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Common Information

$P$ divides his property among his four sons $A, B, C$ and $D$ after donating Rs 20,000 and $10\%$ of his remaining property. The amounts received by the last three sons are in $A.P.$ and the amount received by the fourth son is equal to the total amount donated.

The first son receives as his share Rs 20,000 more than the share of the second son. The last son received Rs 1 lakh less than the eldest son.

Q.

Common Information Question: 1/2

 What is the total donation made by $P$?

 A.

Rs 55,000

 B.

Rs 65,000

 C.

Rs 75,000

 D.

Rs 80,000

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Solution:
Option(D) is correct

All amounts are in thousands of rupees.

Let $P$'s total property be $(20+10p)$

The donation is $20+p$

The $4^{th}$ son's share is $20+p$ -------- (A)

The $1^{st}$ son's share is $120+p$ ------ (B)

The $2^{nd}$ son's share is $100+p$ -------- (C)

Now, as $4^{th},3^{rd}$ and $2^{nd}$ are in $AP$.

The $3^{rd}$ son's share is,

$=\dfrac{(\text{share of $4^{th}$ son})+ (\text{share of $2^{nd}$ son})}{2}$

$=\dfrac{(20+p)+(100+p)}{2}$

$=60+p$ -------- (D)

Now, Sum of all son's share + Total Donation  = Total Property

$\Rightarrow [(A)+(B)+(C)+(D)]+[20+p] $$=20+10p$

$\Rightarrow [(20+p)+(120+p)+(100+p) $$+(60+p)]=9p$

$\Rightarrow 300+4p=9p$ -------- (E)

$\Rightarrow p=60$.

$P$'s total donation is $20+p= 80,000$

Edit: For an alternative and detailed solution, check PRATYUSH ANAND's comment.

Edit 2: Updated solution based on various comments. Added explanation on how to reach equation (D). Also added explanation on how to calculate the value of $p$ {i.e. equation (E)}


(7) Comment(s)


Wajjahath
 ()

the question says donation is "20 and 10p" and not "20+p"


Shireen
 ()

$p$ is $10\%$ of remaining property (i.e. total property $=10p+20,000$) which is used to make calculations easier.

So, donation amount $=20+p$ is correct.


PRATYUSH ANAND
 ()

And total donation= 4th son's(D) share= Rs 80,000



PRATYUSH ANAND
 ()

Hi 2 all,

Let P's total Property be $x$.

And as per prob. indicates 1st son is eldest and 4th son is youngest.

then 2nd ,3rd and 4th sons amount in A.P.

i.e $a+2d$, $a+d$ and $a$ respectively.

Now as per question.

Total Donation $=20k+ 10\%(x-20k)$.

1st son share $=a+2d+20k.$

and 1'st son share - 4th son share $=100k( 1Lakh).$

i.e $(a+2d +20k)-(a)=100k$

$\Rightarrow d=40k.$

1st son's share $=a+ 2d +20k=a+100k$

2nd son's share $=a+ 2d =a+80k$

3rd son's share $=a+d =a+40k$

4th son's share $=a$

4th son share = total donation

i.e $a=20k+ 10/%(x-20k)$

$\Rightarrow x-10a=-180k$ -------(i)

(1st+2nd+3rd+4th)sons share = (Total Property) - (Total Donation)

i.e $4a+220k= x- [20k+ 10\%(x-20k)]$

$\Rightarrow 9x-40a=2380k$ ----------(ii)

On solving (i) and (ii), we will get,

$x=620k$ and $a=80k$

Hence, P's Total Property is Rs 6,20,000.

1st son's(A) share Rs 1,80,000

2nd son's(B) share Rs 1,60,000

3rd son's(C) share Rs 1,20,000

4th son's(D) share Rs 80,000



Shashank
 ()

How did we get the share of 3rd son and how we got $300 + 4p=9p$


Sujal Padhiyar
 ()

Same question, How did we get the share of 3rd son and how we got $300 + 4p=9p$?

Deepak
 ()

Updated the solution to make it clearer. Hope it's fine now.