Let's define the number of coins used:

• Let $x$ be the number of ₹10 coins.
• Let $y$ be the number of ₹5 coins.
• Let $z$ be the number of ₹2 coins.

We are given two pieces of information:

1. The total value of the coins is ₹120: $10x + 5y + 2z = 120$
2. The total number of coins is 25: $x + y + z = 25$

We now have two equations:

1. $10x + 5y + 2z = 120$
2. $x + y + z = 25$

Step 1: Solve one equation for one variable

Let's solve equation (2) for $z$: $z = 25 - x - y$

Step 2: Substitute into the first equation

Substitute $z = 25 - x - y$ into equation (1): $10x + 5y + 2(25 - x - y) = 120$

Step 3: Simplify the equation

$10x + 5y + 50 - 2x - 2y = 120$ $(10x - 2x) + (5y - 2y) + 50 = 120$ $8x + 3y + 50 = 120$ $8x + 3y = 70$

Step 4: Solve for integer values of $x$ and $y$

Now, we need to solve the equation $8x + 3y = 70$.

First, let's try different integer values for $x$:

• If $x = 8$: $8(8) + 3y = 70 \implies 64 + 3y = 70 \implies 3y = 6 \implies y = 2$ Substituting $x = 8$ and $y = 2$ into $z = 25 - x - y$: $z = 25 - 8 - 2 = 15$

So, the solution is $x = 8$, $y = 2$, and $z = 15$.

Conclusion:

The number of ₹10 coins used in the payment is 8.

Would you like further details or have any questions?

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Here are 5 related questions for practice:

1. What if the total payment was ₹150 instead of ₹120?
2. How many coins would be used if only ₹10 and ₹5 coins were available?
3. What if you had an additional coin denomination, say ₹1, in the problem?
4. What is the least number of coins needed to make a payment of ₹120 with ₹10 and ₹5 coins only?
5. How would the solution change if the total number of coins was 30?

Tip: When solving systems of equations with integer solutions, substituting values and simplifying can often lead to quicker results.