**1) a + (a + 2) + (a + 4) = 45 **

**3a + 6 = 45 **

**3a = 39 **

**a = 13 **

__Ans: 13, 15, 17__

**2) ***Let one number be x.*

*Then the other number be = 3x*

**If 15 is added to both numbers, then one of the new numbers becomes twice that of the other new number.**

**When 15 is added to the numbers, the new numbers are x + 15 and 3x + 15**

**⇒ 2(x + 15) = 3x + 15**

**⇒ 2x + 30 = 3x + 15**

**⇒ 2x - 3x = 15 - 30 (Transposing 3x to the LHS and 30 to the RHS)**

**⇒ - x = - 15**

**⇒ x = 15**

**Therefore one of the number = 15**

**⇒ The other number = 3x**

**= 3 x 15**

**= 45**

__The two numbers are 15 and 45.__

**3) First Number at unit place is is x and tens place is y.**

**So original number is 10y + x **

**After interchanging digits number will be 10x + y**

**Number obtained by interchanging the digits exceed the given number by 27.**

**10y + x + 27 = 10x + y **

**10y - y + x - 10x + 27 = 0 **

**9y - 9x + 27 = 0**

**9 (3y - 3x + 9 = 0)**

**3y - 3x + 9 = 0 ---------(1) **

**The sum of digit of a 2 digit number is 9 **

**x + y = 9 ---------(2) **

**By elimination method:**

**Multiplying eq 1 by 1 3y - 3x = 9**

**Multiplying eq 2 by 3 3y + 3x = 27 **

**- 6x = -18 **

**x= - 18/ -6**

**x = 3**

**Substituting value in eq 2**

**x + y = 9**

**3 + y = 9**

**y = 6 **

__The number is 36.__

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