- Multiplication (each question has two 8-digit numbers to multiply)
- Addition (each question has ten 10-digit numbers to sum)
- Square roots (each question has a 6-digit number whose square root must be found to 5 decimal places)
- Multiplication – Basic Method — for Small Numbers
When calculating a multiplication where one of the numbers is small, such as 68435 × 18, it may be fastest to simply add together multiples of the smaller number:
- 5 × 18 = 90 ⇒ …………0
- 9 + 3 × 18 = 63 ⇒ ……….30
- 6 + 4 × 18 = 78 ⇒ ……..830
- 7 + 8 × 18 = 151 ⇒ ……1830
- 15 + 6 × 18 = 123 ⇒ 1231830
In fact when I coach beginners in Mental Math, one of the first things I work on is expanding the knowledge of times tables to other useful numbers — such as 18 — to make this easier to perform.
But when using this basic method for larger multiplications, such as 29136 × 5847, we don’t have enough working memory to calculate each multiple of e.g. 5847 without forgetting the numbers we’ve already calculated! So we need another method — one that’s more efficient in terms of memory.
Below I’ll show you the cross-multiplication method that most advanced mental calculators use for multiplications.
Some nice things about this method are:
- You only need to know your times tables up to 9 × 9
- However large the multiplications, you never have to remember many numbers at once
- It’s very straightforward once you know the simple pattern
To see how this works, we’ll take our example of 29136 × 5847:
1st digit — units place:
To begin, we simply multiply 6 × 7 = 42. Then the rightmost digit of the answer is 2 and we can “carry” the 4 for the next step:
29136 × 5847 2 (carrying 4 from the 42)
2nd digit — factors with 10:
Next we consider 40 × 6 and 7 × 30, as these are the digit products (4 × 6 and 7 × 3) that come with a factor of 10, just like the 40 we remembered from the previous step.
The quickest way is to start with the 4 from the 40 that we carried, then add on the 4 × 6 and 7 × 3:
- 4 + 4 × 6 = 28
- 28 + 7 × 3 = 49
These addition-multiplication pairs are quick to do with practice.
Again we can write down the “9” in the tens place of the final answer, and keep the 4 for the following step.
29136 × 5847 92 (carrying 4 from the 49)
3rd digit — factors with 100:
We continue with 800 × 6, 40 × 30 and 7 × 100, as these are the digit products that come with a factor of 100.
Again — start with the 4 from the 400 that we carried, then add on the other products:
- 4 + 8 × 6 = 52
- 52 + 4 × 3 = 64
- 64 + 7 × 1 = 71
Write down the “1” in the 100s place of the final answer, and keep the 7 for the following step:
29136 × 5847 192 (carrying 7 from the 71)
Notice that in each step, the order of the colors on the top is the mirror image of the colors on the bottom, as each matching pair of digits must represent the same power of 10.
You can add the digits in any order you like, but I find it helpful to always start with the bottom-left and top-right product (here the 8 × 6) and systematically move simultaneously rightwards on the bottom number and leftwards on the top number.
4th digit — factors with 1000:
By now the pattern should be fairly clear, so I’ll continue with minimal commentary:
- 7 + 5 × 6 = 37
- 37 + 8 × 3 = 61
- 61 + 4 × 1 = 65
- 65 + 7 × 9 = 128
29136 × 5847 8192 (carrying 12 from the 128)
5th digit — factors with 10,000:
This time, notice that the “6” has already been multiplied by every digit of the bottom number, so will not be active for the rest of the calculation:
- 12 + 5 × 3 = 27
- 27 + 8 × 1 = 35
- 35 + 4 × 9 = 71
- 71 + 7 × 2 = 85
29136 × 5847 58192 (carrying 8 from the 85)
6th digit — factors with 100,000:
From now on the calculation gets simpler as there are fewer and fewer products of the same magnitude:
- 8 + 5 × 1 = 13
- 13 + 8 × 9 = 85
- 85 + 4 × 2 = 93
29136 × 5847 358192 (carrying 9 from the 93)
7th digit — factors with 1,000,000:
- 9 + 5 × 9 = 54
- 54 + 8 × 2 = 70
29136 × 5847 0358192 (carrying 7 from the 70)
8th digit — factors with 10,000,000:
- 7 + 5 × 2 = 17
As this is the final stage, we don’t have to carry anything, and simply write down the remaining digits:
29136 × 5847 170358192
So that is the standard cross multiplication method used by amateur human calculators, as well as current and past multiplication world-record holders such as Freddis Reyes and Marc Jornet Sanz! (Jeonghee Lee prefers a left-to-right method instead).
To conclude — the method in general is:
- Start with the rightmost digit of each number:
- calculate their product
- write down the units digit
- carry the tens digit for the next stage
- For every subsequent digit of the answer:
- take the carried number from before
- add all products of the same magnitude by working systematically
- write down the units digit in the final answer
- carry the rest for the next stage
In the Memoriad competition, and in the Mental Calculation World Cup, competitors must multiply 8-digit numbers, such as 12345678 × 98702468, and the fastest competitors can do these in 15-30 seconds!
