They could multiply any two numbers simply by halving and doubling and adding up partial sums.

Notes On The Method

The problem 30 divided by 2 1/2 is documented in the Rind Mathematical Papyrus, Problem 76. The document is from ~1650 BC, which is more than 3,600 years ago.

I came across this problem in a text about non-western mathematical history, The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook.

Today this algorithm is taught as the partial quotients method [link to pdf].

Why This Method Works

Let’s do the problem 30 divided by 2.5. Algebraically, that means we want to solve the following equation for the quotient Q.

30/2.5 = Q.

We can multiply both sides by 2.5 to get a multiplication problem.

2.5Q = 30

Now let’s find the partial quotients! That is, let’s add up multiples of 2.5 until we get to 30.

We know that 1 part of 2.5 is just 2.5.

2.5(1) = 2.5

We also know 2 parts is equal to 5.

2.5(2) = 5

And finally 10 parts is equal to 25.

2.5(10) = 25

So we can add up the equations totaling 5 and 25 to get the following.

2.5(5) + 2.5(10) = 5 + 25

Now we factor the left-hand side and add the right hand side.

2.5(2 + 10) = 30

2.5(2 + 10) = 30

In other words, we have found the equation 2.5Q = 30 has the solution Q = 2 + 10 = 12.

Proof

Let’s say we want to solve X/Y, where the dividend is X and the divisor is Y.

We want to find the quotient Q. So we have Q = X/Y.

We can re-arrange the equation by multiplying both sides by Y. So we have the following equation.

YQ = X

Suppose we can find multiples of Y, with the partial quotients Q i and the partial sums X i , where the sum of X i is equal to X.

YQ 1 = X 1

YQ 2 = X 2

…

YQ n = X n

We can add up the equations, and factor out the Y, to get the following.

Y(Q 1 + … + Q n ) = X 1 + … + X n

By construction the right hand side is equal to X.

Y(Q 1 + … + Q n ) = X

For a division problem, the quotient Q satisfies the following equation.

Y(Q) = X