Chapter 3.1, Problem 10P

To determine

To Find: All possibilities for N such that the average of N, K, and 35 is 30.

Answer to Problem 10P

Final Answer: All the possible values of N satisfying the given condition are {50,41,32,23,14}

Explanation of Solution

Given Information: It is given that N is a two digit positive integer and on interchanging its digits we get an integer K such that the average of N, K, and 35 is 30.

Formula used: A two digit number is expressed as $10x+y$ if its unit digit is $y$ and tens digit is $x$ and on interchanging its digits it becomes $10y+x$

Average of 3 numbers a, b, and c (say) = $\frac{a+b+c}{3}$

Calculation: According to the question, we have

N= $10x+y$ ........ (1)

K= $10y+x$ ........(2)

The average of N, K, and 35 is 30

So $\frac{N+K+35}{3}=30$

From (1) and (2), we have

  $\frac{(10x+y)+(10y+x)+35}{3}=30$

Or  $11x+11y+35=90$

Or  $11x+11y=55$

Or  $x+y=5$ ...........(3)

From (3) we have the sum of digits of N is equal to 5

And since N is a two digit positive integer so x and y must be an integer but x cannot be zero.

So the possible values that N can have is 50, 41, 32, 23, 14.