To practice this you can use the Memoriad software — although I recommend to start with smaller products of 3- and 4-digit numbers before working your way up.
How to Add and Subtract Fractions with Mental Maths
While addition and subtraction of integers is usually straightforward, there are more steps involved when you add or subtract fractions.
A fraction consists of a number—the numerator—divided by another number—called the denominator. It is usual for both these numbers to be positive integers (whole numbers).
For example, in 415, the numerator is 4, and the denominator is 15.
The basic formula for adding fractions is:
In words, this means that you multiply each numerator by the opposite demonimator, and add these results to get the new numerator. The new denominator is the product of the original denominators.
Subtraction uses the same calculation, except using a minus rather than a plus:
On this page, I’ll use examples using addition calculations only.
As an example:
Explanation of Formula
You can use this formula without understanding it, but helps you memorize it, and be creative with it, if you do understand it.
Imagine you have two pizzas of equal size:
- Pizza 1 is cut into b equal slices, of which you’ll eat a of them.
- Pizza 2 is cut into d equal slices, of which you’ll eat c of them.
In total you will eat ab+cd pizza. How much is this?
Imagine that you would carefully cut each slice of the first pizza into d slices. The slices are much smaller now—the pizza is divided into bd slices—and you’ll eat ad of them.
Then cut each slice of the second pizza into b slices. This pizza is also now divided into bd slices—and you’ll eat bc of them.
In total, you ate ad+bc slices, and each one was 1bd of a whole pizza.
A fraction is simplified if there are no prime numbers that divide into both the numerator and denominator. For example, 4060 is not simplified, because 2 divides into both 40 and 60. In fact, so does 5, and even some larger non-prime numbers, like 20. If you divide the top and bottom of the fraction by 20, the fraction becomes 23, which is the simplified form.
On this page, I’ll assume that you need to add or subtract fractions that are already simplified, which is usual for mental math competitions. There is a note at the end to describe what you should do if they are not simplified.
A fraction is improper, if the numerator is larger than the denominator. For example, 143 is an improper fraction. Improper fractions can be written as mixed fractions—with an integer part and a proper fraction part. For example, 143=423
In mental calculation competitions, you must give all answers in mixed form. Improper fractions are marked incorrect!
To convert an improper fraction to a mixed fraction, first divide the numerator by the denominator and obtain the remainder:
14÷3=4 rem. 2
The integer part is the result of the division—4—and the remainder—2—is the numerator for the mixed fraction.
As a complete example:
Cases that Require Simplification
If two numbers—b and d—don’t share any factors, they are called co-prime. This is the same as saying that bd would be a simplified fraction.
If the two denominators—b and d—are co-prime, then it is guaranteed that the resulting fraction will not need simplification. But otherwise, you will also need to try to simplify the final fraction.
In mental calculation competitions, you must give all answers in simplified form. Unsimplified fractions are marked incorrect!
8 and 24 are not co-prime, as both are even, so we must simplify at the end:
This calculation involved some fairly large numbers, and it could be worse if the original fractions had larger denominators! Luckily, there is a shortcut:
- Find any number that the denominators both divide into—the smaller the better. In the example above, you could use 48, but the best would be to use 24.
- Express both fractions using this new denominator: 924+724. In this case, the second fraction did not need to change, but in the first one, the deminomator had been multiplied by 3 to get from 8 to 24. So it was necessary to multiply its numerator the same way: 3×3=9.
- Simply add the numerators, and place that above the new denominator.
Notice that sometimes—like here—we need to do a simplification step at the end, even though we already simplified at an earlier stage.
Proof that Addition of Simplified Fractions with Co-prime Denominators Never Requires Simplification
You can skip this paragraph if you are not currently curious about the mathematics behind the method.
Does the result from the formula, ad+bcbd, require simplification?
Assume that b and d share no prime factors—meaning they are co-prime. Does any factor of b divide into the numerator ad+bc?
Certainly it will divide into bc. But it doesn’t divide into ad, because b shares no prime factors with d, and nor with a (as ab was already simplified). Since it does divide into bc but not into ad, it can’t divide into their sum.
By the same argument, the numerator also shares no prime factors with d.
Therefore there are no prime numbers—and thus no integers of any type—that we can divide by to simplify ad+bcbd.
Final Summary for Mental Calculation
When adding or subtracting fractions using mental math:
- Check—or assume—that the fractions can’t be simplified.
- Check whether the denominators share any factors.
- If they don’t share any factors, use the general formula, and leave your answer as a mixed fraction.
- If they do share factors, you can either use the general method (simpler) or just change the fractions manually to have the same denominator (easier arithmetic). Then simplify the final answer, if necessary.
- Remember not to write down any intermediate steps if training for a competition!
As a final example:
- Using the general formula:
- Alternatively, by changing fractions manually:
Here, we must check whether 2512 can be simplified further, but it cannot. So it is the final answer.
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