GATE Civil 2024 Set 1 | GA | Question: 2

Question

In a locality, the houses are numbered in the following way:

The house-numbers on one side of a road are consecutive odd integers starting from $301$, while the house-numbers on the other side of the road are consecutive even numbers starting from $302$. The total number of houses is the same on both sides of the road.

If the difference of the sum of the house-numbers between the two sides of the road is $27$, then the number of houses on each side of the road is

• A. $27$
• B. $52$
• C. $54$
• D. $26$

Comments

easy way:

Answer 1

let's say both the side has $n$ houses

$S_1= 301, 303,... , (300+2n-1)$

$S_2= 302,304,..., (300+2n)$

$\sum S_2 -\sum S_1= 27$

$(302 + 304+ 306+ \dots+n )-(301 + 303+ 305 + .. +n )  =27$

$(302-301) + (304-303)+ \dots+ (300+2n-300-2n+1)=27$

$\underbrace{1+ 1+1+1+1+\dots +1}_{\text{n times}}$ = $27$

$n.1 = 27$

now from here we can see that $n$ must be $27$

Comments

By seeing the question, we can immediatley sense that:

if only 1 house on each side, then difference is 1 i.e., (302-301)

if 2 houses on each side, then difference is 2 i.e., (302-301) + (304-303)

if 3 houses on each side, then difference is 3 i.e., (302-301) + (304-303) + (306-305)

$\vdots$

if 27 houses on each side, then difference is 27.

---

Solving by summation formula,

Let the number of last house on one side be $x$, then the number on last house on the other side will be $x+1$;

$\implies \dfrac{n}{2}(302 + x+1) -\dfrac{n}{2}(301+x)=27$

$\implies n(2)=27*2$

$\implies n=27$

Answer 2

Let try option by option. 

Option A: 27

The houses starting with 301 will go on till 353 that is {301 , 303 , 305 , 307 , 309 ...... , 353} . The summation of the odd numbered houses will be 8829.

The houses starting with 302 will go on till 354 that is {302 , 304 , 306 , 308 , 310 ...... , 354} . The summation of the even numbered houses will be 8856.

The difference will be 8856-8829 that is 27. Therefore, the Correct Answer is Option A) 27 for this question.

 

PS: No need to check other options.

Comments

The reason for answer is 27 is not included.

---

Choosing option A: 27 and verifying whether it is correct or not - is it the procedure you are following in the answer ?

---

@Shaik Masthan yeahh may be he is doing the same that u r saying..

---

@Thadymademe,

The houses starting with 301 will go on till 353

How you know it will end at 353 ? Still the answer needs an edit. 

 

By the way, is there anything wrong in the answer posted @shiva ?

---

yeah idk why he downvoted my ans lol :)

---

No bro,  answer is correct .

@Thadymademe  got the last digit 353 and 354 on both sides by number of terms/house no on the both side are 27.

an=301+(27-1).2

an= 301+52

an=353

 

 

as usual he got on other side last digit/house no 354

 

---

After my comment, that answer has been edited by me with the explanation. @Ankush भैया

Answer 3

When an Arithmetic Progression (AP) series is multiplied/divided by a constant or a constant is added/subtracted to/from each term of the series, the resulting series remains an AP

Thus, from the given series, if we subtract 300 from each term, 
We get 1,3,5,7... on one side of the road and 2,4,6,8... on other side of the road.
Number of houses on both sides is equal.
Thus, let number of houses be n.

sum of first n odd natural numbers  = n²
sum of first n even numbers  = n(n+1)

Difference of the sum is given 27.
Thus, n(n+1) -  n² = 27
Solving , we get n = 27.