This method requires that the student knows the squares of $1$ to $10$. However the method will work for larger numbers as well.
II. Squares Of Numbers Around 50
We will illustrate this method using a few examples. Let us start with a number which is slightly less than $50$, for example $43$.
We start with the fact that $43$ is $7$ short of $50$. Let us name this number $7$ as $d$. Here are the steps we follow to find the square of $43$.
Take the number $25$.
Reduce it by $d$ ($7$ in this case), which gives us $25 - 7 = 18$
Add two zero's to the end of the result, which gives us $1800$.
Add the square of $d$ (in this case $7^2$ is $49$) to this result which gives us $1849$.
The square of $43$ is $1849$
Let us take one more example, this time a number slightly higher than $50$, for example $59$
Notice that the number $59$ is greater than $50$ by $9$, and let us name the difference $9$ as $d$ .
Take the number $25$.
Increase it by $d$ ($9$ in this case) to give us $25 + 9 = 34$
Add two zeros at the end, to give us $3400$.
Add the square of $d$ (in this example $9^2$ is $81$) to this number, to give us $3481$
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Now let us take an example of a number which differs from $50$ by a slightly larger value.
For example, $67$, which is greater than $50$ by $17$.
Start with $25$
$25 + 17 = 42$ (Since the given number is larger than $50$, we are adding the difference)
$17^2 = 289$
$4200 + 289 = 4489$
The square of $67$ is $4489$
To summarize the steps:
- Find the difference of the given number with $50$
- Start with $25$
- Subtract the difference $d$ if the given number is less than $50$, add otherwise.
- Place two $0$s to the right of the result.
- Add the square of the difference with the result.
III. Squares Of Numbers Around 100
For this method, let us look at the steps involved before we get into the examples.
To find the square of a number which is around $100$ (either greater or less) we can use the following steps:
- Start with the given number (let's say $N$)
- Find the difference of the given number $d$ with $100$
- Subtract the difference from $N$ if the given number less than $100$, add otherwise
- Append two $0$s to the result.
- Add the square of the difference $d$ to the result.
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Let us take an example of a number greater than $100$, let's say $112$
The difference of this number from $100$ is $12$.
We start with $112$ and add $12$ to it (because the given number is greater than $100$) to get $124$.
By appending two $0$s we get $12400$
By adding the square of the difference ($12^2 = 144$) we get $12544$ which is the square of $112$
Let us take another example, this time, a number slightly less than $100$, let say $86$.
The difference of the given number with $100$ is $14$.
We start with $86$ and subtract $14$ from it to get $72$ (because the given number is less than $100$)
Append two $0$s to $72$ to get $7200$.
Add the square of $14$, that is $196$ to $7200$ to get $7396$ as the answer.
$86^2 = 7396$
IV. Combining The Two Methods
To make the best use of the two techniques, it is recommended that one memorizes the square from $1$ to $25$. And this method can be used to find the squares of any number between $1$ and $125$
$1 - 25$ - Get a straight answer
$26 - 75$ - Use the first method
$76 - 125$ - Use the second method
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V. Practice Time
Use the following widget to practice and improve your speed. You can set the level $(10 - 25)$ of your practice based on how much of the square values you have memorized. You need to have memorized a minimum of $10$ and a maximum of $25$ values.
--------- Reference to widget: 00aa9a80-6292-4b8b-bbd3-3367d335e816 ---------