Question

The product of two 2 digit numbers is 1450. If the product of the ten’s place digits in these

numbers is 10 and the product of one’s place digits in these numbers is 40. Find the numbers.
​

Answer 1

Given :

Product of two 2 digit numbers = 1450

Product of ten's place digit = 10

Product of one's place digit = 40

To find :

Find the numbers .

Solution :

Let the first number be ( 10x$_1$ + y$_1$ ) and the second number be ( 10x$_2$ + y$_2$ ) .

Let x$_1$ be 2 and x$_2$ be 5 as the product of these two numbers are 10 .

Then , y$_1$ can be 8 or 5 and accordingly , the value of y$_2$ will be 5 or 8 , as the product of 8 and 5 is 40 .

now ,

( 10x$_1$ + y$_1$ ) * ( 10x$_2$ + y$_2$ ) = 1450

=> 100x$_1$x$_2$ + 10x$_1$y$_2$ + 10x$_2$y$_1$ + y$_1$y$_2$ = 1450

putting the assumed value of x$_1$ and x$_2$ , and y$_1$y$_2$ = 40 ,

=> 1000 + 20y$_2$ + 50y$_1$ + 40 = 1450

=> 20y$_2$ + 50y$_1$ = 410

=> 2y$_2$ + 5y$_1$ = 41

To statisfy the above relation y$_1$ will be 5 and y$_2$ will be 8 .

Thus , the numbers are 25 and 58 .

The numbers are 25 and 58 .

Answer 2

Given : The product of two 2 digit numbers is 1450. If the product of the ten’s place digits in these  numbers is 10 and the product of one’s place digits in these numbers is 40

To find : the numbers.

Step-by-step explanation:

Let say two numbers are

AB   & CD

Value of AB = 10A + B

Value of CD = 10C + D

(10A + B)(10C + D)  =  1450

=> 100AC + 10AD + 10BC + BD  = 1450    Eq1

product of the ten’s place digits in these  numbers is 10

=> AC  = 10

Substitute in Eq 1

=> 100(10)  + 10AD + 10BC + BD = 1450

=> 10AD + 10BC + BD = 450

Product of one’s place digits in these numbers is 40.

=> BD = 40

=> 10AD + 10BC + 40 = 450

=> 10AD + 10BC  = 410

=> AD + BC  = 41

AC = 10

BD = 40

AD + BC =  41

=> A(40/B)  + B(10/A)  = 41

=> 40A²  + 10B²  = 41AB

=>  40A² -  41AB + 10B² = 0

=> 40A²  - 25AB - 16AB + 10B² = 0

=> 5A(8A - 5B)  - 2B(8A - 5B) = 0

=> (5A - 2B) (8A - 5B) = 0

=> 5A - 2B = 0

=>  A = 2  , B = 5      or  A = 5 , B = 8

=> C = 5  , D  = 8      or   C = 2  , D = 5

AB  = 25    CD = 58

or

AB = 58    CD = 25

Hence two numbers are 25 & 58

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