Try this beautiful Pen & Note Books Problem from Algebra, from PRMO 2017.

## Pen & Note Books – PRMO 2017, Question 8

A pen costs $Rs 11$ and a notebook costs $Rs. 13 .$ Find the number of ways in which a person can spend exactly Rs.1000 to buy pens and notebooks.

- $9$
- $7$
- $11$

**Key Concepts**

Algebra

Equation

multiplication

## Check the Answer

Answer:$7$

PRMO-2017, Problem 8

Pre College Mathematics

## Try with Hints

Given A pen costs Rs.\(11\) and a note book costs Rs.\(13\)

$11 x+13 y=1000$………………….(1)

Can you now finish the problem ……….

Now $11 x+13 y=1000$

$\Rightarrow 11 x=1000-13 y=(1001-11 y)-(2 y+1)$

=$11(91-y)-(2 y+1)$

$\Rightarrow 11 | 2 y+1$

Let $2 y+1=11(2 k-1), k \in I^{+}$

$\Rightarrow y=11 \mathrm{k}-6$

therefore $11 x=11(97-11 k)-11(2 k-1)$

$\Rightarrow x=98-13 k$

But $x>0 \Rightarrow k<\frac{98}{13} \Rightarrow k \leq 7$

Can you finish the problem……..

But $x>0 \Rightarrow k<\frac{98}{13} \Rightarrow k \leq 7$

therefore for each $\mathrm{k} \in{1,2, \ldots ., 7},$ we get a unique pair $(\mathrm{x}, \mathrm{y})=(98-13 \mathrm{k}, 11 \mathrm{k}-6)$ satisfying equation

Hence 7 ways are possible.

## Other useful links

- https://www.cheenta.com/ordered-pairs-prmo-2019-problem-18/
- https://www.youtube.com/watch?v=VLyrlx2DWdA