Question

In a cricket tournament 16 school teams participated. A sum of Rs.8000 is to be awarded among themselves as prize money. If the last placed team is awarded Rs.275 in prize money and the award increases by the same amount for successive finishing places, amount will the first place team receive is
a) Rs720
b) Rs725
c) Rs735
d) Rs780

Answer 1

Hint: We will first assume that the team at 1 place will get Rs.a. We will form a series of AP in which the common difference will be d. We get first relation from AP as ${{T}_{1}}=a+\left( n-1 \right)d$ have ‘n’ = Number of teams is 16. We will also use formula to find the sum of AP which is ${{S}_{n}}=\dfrac{n}{2}\left\{ 2a+\left( n-1 \right)d \right\}$, from this we will get a second relation. Using these two relations, we will find the value of ‘a’.

Complete step-by-step answer:
It is given in the question that in a cricket tournament there are 16 teams participating. A sum of Rs8000 is to be awarded among all teams of prize money. It is also given that the last place team is awarded Rs275 and the award increases by the same amount for successive finishing places, then we have to find the amount that the first place team received.
Let us assume that the first place team got Rs.a. As it is clear that the amount of money is increased by the same amount for successive finishing places. So, we can say that the amounts are forming AP series.
Let us assume that the constant amount be Rs.d. So, we get an initial amount = Rs275.
\[\text{Number of teams}=16\] and
Sum of all the prizes to be distributed $\left( {{S}_{n}} \right)$ = Rs. 8000. Using the formula ${{T}_{1}}=a+\left( n-1 \right)d$, we have $n=16$ and ${{T}_{1}}=275$, therefore,
$275=a+\left( 16-1 \right)d$, that is,
$275=a+15d$.
We know that sum of all the terms of an AP is given by ${{S}_{n}}=\dfrac{n}{2}\left\{ 2a+\left( n-1 \right)d \right\}$, we have ${{S}_{n}}=8000$, $n=16$, therefore, putting these values, we get,
$8000=\dfrac{16}{2}\left\{ 2a+\left( 16-1 \right)d \right\}$, solving further, we get,
$8000=8\left\{ 2a+15d \right\}$ or
$1000=\left\{ 2a+15d \right\}$ .
 Now, we got 2 relations and 2 variables. Therefore subtracting $275=a+15d$ from $1000=\left\{ 2a+15d \right\}$, we get the value of a as follows,
$1000-275=\left\{ 2a+15d \right\}-\left( a+15d \right)$, therefore on solving we get,
$2a-a+15d-15d=725$, that is,
$a=725$.
So, the team which comes in first place in the tournament will get Rs725 as prize money.
Therefore, option b) is correct.

Note: Students may miss-understand this question as they may think that the last team got Rs275 so, the team at 15^th place will get Rs275 + Rs275 = Rs550 and similarly the team at 14^th place will get Rs825 and so on. But this is wrong because if we calculate the sum of prize money for the first team, we will get it as Rs4400 and this is not possible considering the options and the total prize money of Rs8000.