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**Easy way of multiplying numbers ending with 5 by itself.**

If we multiply some number ending with 5, suppose 25 by 25, then first multiply the last digits 5 by 5 which gives 25 and then multiply the first digit 2 by a number 1 more than 2 which is 3, this gives (2*3=6). so, the answer is 625.

Another example could be 85 multiplied by 85. Again, last digits 5*5=25 and then the first digit 8 multiplied by a number 1 more than 8 i.e. 9 gives us (8*9=72). So, the answer would be 7225.

**Graphical presentation of the Pythagoras theorem: **This shows the relationship between the three sides of a right angle triangle. The theorem says: a^2+b^2=c^2 (where c is the hypotenuse; a and b are the other 2 sides of the triangle).

An easy way of understanding (a+b)^2=a^2+2ab+b^2. Here, size of a and b does not make any difference, any of them can be bigger or smaller. But, the result would be always same.

**Easy way of solving subtraction problems where the top number has lots of zeroes and any number in the bottom. **

Example:

6,000

– 2,798.

———

In this example, if we solve by the normal procedure, it will take lot of time and carry forwards efforts and there are more chances of mistakes by kids.

An easy of solving this problem would be by subtracting 1 from both the numbers. Subtracting 1 from both numbers would be easy.

Then the problem would look like this:

5,999

– 2,797

———

3,202

This does not involve any carry forwards and therefore easy to solve it. Subtracting any number from 9 is also easy. So, the same problem can be solved in less time and with less efforts with minimal chances of mistakes.

**Divisibility Factors:**

**Divisbility by 3**: If the total of all the digits in a number is divisible by 3, then the number is divisible by 3.

Example: 327, total of all digits is 3+2+7=12, which is divisible by 3. So, the number 327 is divisible by 3. And the quotient is 109.

Another example: 428, total of all the digits, 4+2+8=14, which is not divisible by 3, so 428 is not divisible by 3.

**Divisibility by 4**: If the last 2 digits of a number is divisible by 4, then the number is divisible by 4.

Example: 1428, last 2 digits are 28, which is divisible by 4. So, 1428 is divisible by 4.

Another example: 1706, last 2 digits is 06, which is not divisible by 4. So, 1706 is not divisible by 4.

**Another easy way of subtraction by adding: **

If suppose we have to pay in pence and we have 10 or 20 pound notes. Then the problem would look like this.

10.00

– 2.94

——–

If we solve this by normal method, this will be a bit complicated for kids and time taking.

To make it easy, we can add some number to the lower number to make it a whole number and add the same number to the upper number too. In the above problem, we will add 6p or 0.06 to both the upper and lower numbers.

The problem will look like this now:

10.06

– 3.00

——–

Now, it is more easy because the lower number is a whole number. The simple answer would be 7.06.

**Fractions: Finding some fractions in between 2 fractions**

Ex: We have to find some fractions between the fractions 1/4 and 4/7

To do this, convert the 2 fractions into equivalent fractions with common denominator which will be the Lowest Common Multiple (LCM) which can be calculated as 4*7=28 in the present case.

Now, the equivalent fractions:

1/4=7/28 (By multiplying 7 in both the top and bottom numbers)

4/7=16/28 (By multiplying 4 in both the top and bottom numbers)

Now, the fractions in between both the equivalent fractions can be 8/28=2/7, 9/28, 10/28=5/14, 11/28, 12/28=3/7, 13/28, 14/28=1/2, 15/28 etc.

**Divisibility by 7 rule**

Double the last digit, then subtract this number from the number made by the remaining digits, this number should be divisible by 7. (We can apply this rule to that answer again)

Examples:

595: double of 5 is 10, 59-10=49, 49/7=7, so 595 is divisible by 7

812: double of 2 is 4, 81-4=77, 77/7=11, 812 is divisible by 7

1167: double of 7 is 14, 116-14=102, double of 2 is 4, 10-4=6, 6 is not divisible by 7, so 1167 is not divisible by 7

907: double of 7 is 14, 90-14=76, 76 is not divisible by 7, so 907 is not divisible by 7