The sum of a 3-digit number and a 1-digit number is 217. The product of the numbers is 642. If one number is between 200 and 225, what are the numbers?

Let, the numbers be "x" and "y". Consider "x" as the triple digit number. Now, according to the question, x + y = 217..........................equation (1)
x * y = 642............................equation (2)
Now, Taking equation (2), x * y = 642 y = 642 / x..................................equation (3)
Now, Taking equation (1), x + y = 217
Substituting the value of y from equation (3), we get,
x + x 642 ​ = 217
x x ∗ x + 642 ​ = 217
x 2 + 642 = 217 ∗ x
x 2 + 642 = 217 x
x 2 + 642 − 217 x = 0
x 2 − 217 x + 642 = 0
x 2 − 3 x − 214 x + 642 = 0
x ( x − 3 ) − 214 ( x − 3 ) = 0
( x − 3 ) ( x − 214 ) = 0
Using zero product property, EITHER, x - 3 = 0 x = 3 OR, x - 214 = 0 x = 214 Since, "x" is the triple digit number, x = 214. Now, Taking equation (2), x * y = 642 Substituting the value of "x" in the equation, we get, (214) * y = 642 y = 642 / 214 y = 3 **
** So, the numbers are 214 and 3.

Answered by rabinshrestha41 | 2024-06-10

The two numbers that satisfy the conditions are 214 (the 3-digit number) and 3 (the 1-digit number). Their sum is 217, and their product is 642, which meets the criteria given in the problem. The solution confirms that 214 falls between 200 and 225, as required.
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Answered by rabinshrestha41 | 2024-10-01