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63 Rh (1) The area of the floor is 3xy square feet ; the number of square yards of carpet required is {3xy}{9} = {xy}{3} (2) The perimeter of the room is 2(3x + y) feet ; the area of the walls is 2a(3x +y) square feet ; number of square yards of paper required is {2a(3x + y)}{9}.

Ex. 4. The digits of a number beginning from the left are a, b, c ; what is the number ?

Here c is the digit in the units' place ; b standing in the tens' place represents b tens ; similarly a represents a hundreds.

The number is therefore equal to a hundreds + b tens + c units = 100a+ 10 b + c.

If the digits of the number are inverted, a new number is formed which is symbolically expressed by 100c + 10b + a.

Ex. 5. What is (1) the sum, (2) the product of three consecutive numbers of which the least is n ?

The numbers consecutive to n are n+ 1, n + 2. the sum = n + (n + 1) + (n + 2) = 3 n + 3. And the product = n(n + 1) (n + 2).

We may remark here that any even number may be denoted by 2n, where n is any positive whole number; for this expression is exactly divisible by 2.

Similarly, any odd number may be denoted by 2n + 1 ; for this expression when divided by 2 leaves remainder 1.

EXAMPLES VIII. b.

1. Write four consecutive numbers of which x is the least. 2. Write three consecutive numbers of which y is the greatest. 3. Write five consecutive numbers of which x is the middle one. 4. What is the next even number after 2n ? 5. What is the odd number next before 2x + 1 ? 6. Find the sum of three consecutive odd numbers of which the middle one is 2n + 1.