Question

What number when multiplied by itself becomes 83.7225?
A) 9.15
B) 9.25
C) 9.45
D) 9.05

Answer 1

Hint:
In order to find the number which gives 83.7225 when multiplied by itself we need to find the square root of the given number. At first we need to group the digits of the given number in twos rom the last digit and first we need to find the perfect square near to the first the two digits and further we need to multiply the divisor by 2 and we need find a number y such that when 18y is multiplied by y we get a number near to the obtained number and the process to continued till the remainder is zero.

Complete step by step solution:
We are given that when a number x is multiplied by itself we get 83.7225
And we need to find the value of x
$
   \Rightarrow x*x = 83.7225 \\
   \Rightarrow {x^2} = 83.7225 \\
   \Rightarrow x = \sqrt {83.7225} \\
 $
Now we need to find the square root of the given number
Let find the square using long division method
At first we need to group the digits of the given number in twos from the last digit
$ \Rightarrow \left| \!{\overline {\,
 {\overline {83} .\overline {72} \overline {25} } \,}} \right. $
Now consider the first group of numbers , that is 83
Now we need to check the nearest square number to 83
We know that the nearest square number to 83 is ${9^2} = 81$
Hence
$
  {\text{ 9}} \\
   \Rightarrow 9\underline {\left| \!{\overline {\,

  \overline {83} .\overline {72} \overline {25} \\
  81 \\
  \,}} \right. } \\
  {\text{ }}2 \\
 $
Now lets bring the next group of numbers down
And now multiply 2 to the number 9 and we can use it as the next divisor
\[
  {\text{ 9}}{\text{.}} \\
   \Rightarrow 9\underline {\left| \!{\overline {\,

  \overline {83} .\overline {72} \overline {25} \\
  81 \\
  \,}} \right. } \\
  {\text{18y }}\left| \!{\overline {\,
 {272{\text{ }}} \,}} \right. \\
 \]
Now we need to find a number y such that multiplying 18y with y gives a number near to 272
So when we multiply 181 into 1 we get 181
When we multiply 182 into 2 we get 364
Hence the number nearer to 272 is 181
Therefore we get y to be 1
\[
  {\text{ 9}}{\text{.1}} \\
   \Rightarrow 9\underline {\left| \!{\overline {\,

  \overline {83} .\overline {72} \overline {25} \\
  81 \\
  \,}} \right. } \\
  {\text{181 }}\left| \!{\overline {\,

  272 \\
  181{\text{ }} \\
  \,}} \right. \\
  {\text{ }}\overline {091} \\
 \]
Now lets bring the next group of numbers down
And now multiply 2 to the last digit of the number 181 and we can use it as the next divisor
\[
  {\text{ 9}}{\text{.1}} \\
   \Rightarrow 9{\text{ }}\underline {\left| \!{\overline {\,

  \overline {83} .\overline {72} \overline {25} \\
  81 \\
  \,}} \right. } \\
  {\text{181 }}\left| \!{\overline {\,

  272 \\
  181{\text{ }} \\
  \,}} \right. \\
  {\text{182y }}\left| \!{\overline {\,
 {9125} \,}} \right. \\
 \]
Now we need to find a number y such that multiplying 182y with y gives a number near to 9125
So when we multiply 1821 into 1 we get 1821
When we multiply 1822 into 2 we get 3644
When we multiply 1823 into 3 we get 5469
When we multiply 1824 into 4 we get 7296
When we multiply 1825 into 5 we get 9125
Therefore we get y to be 5
\[
  {\text{ 9}}{\text{.15}} \\
   \Rightarrow 9{\text{ }}\underline {\left| \!{\overline {\,

  \overline {83} .\overline {72} \overline {25} \\
  81 \\
  \,}} \right. } \\
  {\text{181 }}\left| \!{\overline {\,

  272 \\
  181{\text{ }} \\
  \,}} \right. \\
  {\text{1825 }}\left| \!{\overline {\,

  9125 \\
  \underline {9125} \\
  0 \\
  \,}} \right. \\
 \]
Hence now we get the square root of 83.7225 to be 9.15

Therefore the correct option is A.

Note:
1) Here we get rid of the decimal in the calculation part just to avoid mistakes but students must be careful enough to write the decimal point while writing the final answer.
2) Square root of a whole number can also be found by using prime factorization.
3) Since we are given options we can even find the square of each option and select the answer but it consumes more time.