Multiply two numbers ending in 5

Can you multiply 35 by 115 in your head? Learn this neat math trick of how to multiply two numbers that end in the digit 5. You can use it to estimate many other problems. The video also contains a proof of why the method works.

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Alberto Nishikawa

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[NZC_Meow]

I also know a trick:
If there is 2 x 5 between any multiplication then you can forget it and put a 0
For example : 2 x 6 x 9 x 5
Usually we multiply 2 with 6 and get 12 and then multiply 12 with 9 and then the answer with 5. But you can forget the 2 and 5 and multiply 6 with 9. The answer is 54. And the 2 and 5 become 0. You add the 0 after 54. So the answer is 540. In this way you can do maths very fast.

[snowfloofcathug]

Awesome! And that about the cost of gas, you can even take it one step further and add the remaining 0.42 which you calculate from 14x0.03, so just use that mind trick, and then fix up the rest of the digits, amazing! Love it

[MK-jike]

There's a (possibly) easier method using the difference of two squares identity: a^2 - b^2 = (a - b)(a + b).
Let a be the average of the two numbers and b is half the difference between them.
e.g. 15 x 35 = (25 - 10)(25 + 10) = 25^2 - 10^2 = 625 - 100 = 525
65 x 75 = 70^2 - 5^2 = 4900 - 25 = 4825
35 x 115 = 75^2 - 40^2 = 5625 - 1600 = 4025
18 x 46 = (32 -14)(32 + 14) = 32^2 - 14^2 = 1024 - 196 = 828 - tbh that one was a bit harder to work out in my head

It works well for finding the squares of numbers too by flipping the formula: a^2 = (a - b)(a + b) + b^2.
e.g 45^2 = (45 - 5)(45 + 5) + 5^2 = 40 x 50 +25 = 2025
97^2 = 100 x 94 + 9 = 9409

[barryjackson]

Difference in squares is useful here. Ex: 525 is 625(25 squared) minus 100(10 squared). 25 squared minus 10 squared =(25+10)(25-10) or 35x15=525.

[itssum]

This seems a lot harder than just multiplying the fucking numbers.

[MrHatoi]

The reason this works is pretty obvious. If we split the terms into (x+5)(y+5), we then get xy + 5x + 5y + 25. The 25 is added in the last step, so we'll ignore that. Also, x and y are both guaranteed to be divisible by 10, so we can use the trick for multiplying numbers divisible by 10, [(x/10)(y/10)]*100. Now we have 5x + 5y left. If we divide by 100, we get [(x/10)+(y/10)]/2. You'll notice that this is the average of x/10 and y/10. But now, we need to multiply that result by 100 to keep the end value the same. Putting that back into the expression, we get:
[(x/10)(y/10)]*100 + [(x/10)+(y/10)]/2*100 + 25

Since x and y are both divisible by 10, dividing them by 10 is the same as cutting off the last digit, which is what was done in the video. Therefore, to make this clearer, we will replace x/10 with a and y/10 with b.
100ab + (a+b)/2*100 + 25

And if we condense it further, we get:
100[ab+(a+b)/2] + 25

Notice this is exactly what he did in the video. Get the product of the numbers with their last digits cut off, add it to their average, multiply it by 100 and add 25. FYI I didn't watch the rest of the video before writing this.

[vishaljhaveri]

1. Ignore the 5's.
2. Add the Product and Average of the leading digits. (Product+Average)
3. Multipy the resultant by 100.
4. Add 25. You will get the answer.

Overall:
((Product+Avg)*100)+25
Note: - Product and Average of leading digits .
Ignore the 5 in the whole process.

