Question:
<p>The sum of the digits of a two-digit number is 15. The number obtained by interchanging the digits exceeds the given number by 9. The number is</p>
<p style="display:inline">87</p>
<p style="display:inline">96</p>
<p style="display:inline">69</p>
<p style="display:inline">78</p>
Step-by-Step Solution
Key Concept: Represent a two-digit number using its decimal expansion (10x + y) to translate the digit relationships into a system of linear equations.
<p>Let us assume the tens and the unit digits of the required number be x and y respectively<br />
<span class="math-tex">$\therefore$</span> Required number = (10x + y)<br />
According to the given condition in the question,<br />
we have<br />
x + y = 15 .....(i)<br />
By reversing the digits, we obtain the number = (10y + x)<br />
<span class="math-tex">$\therefore$</span> (10y + x) = (10x + y) + 9<br />
10y + x - 10x - y = 9<br />
9y - 9x = 9<br />
y - x = 1 .....(ii)<br />
Now, on adding (i) and (ii) we get:<br />
2y = 16<br />
<span class="math-tex">$\therefore$</span> y = <span class="math-tex">$\frac{16}{8}$</span> = 8<br />
Putting the value of y in (i), we get:<br />
x + 8 = 15<br />
x = 15 - 8<br />
x = 7<br />
<span class="math-tex">$\therefore$</span> Required number = (10x + y) = 10 <span class="math-tex">$\times$</span> 7 + 8 = 70 + 8 = 78</p>
Correct Answer: D