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# I have a total of Rs 300 in coins of denomination Rs 1, Rs 2 and Rs 5. The number of Rs. 2 coins is 3 times the number of Rs 5 coins. The total number of coins is 160. How many coins of each denomination are with me?

Last updated date: 10th Aug 2024
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Hint:
Here, we will assume the number of Rs. 5 coins to be some variable. We will use the condition to find the values of Rs 2 and Rs 1 in terms of variables. We will multiply the number of coins by their denomination and add all of them together to form an equation. We will then solve the equation to get the value of the variable and hence, the number of coins of each denomination with the person.

Complete step by step solution:
According to the question,
Total number of coins $= 160$
Given total value of coins $= {\text{Rs}}300$
Now, it is given that the coins are in the denomination of Rs 1, Rs 2 and Rs 5.
Also, the number of Rs. 2 coins is 3 times the number of Rs. 5 coins
Let the number of Rs.5 coins be $x$.
Therefore, number of Rs. 2 coins $= 3x$
Now, the number of Rs 1 coins will be equal to the difference between the total number of coins and the sum of Rs. 5 and Rs 2 coins.
Hence, number of Rs.1 coins $= 160 - 4x$
Now, for the total value of coins, we will multiply the number of coins by their respective value.
Total value of coins $= 1\left( {160 - 4x} \right) + 2\left( {3x} \right) + 5\left( x \right)$…………………………………$\left( 1 \right)$
But according to the question,
Given the total value of coins$= {\text{Rs}}300$.
So, substituting the total value of coins in equation $\left( 1 \right)$, we get
$\Rightarrow 300 = 160 - 4x + 6x + 5x$
Adding and subtracting the like terms, we get
$\Rightarrow 7x = 140$
Dividing both sides by 7, we get
$\Rightarrow x = 20$
Therefore, the number of Rs 5 coins$= x = 20$
Also, the number of Rs 2 coins $= 3x = 3 \times 20$
Multiplying the terms, we get
$\Rightarrow$ The number of Rs 2 coins $= 60$
And, the number of Rs 1 coins $= 160 - 4x = 160 - 4\left( {20} \right)$
Multiplying the terms, we get
$\Rightarrow$ The number of Rs 1 coins $= 160 - 80 = 80$

Hence, the person has 20 coins of denomination Rs 5, 60 coins of denomination Rs 2 and 80 coins of denomination Rs 1.

Note:
A denomination means the unit value of financial instruments such as coins, currency notes, securities, bonds, etc. While solving the questions related to the denominations of a coin, we must know the difference between the face value and the value of money. This means that if we are given that we have a Rs 2 coin, then this is the face value of that coin. But, if we are given that we have 20 coins of the denomination Rs 2. This is the number of coins of that specific denomination. Hence, if we want to know what the value of money with us is then, we would multiply the face value of the denomination by the number of coins of that specific denomination. In this case,
The value of money $= 2 \times 20 = {\text{Rs}}40$