Shaan got a total of Rs. 912 in the denomination of equal numbers of Rs. 1, Rs. 5 and Rs. 10 coins. How many coins do Shaan possess?
171
Understanding the Coin Denomination Problem
The question describes a scenario where Shaan has a collection of coins totaling Rs. 912. The unique condition is that Shaan has an equal number of coins for three different denominations: Rs. 1, Rs. 5, and Rs. 10.
Our goal is to find the total number of coins Shaan possesses.
Setting Up the Equation for Total Value
Let's denote the number of coins for each denomination as 'n'. Since Shaan has an equal number of Rs. 1, Rs. 5, and Rs. 10 coins, he has:
- 'n' number of Rs. 1 coins
- 'n' number of Rs. 5 coins
- 'n' number of Rs. 10 coins
The total value from each type of coin can be calculated by multiplying the number of coins by their respective denomination value:
- Value from Rs. 1 coins = $n \times 1 = n$
- Value from Rs. 5 coins = $n \times 5 = 5n$
- Value from Rs. 10 coins = $n \times 10 = 10n$
The total value of all the coins is the sum of these individual values, which is given as Rs. 912. So, we can write the equation:
$$ \text{Total Value} = (\text{Value from Rs. 1 coins}) + (\text{Value from Rs. 5 coins}) + (\text{Value from Rs. 10 coins}) $$
$$ 912 = n + 5n + 10n $$
Solving for the Number of Coins of Each Denomination
Now, we simplify the equation to find the value of 'n':
$$ 912 = n + 5n + 10n $$
Combine the terms on the right side:
$$ 912 = (1 + 5 + 10)n $$
$$ 912 = 16n $$
To find 'n', divide both sides of the equation by 16:
$$ n = \frac{912}{16} $$
Let's perform the division:
| Step | Calculation |
|---|---|
| Divide 912 by 16 | $912 \div 16$ |
| 16 goes into 91 five times (16 * 5 = 80) | $91 - 80 = 11$. Bring down 2, making it 112. |
| 16 goes into 112 seven times (16 * 7 = 112) | $112 - 112 = 0$. Remainder is 0. |
So, $n = 57$.
This means Shaan has 57 coins of Rs. 1, 57 coins of Rs. 5, and 57 coins of Rs. 10.
Calculating the Total Number of Coins
The question asks for the total number of coins Shaan possesses. This is the sum of the number of coins of each denomination:
$$ \text{Total Coins} = (\text{Number of Rs. 1 coins}) + (\text{Number of Rs. 5 coins}) + (\text{Number of Rs. 10 coins}) $$
$$ \text{Total Coins} = n + n + n $$
$$ \text{Total Coins} = 3n $$
Substitute the value of n = 57:
$$ \text{Total Coins} = 3 \times 57 $$
Now, multiply 3 by 57:
| Calculation | Result |
|---|---|
| $3 \times 50$ | 150 |
| $3 \times 7$ | 21 |
| $150 + 21$ | 171 |
So, the total number of coins is 171.
Verifying the Total Value
Let's quickly check if 57 coins of each denomination add up to Rs. 912:
- Value from 57 Rs. 1 coins = $57 \times 1 = 57$
- Value from 57 Rs. 5 coins = $57 \times 5 = 285$
- Value from 57 Rs. 10 coins = $57 \times 10 = 570$
Total Value = $57 + 285 + 570 = 912$. This matches the given total value, confirming our calculation for 'n' is correct.
Therefore, Shaan possesses a total of 171 coins.
Revision Table: Key Concepts
| Concept | Explanation |
|---|---|
| Denomination | The face value of a coin or currency note (e.g., Rs. 1, Rs. 5, Rs. 10). |
| Equal Number of Coins | Having the same count of items for different categories. In this problem, the count of Rs. 1, Rs. 5, and Rs. 10 coins is identical. |
| Total Value | The sum obtained by adding the values of all items. Calculated as (Number of items) × (Value per item). |
| Algebraic Equation | A mathematical statement showing two expressions are equal, often used to represent relationships and solve for unknown quantities. |
Additional Information: Solving Coin Problems
Coin problems often involve setting up equations based on the number of coins and their total value. Here are common types and approaches:
- Problems with Equal Numbers: As seen here, if the number of coins of each type is the same, represent this number by a single variable (e.g., 'n') and form an equation based on the total value.
- Problems with Different Numbers but Known Relationships: Sometimes, you might be told there are twice as many Rs. 5 coins as Rs. 1 coins, or a certain number more of one type than another. Use variables to represent the counts based on these relationships (e.g., if Rs. 1 coins are 'x', Rs. 5 coins are '2x' or 'x+k').
- Problems with a Fixed Total Number of Coins: If the total count of coins is given, and you have two types, you can represent the count of one type as 'x' and the count of the other type as '(Total Count - x)'. Then, form a value equation.
In all cases, the core idea is to translate the word problem into mathematical equations and solve for the unknown quantities.
Important Questions from Linear Equation in 1 Variable
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