Question

What is meant by playing a number

Answer 1

A number is said to be in a generalized form if it is expressed as the sum of the product of its digits with their respective place values.

Thus, a two-digit number having a and b as its digits at the tens and the ones places respectively is written in the generalized form as 10a + b, i.e., in general, a two-digit number can be written as 10a + b, where ‘a’ can be any of the digits from 1 to 9 and ‘b’ can be any of the digits from 0 to 9.

Similarly, a three-digit number can be written in the generalized form as 100a + 10b + c, where ‘a’ can be any one of the digits from 1 to 9 while ‘b’ and ‘c’ can be any of the digits from 0 to 9.

For example:

The generalized forms of a few numbers are given below:

56 = 10 × 5 + 6;

37 = 10 × 3 + 7;

90 = 10 × 9 + 0;

129 = 100 × 1 + 2 × 10 + 9;

206 = 100 × 2 + 10 × 0 + 6;

700 = 100 × 7 + 10 × 0 + 0.

Word problems on playing with numbers:

1. What is the original number, if the sum of the digits of a two-digit number is seven. By interchanging the digits is twenty seven more than the original number?

Solution:

Let the original number be 10a + b.

Then, ‘a’ is the tens digit and ‘b’ is the units digit.

Since the sum of the digits is 7,

Therefore a + b = 7,

i.e., b = 7 - a..

So, the original number is 10a + (7 - a).

Therefore, the number obtained by interchanging the digits is

10(7 - a) + a,

and so we have {10(7 - a) + a} — {10a + (7 - a)} = 27.

Solving this equation, we get

a = 2.

And so, b = 7 - 2

= 7 - 2

= 5.

Hence, the original number is 10a + b = 20 + 5 = 25.