Question

Form a pair of linear equations in two variables for the following information.
“There are some $50$ paisa and some $25$ paisa coins in a bag. The total number of coins are $140$ and the value of coins is rupees $50$.”

Answer 1

Hint:Here, we have two unknowns, i.e., number of $25$ paisa coins and number of $50$ paisa coins. We will assume these as unknown variables. We have been given the total value of coins and the total number of coins. We will thus use these two given values to make a pair of linear equations in two variables.

Complete step-by-step answer:
Let the Number of $25$ paisa coins $ = x$
Let the Number of $50$ paisa coins $ = y$
Now, we have two unknowns. So, we will have to have two equations that will describe the relationships between the unknowns and, if asked, we can find the number of $25$ paisa and $50$ paisa coins after solving these two equations.
Let’s think about the information that are given to us in the problem. We are given the total number of coins in the bag. Also, we have been given the total value of coins. Each of these pieces of information will produce an equation.
“The total number of coins are $140$”:
$ \Rightarrow x + y = 140$ ………………………………… Eqn I
“the value of coins is rupees $50$”:
$ \Rightarrow $ Total value of $25$ paisa coins $ + $ Total value of $50$ paisa coins $ = 5000$
($\because 50$rupees $ = 5000$paisa)

Mathematically,
          $ \Rightarrow 25x + 50y = 5000$ $ \Rightarrow 25(x + 2y) = 5000$
$ \Rightarrow x + 2y = 200$ ………………………………… Eqn II
Hence, the pair of linear equations in two variables for the given information is:
$x + y = 140$
$x + 2y = 200$

Note:If we are asked to get the number of coins of both types, we can also use a direct formula as well. We can use a trick if this Two-Coin problem would have been asked in a competitive exam.
Alternative method:
Let total value of coins (in rupees) $ = V$
Let total number of coins $ = N$
Let value of coin_1 (in rupees) $ = a$
Let value of coin_2 (in rupees) $ = b$
Let the number of coin_1 $ = x$
Let the number of coin_1 $ = y$
Then,
Number of coin_1 $ = x$$ = \dfrac{{bN - V}}{{b - a}}$…………. Eqn I\[\]
Number of coin_2 $ = y$$ = N - x$…………. Eqn II
In this problem,
$V = 50$
$N = 140$
Let $x$ be the number of $50$ paisa (i.e., $0.50$ rupee) coins. Thus, $a = 0.50$.
Let $y$ be the number of $25$ paisa (i.e., $0.25$ rupee) coins. Thus, $b = 0.25$.
As per equation I,
Number of $50$ paisa coins $ = x$$ = \dfrac{{bN - V}}{{b - a}}$
\[ \Rightarrow x = \dfrac{{0.25 \times 140 - 50}}{{0.25 - 0.50}} = \dfrac{{35 - 50}}{{ - 0.25}} = \dfrac{{ - 15}}{{ - 0.25}} = \dfrac{{1500}}{{25}}\]$ = 60$
\[ \Rightarrow x = 60\]
As per equation II,
Number of $25$ paisa coins $ = y$$ = N - x$
\[ \Rightarrow y = 140 - 60 = 80\]
\[ \Rightarrow y = 80\]
$\therefore $ Number of $25$ paisa coins $ = 80$
$\therefore $ Number of $50$ paisa coins $ = 60$
Cross-check
Total Value (in Rupees) $ = $ no. of $25$ paisa coins $ \times $ $0.25$ $ + $ no. of $50$ paisa coins $ \times $ $0.50$
          $ = 80 \times 0.25 + 60 \times 0.50$
          $ = 20 + 30$
          $ = 50$
Thus, our answer is correct. (Hence proved)