Question

If 378 coins consist of one rupee, 50 paise and 25 paise coin whose values are in ratio 13:11:7, then find the number of 50 paise coins.

Answer 1

Hint: The given question deals with the concept of Ratio and proportion. In order to find the number of 50 paise coins, we will at first assume the number of coins for three different variables. Then we will equate the given ratio with the number of coins multiplied to its corresponding value. After which we will use substitution to find the answer.

Complete step by step solution:
Let the number of one rupee coins\[ = x\],
And the number of 50 paise coins\[ = y\],
Also, the number of 25 paise coins\[ = z\].
Given that, \[x + y + z = 378 - - - - - (1)\]
Now, the given ratio of values of coins is \[13:11:7\].
Here, we know that 50 paisa \[ = \dfrac{1}{2}\] rupee
And 25 paisa is \[ = \dfrac{1}{4}\] rupee
Therefore we can write,
\[ \Rightarrow 13:11:7 = x \times 1:y \times \dfrac{1}{2}:z \times \dfrac{1}{4}\]
\[ \Rightarrow 13:11:7 = x:\dfrac{y}{2}:\dfrac{z}{4}\]
\[ \Rightarrow \dfrac{x}{{13}}:\dfrac{y}{{2 \times 11}}:\dfrac{z}{{4 \times 7}} = k\]

Therefore, from the above equation,
\[x = 13k\],\[y = 22k\] and \[z = 28k\]
Now, substituting these values in equation (1)
We get,
\[ \Rightarrow x + y + z = 378 = 13k + 22k + 28k = 378\]
Which gives,
\[ \Rightarrow 63k = 378\]
Therefore, \[k = \dfrac{{378}}{{63}}\]
Which gives, \[k = 6 - - - - - (2)\]
Now, we know that the value of 50 paise coins is \[y = 22k\].
Therefore, from equations (2)
\[y = 22k = 22 \times 6 = 132\] which is our required answer.

Hence, the number of 50 paise coins are $132$.

Note: The most important step of the given question is equating the ratio of values to variable k. By doing so we find the values of the assumed variables in terms of k and then we can easily substitute the value with the total number of coins to find the value of k.Almost, all questions of ratio and proportion related to coins and values can be solved by using the above used method.