Question

Babubhai borrows Rs. 4000 and agrees to repay with a total interest of Rs. 500 in 10 instalments, each instalment being 10 less than the preceding instalment by Rs. 10. What should be the first and the last instalment?
[a] 495, 405
[b] 400,500
[c] 365,315
[d] 400,450

Answer 1

Hint: Assume that the first instalment of Babubhai be a. Observe that the instalments made by Babubhai form an A.P with the first term as a and the common difference -10. Using ${{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$ find the total amount paid by Babubhai over ten instalments in terms of a. Using Interest = Amount paid – Amount borrowed find the total interest paid by Babubhai in terms of a. Equate this expression to 500 as Babubhai pays a net interest of Rs. 500. Hence form an equation in a. Solve the expression to get the value of a. Hence find the value of the first instalment. Using ${{a}_{n}}=a+\left( n-1 \right)d$, find the last instalment made by Babubhai.

Complete step-by-step answer:
Let the first instalment made by Babubhai be a.
Hence the instalments made by Babubhai are a, a-10, a-20,…, which forms an A.P with the first term as a and common difference -10.
Now we know that in an A.P ${{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]$
Hence the total amount(in Rs) paid by Babubhai over the course of 10 instalments $=\dfrac{10}{2}\left( 2a+9\left( -10 \right) \right)=5\left( 2a-90 \right)=10a-450$
Hence the interest over the total amount paid by Babubhai $=10a-450-4000=10a-4450$.
But given that the interest paid by Babubhai = Rs. 500.
Hence, we have
$10a-4450=500$
Adding 4450 on both sides, we get
$10a=4950$
Dividing by 10 on both sides, we get
$a=495$
Hence the first instalment paid by Babubhai = Rs. 495
Also, we know that ${{a}_{n}}=a+\left( n-1 \right)d$
Hence we have ${{a}_{10}}=495+\left( 9 \right)\left( -10 \right)=495-90=405$
Hence the first and the last instalment paid by Babubhai respectively are Rs. 495 and Rs. 405
Hence option [a] is correct.

Note: [1] In these types of questions we need to understand that the given question involves the formation of an A.P. Many students fail to see that the instalments form an A.P and hence are not able to solve the question. We also need to understand that we can completely determine an A.P using a,n and d. Hence we search for the missing value among a,n and d. Here I was missing. So we began to find the value of a. Once a was determined, the complete sequence was determined.