R	pays	₹100	to	P	with	₹5,	₹2	and	₹1	coins.	The	total	number	of	coins	used	for	paying	are	40.	What	is	the	number	of	coins	of	denomination	₹5	in	the	payment?	

Let the number of ₹5 coins be \( x \), the number of ₹2 coins be \( y \), and the number of ₹1 coins be \( z \).

We are given two conditions:
1. The total value of coins is ₹100:
   \[
   5x + 2y + z = 100
   \]
2. The total number of coins used is 40:
   \[
   x + y + z = 40
   \]

Now, we have a system of two equations:
1. \( 5x + 2y + z = 100 \)
2. \( x + y + z = 40 \)

To solve, subtract the second equation from the first:
\[
(5x + 2y + z) - (x + y + z) = 100 - 40
\]
\[
4x + y = 60
\]
This simplifies to:
\[
y = 60 - 4x
\]

Substitute \( y = 60 - 4x \) into the second equation:
\[
x + (60 - 4x) + z = 40
\]
\[
x + 60 - 4x + z = 40
\]
\[
-3x + z = -20
\]
\[
z = 3x - 20
\]

Now, substitute \( z = 3x - 20 \) into the first equation:
\[
5x + 2(60 - 4x) + (3x - 20) = 100
\]
Simplify the equation:
\[
5x + 120 - 8x + 3x - 20 = 100
\]
\[
0x + 100 = 100
\]
This equation is always true, so there are infinitely many solutions. However, the variables \( x \), \( y \), and \( z \) must all be non-negative integers. Thus, we solve for values of \( x \), \( y \), and \( z \) that satisfy this condition.

From \( z = 3x - 20 \), \( z \geq 0 \), so \( 3x \geq 20 \), meaning \( x \geq \frac{20}{3} \), or \( x \geq 7 \).

Checking integer values of \( x \):

- If \( x = 7 \), then:
  \[
  y = 60 - 4(7) = 60 - 28 = 32
  \]
  \[
  z = 3(7) - 20 = 21 - 20 = 1
  \]
  This gives a valid solution: \( x = 7 \), \( y = 32 \), \( z = 1 \).

Thus, the number of ₹5 coins is \( \boxed{7} \).

Would you like more details on the solving process?

Here are 5 additional questions to explore:
1. How would the solution change if the total amount was ₹200?
2. Can you solve for the number of coins if only ₹2 and ₹1 coins were used?
3. How can this method be applied to different coin denominations?
4. What if there were only 30 coins used to pay ₹100?
5. How can this system of equations be solved graphically?

**Tip:** When dealing with systems of equations, try simplifying by eliminating one variable to make solving easier.

R pays ₹100 to P with ₹5, ₹2 and ₹1 coins. The total number of coins used for paying are 40. What is the number of coins of denomination ₹5 in the payment?

Solve a system of equations to find the number of ₹5 coins used to pay ₹100 with a total of 40 coins. This problem involves linear equations and algebraic manipulation, suitable for students in Grades 8-10.

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