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Arithmetic Problems

Mental calculation tips

About this web app

- Use the digit buttons to answer the question. Hit the 'OK' button to submit your answer.
- If you're on an iPhone, you can tilt your phone for a different layout.

Enter your solution and press 'Enter' to submit

When you calculate in your mind, often you have to take several steps to a solution.

For example:

12 x 36 =

Step 1: 10 x 36 = 360

Step 2: 2 x 36 = 72

Step 3: 360 + 72 = 432

It can be useful to say the outcome of the steps in your mind as you calculate them. So when you calculate 12 x 36, you would say 360, plus 72, equals 432.

By mentally saying these steps, they'll be stored more firmly in your short term memory.

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With the method described below, it becomes relatively easy to calculate big numbers in your mind. With a little practice, you can mentally add and subtract numbers in the millions.

Start at the left and calculate one digit at a time.

For example:

4629 + 3463 =

Step 1, the first digit form the left: 4 + 3 = 7. Say "seven."

Step 2, the second digit from the left: 6 + 4 = 10. Because 10 is not a single digit, you have to add the first digit to the one from step 1. 7 + 1 = 8. So now the first digit is 8, and the second digit is 0. Say "eight zero."

Step 3, the third digit from the left: 2 + 6 = 8. Say "eight zero eight."

Step 4, the fourth digit from the left: 9 + 3 = 12. Once again, you have to add 1 to the previous step's digit. Say "eight zero nine two."

And that's the solution: 4629 + 3463 = 8092

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The method for subtraction is much like the one for addition. You start from the left again. With the addition method, you calculate one digit at the time, and add 1 to the previous digit when the current digit becomes a number of 10 or greater. With subtraction, instead of having to add 1 to the previous digit, you sometimes have to subtract 1 from the previous digit, when the current digit becomes a number below zero. You will see this in the following example:

4629 – 3463 =

Step 1, the first digit from the left: 4 – 3 = 1. Say "one."

Step 2, the second digit from the left: 6 – 4 = 2. Say "one two."

Step 3, the third digit from the left: 2 – 6. You see right away this will become a negative number. So you'll have to subtract 1 from the previous digit. So far, in step 1 and 2, we've memorized "one two." Subtract 1 from the last digit. Say "one one." You can now add the 1 you've subtracted from the previous digit as a 10 to the current digit. So 2 – 6 become 12 – 6 = 6. Say "one one six."

Step 4, the fourth digit from the left: 9 – 3 = 6. Say "one one six six." This is the solution: 4629 – 3463 = 1166

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If a number ends with one or more zeroes: first cut the zeroes, then paste them behind the outcome.

For example:

200 x 80000 =

Step 1: cut all six zeroes.

Step 2: 2 x 8 = 16

Step 3: paste all six zeroes behind the outcome: 16000000

200 x 80000 = 16000000

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First add the required number to get a round number, multiply, then subtract the added number.

For example:

7 x 96 = (7 x 100) – (7 x 4) = 700 – 28 = 672

To find a x 5, calculate a x 10 and divide by 2.

For example:

5 x 67 = 670 ÷ 2 = 335

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Often you'll use combinations of the methods described above.

For example:

56 x 755 =

Step 1: 56 x 755 = (50 x 755) + (6 x 755)

First calculate 50 x 755

Cut the zero: 5 x 755 = 7550 ÷ 2 = 3775

Paste the zero back: 37750

Step 2: 6 x 755 = (5 x 755) + 755 = 3775 + 755 = 4530

Step 3: add the outcomes of step 1 and 2: 37750 + 4530 = 42280

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Find the 10s and/or 100s.

For example:

432 ÷ 18 =

First see how many times 10 x 18 goes into the number you want to divide (the 'dividend'). 20 x 18 = 360, that fits. 30 x 18 = 540, that doesn't fit. So take the 20 x 18 and subtract that from the dividend. 432 - 360 = 72

Now calculate how many times the divisor (18, in this case) goes into the remainder. 72 ÷ 18 = 4

We've found 20 and 4, so 432 ÷ 18 = 20 + 4 = 24

Let's try another example, the same method but with a twist.

504 ÷ 18 =

Once again, first see how many times 10 x 18 goes into the dividend. 20 x 18 = 360, that fits. 30 x 18 = 540, that doesn't fit. But we can see that, although 540 (that's 30 x 18) doesn't fit in the dividend (504), it's closer to it than 360 (that's 20 x 18). So this time it's easier to calculate how many times 18 goes into the difference between 504 and 30 x 18, and subtract that from 30.

(30 x 18) - 504 =

540 - 504 = 36

Now we divide this difference by 18:

36 ÷ 18 = 2

We now know that 504 = (30 x 18) - (2 x 18) = 28 x 18

And that's the solution:

504 ÷ 18 = 28

To find the square of two-digit number mn, you can use this method:

10 x m x (mn + n) + n²

This method only works for two-digit numbers. Replace m with the first digit of the number, and n with the second digit.

For example: 37²

10 x 3 x (37 + 7) + 7² =

1320 + 7² =

1320 + 49 = 1369

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About this web app

'Arithmetic Problems' is a web application that's optimized for iPhone, but it also works on normal computers. It generates an endless supply of arithmetic problems, based on random numbers. You can choose different difficulty levels and different kinds of problems. The application is meant for training mental calculation.

Performing mental calculations is an excellent way to train your brain. According to this Wikipedia article "Mental calculation is said to improve mental capability, increases speed of response, memory power, and concentration power."