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10 coins are four times the number of [tex]₹ 5[/tex] coins, eight times the number of [tex]₹ 2[/tex] coins, and sixteen times the number of [tex]₹ 1[/tex] coins. If all the money in the money box adds up to [tex]₹ 555[/tex], find the number of [tex]₹ 10[/tex] coins in it.

Hint: Let the number of [tex]₹ 10[/tex] coins be [tex]x[/tex]. Then, [tex]10x + 5\left(\frac{x}{4}\right) + 2\left(\frac{x}{8}\right) + 1\left(\frac{x}{16}\right) = 555[/tex].



Answer :

GinnyAnswer
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].

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