To solve this problem, let's follow the given hint. We'll set up and solve the equation step by step.
1. Define the Variables:
Let the number of ₹ 10 coins be [tex]\( x \)[/tex].
2. Relationships Between Coins:
- The number of ₹ 5 coins is [tex]\( \frac{x}{4} \)[/tex].
- The number of ₹ 2 coins is [tex]\( \frac{x}{8} \)[/tex].
- The number of ₹ 1 coins is [tex]\( \frac{x}{16} \)[/tex].
3. Total Value of the Coins:
The total monetary value of the coins can be expressed as:
- Value from ₹ 10 coins: [tex]\( 10x \)[/tex]
- Value from ₹ 5 coins: [tex]\( 5 \times \frac{x}{4} \)[/tex]
- Value from ₹ 2 coins: [tex]\( 2 \times \frac{x}{8} \)[/tex]
- Value from ₹ 1 coins: [tex]\( 1 \times \frac{x}{16} \)[/tex]
4. Set Up the Equation:
According to the problem, the sum of the values of all the coins is ₹ 555. Therefore, the equation is:
[tex]\[
10x + 5\left(\frac{x}{4}\right) + 2\left(\frac{x}{8}\right) + 1\left(\frac{x}{16}\right) = 555
\][/tex]
5. Simplify the Equation:
Let's simplify each term:
[tex]\[
10x + \frac{5x}{4} + \frac{2x}{8} + \frac{x}{16}
\][/tex]
Combine the terms by converting everything to have a common denominator of 16:
[tex]\[
10x = \frac{160x}{16}
\][/tex]
[tex]\[
5 \left( \frac{x}{4} \right) = \frac{5x}{4} = \frac{20x}{16}
\][/tex]
[tex]\[
2 \left( \frac{x}{8} \right) = \frac{2x}{8} = \frac{4x}{16}
\][/tex]
[tex]\[
1 \left( \frac{x}{16} \right) = \frac{x}{16}
\][/tex]
Now add these fractions:
[tex]\[
\frac{160x}{16} + \frac{20x}{16} + \frac{4x}{16} + \frac{x}{16} = 555
\][/tex]
Combining the numerators:
[tex]\[
\frac{160x + 20x + 4x + x}{16} = 555
\][/tex]
[tex]\[
\frac{185x}{16} = 555
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
Isolate [tex]\( x \)[/tex] by multiplying both sides of the equation by 16:
[tex]\[
185x = 555 \times 16
\][/tex]
[tex]\[
185x = 8880
\][/tex]
Divide both sides by 185:
[tex]\[
x = \frac{8880}{185}
\][/tex]
[tex]\[
x = 48
\][/tex]
Thus, the number of ₹ 10 coins is [tex]\( \boxed{48} \)[/tex].
