# Ex.5.3 Q16 Arithmetic Progressions Solution - NCERT Maths Class 10

Go back to  'Ex.5.3'

## Question

A sum of $$\rm{Rs}\, 700$$ is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is $$\rm{Rs}\,20$$ less than its preceding prize, find the value of each of the prizes.

Video Solution
Arithmetic Progressions
Ex 5.3 | Question 16

## Text Solution

What is Known?

$$7$$ cash prizes are given, and each prize is $$\rm{Rs}\,20$$ less than its preceding prize.

What is Unknown?

Value of each of the prizes

Reasoning:

General form of an arithmetic progression is $$a,\left( {a + d} \right),\left( {a + 2d} \right),\left( {a + 3d} \right),\dots$$

Sum of the first $$n$$terms of an AP is given by $${S_n} = \frac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$$

Where $$a$$ is the first term, $$d$$ is the common difference and $$n$$ is the number of terms.

Steps:

Let the cost of 1st prize be $$x$$.

Then the cost of 2nd prize $$=x-20$$

And the cost of 3rd prize $$= x - 40$$

Prizes are $$x,{\rm{ }}\left( {x - 20} \right),{\rm{ }}\left( {x - 40} \right),\dots$$

By observation that the costs of these prizes are in an A.P., having common difference as $$−20$$ and first term as $$x$$.

\begin{align} a&=x \\ d& =-20 \\\end{align}

Given that, $${S_7} = 700$$

$${S_n}= \frac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$$

\begin{align}\frac{7}{2}\left[ {2x + \left( {7 - 1} \right)d} \right] &= 700\\\left[ {2x + \left( 6 \right) \times \left( { - 20} \right)} \right] &= 200\\x + 3 \times \left( { - 20} \right) &= 100\\x - 60 &= 100\\x &= 160\end{align}

Therefore, the value of each of the prizes was $$\rm{Rs}\,160,$$ $$\rm{Rs} \,140,$$ $$\rm{Rs}\, 120, \;\rm{Rs}\,100, \;\rm{Rs}\,80, \;\rm{Rs}\,60,$$ and $$\rm{Rs} \,40.$$

Learn from the best math teachers and top your exams

• Live one on one classroom and doubt clearing
• Practice worksheets in and after class for conceptual clarity
• Personalized curriculum to keep up with school