Question

A Two digit number can be obtained by either multiplying the sum of the digits by 8 and then subtracting 5 or by multiplying the difference of the digit by 16 and then adding 3. Find the number.
A) 100
B) 22
C) 76
D) 83

Answer 1

Hint: Break the numbers in tens and ones and then by given two property of the number in the question try to formulate two equations which can be used to find the values of the ones and tens digit.

Complete step-by-step answer:
Let the two digit number be 10x + y where x is the tens digit and y is the ones digit.
Now as per the question we can write the number as
\[\begin{array}{l}
 \Rightarrow 10x + y = 8(x + y) - 5\\
 \Rightarrow 10x + y = 8x + 8y - 5\\
 \Rightarrow 10x + y - 8x - 8y = - 5\\
 \Rightarrow 2x - 7y = - 5................................................(i)
\end{array}\]
 And also by the second property it can be written as
\[\begin{array}{l}
 \Rightarrow 10x + y = 16(x - y) + 3\\
 \Rightarrow 10x + y = 16x - 16y + 3\\
 \Rightarrow 10x + y - 16x - 16y = 3\\
 \Rightarrow - 6x + 17y = 3...............................................(ii)
\end{array}\]

Now for solving the equation by elimination method we have to multiply (i) by 17 and (ii) by 7, By doing so we get the following equations
\[\begin{array}{l}
 \Rightarrow 34x - 119y = - 85.........................................(iii)\\
 \Rightarrow - 42x + 119y = 21..........................................(iv)
\end{array}\]
Now by adding (iii) aand (iv) we get,
\[\begin{array}{l}
 \Rightarrow 34x - 119y - 42x + 119y = - 85 + 21\\
 \Rightarrow - 8x = - 64\\
 \Rightarrow 8x = 64\\
 \Rightarrow x = \frac{{64}}{8}\\
 \Rightarrow x = 8
\end{array}\]
So we have the tens digit as 8

Now putting the value of x in (i), We get
\[\begin{array}{l}
 \Rightarrow 2x - 7y = - 5\\
 \Rightarrow 2 \times 8 - 7y = - 5\\
 \Rightarrow 16 - 7y = - 5\\
 \Rightarrow - 7y = - 21\\
 \Rightarrow 7y = 21\\
 \Rightarrow y = 3
\end{array}\]
Therefore, the ones digit is 3

According to our assumption the number was 10x+y
i.e., \[\begin{array}{l}
 = 10 \times 8 + 3\\
 = 80 + 3\\
 = 83(ans)
\end{array}\]
Therefore option D is correct.


Note: Breaking the number in tens and ones i.e., 10x+y was the key step. Please be conscious of signs while solving the linear equations in 2 variables. One often makes mistakes there and then the part to correct the answer is diverted. Also I have used the elimination method while solving, you can use any other method for solving linear equations.