Proof:
(10x+5)(10y+5)
100xy+50(x+y)+25
100(xy+ (x+y)/2) + 25
ie.
100(PROD+AVG) + 25

[Hungtran]

I have my own super cheating thing that also about this 5-5 multiplication, the word form is long but it's easy to do:
1-Find the product of the smaller leading digit and (the total of the bigger leading digit and 1)
2-Multiply by 100
3-Add the (product of 50 and the difference of the leading digits)
4-Add 25
Example:75*165
7*(16+1)=7*17=119
119*100=11900
11900+50*(16-7=9)=12350
12350+25=12375
75*165=12375
If you don't believe it, then i move from the simple one:
15*15(same leading digits)
1*(1+1)=2
2*100=200
200+50*(1-1)=200
200+25=225
15*15=225
Or:
15*35=525(the bigger leading digit always plus 1 not the smaller one)
1*(3+1)=4
4*100=400
400+50*(3-1)=500
500+25=525
15*35=525
The result
SLD=Smaller leading digit
BLD=Bigger leading digit
So, if you accidentally add 1 to the smaller leading digit, the result is wrong!:
Example:35*15
3*(1+1)=6
6*100=600
600+50*(3-1)=700
700+25=725
But 35*15 is 525 not 725 so
That's why you should always add 1 to the bigger leading digit ONLY if the multiplication has the different leading digits from each number!

Using Basic Algebra, we can prove this:

Since both the numbers are multiples of 5, they can be expressed as below :

First Number= (10a+5)
Second Number = (10b+5)

Product of the two numbers :

First Number * Second Number = (10a+5)*(10b+5)
=100ab+50(a+b)+25
=100(ab +(a+b/2)) + 25


which means 100 multiplied by sum of product of the numbers and average of the two numbers.

[amritchandan]

Calculating average takes lot more time

[reinamaeda]

Sounds interesting, but I'd like more details for 'average' calculation which I am not exactly clear about...

[firstnamelastname-oyes]

Any limits to this? Does it break when the leading digits go over a certain count, or are a certain set?

[jsoren]

Last step: take the difference between 18 and 15 cents, which is 3 cents, and multiply by 14. This gives you 42 cents which you add to the 44.10 for 44.52.

[topodemosaprender]

You learn something new every day... That's the beauty of math!!!

[noswanson]

An easier way to visualize this is to realize that if you were to divide a multiple of 5 by 10, it becomes adding a half.

For example = 42 * 15 = 10 * 42 * 1.5

In the above example, you are multiplying by 1 and a half. So, mentally you could take 42 + 21 = 63 and multiply by 10, to get 630.

However, this can be easily expanded by using any multiplication with multiples of 5. As simple as 54 * 5 = 270 (either by multiplying outright, or dividing by 2 and multiplying by 10).

This video takes the same principle, essentially. 45 * 45 for example is the same as 10 * 45 * 4.5 So, you multiply 45 by 4 (as stated above) to get 180, add half of 45 (22.5) 202.5 and multiply by 10, to get 2025.

Multiplication with 15 is also to think about when realizing, you can also divide by 2, and multiply by 3, and then by 10. 18 * 15 = 270 (divide 18 by 2, multiply by 3, and then multiply by 10).

There are lots of cool tricks out there. I just had noticed the commonality with the (adding the average) as essentially being the same as what I've been doing for years with thinking of the 5 as a half of 10, and then adding that on.

[lechaiku]

the easier way without multiplying by 100 and avoiding decimals

15
x 35

1 x 3 || 5 x 5
3 || 25
+ 2 0 add the average of 10 + 30 (20) and put just before the unit's place

= 525


65
x 75

6 x 7 || 5 x 5
42 || 25
+ 6 5 ----> add the average of 60 + 70 (65) and put just before the unit's place

= 4875

[bijanchaudhuri]

Very useful and explained nicely algebrically. Thanks.

[ericveneto]

Easier math if you use $3.20/gal for the gas. And it even improves your estimate.

[cristobalcampos]

Cristobals method of Multiplying Endings of 5

15×35

1
3
1
=5 25 (even)

35×115
3
33
+4
40 25 (even)

[Nguyendelacruzsingh]

i cant understand the average, how to solve that?