Introduction to Coin Value Problems and the Use of System Equations

Coin value problems are common in mathematics, often used to exercise logical and algebraic skills. One classic example involves determining the number of different denominations of coins that add up to a specific value. This article aims to break down one such problem involving 20 paise and 25 paise coins totalling a sum of Rs. 72 using the mathematical tool of a system of equations.

Problem Statement

A total of 325 coins, made up of 20 paise and 25 paise coins, sum up to Rs. 72. To find out the number of 25 paise coins, we can use a system of equations to solve this problem.

Solving the Problem

Let's denote x as the number of 20 paise coins and y as the number of 25 paise coins. The problem can be formalized into the following two equations:

The total number of coins: x y 325 The total value of the coins in rupees where 20 paise 0.2 rupees and 25 paise 0.25 rupees: 0.2x 0.25y 72

To simplify the second equation, we can multiply by 100 to remove the decimals:

2 25y 7200

Now we have the system of equations:

x y 325 2 25y 7200

To solve for x, we can express it in terms of y from the first equation:

x 325 - y

Substituting this into the second equation gives:

20(325 - y) 25y 7200

Expanding and simplifying:

6500 - 20y 25y 7200

6500 5y 7200

5y 7200 - 6500

5y 700

y 140

Thus, the number of 25 paise coins is y 140.

Substitute y 140 back into the first equation to find x:

x 140 325

x 325 - 140

x 185

Final Answer: The number of 25 paise coins is 140.

Laboratory Style Solution

Here's another method to solve this problem. Let n be the number of 20 paise coins and 324-n be the number of 25 paise coins. The given equations become:

0.2n 0.25(324 - n) 71

20n 25(324 - n) 7100

20n 8100 - 25n 7100

1000 5n

n 200

Then, the number of 25 paise coins is 324 - 200 124.

Checking: 200 124 324 and 0.20(200) 0.25(124) 40 31 71.

Answer: Number of 25 paise coins 124.

Another Approach

Let 20 paise coins x and 25 paise coins 324 - x. The equation becomes:

0.2x 0.25(324-x) 71

2 25(324-x) 7100

2 8100 - 25x 7100

-5x -1000

x 200

20 paise coins 200

25 paise coins 324 - 200 124

Answer: 20 paise coins 200, 25 paise coins 124.

Yet Another Method

Let x and y be the respective number of coins of 20 paise and 25 paise to make a sum of Rs 71.

225y 7100 Eq.1 xy 324 Eq.2

Multiplying both sides of Eq.2 by 20:

220y 6480 Eq.3

Subtracting Eq.3 from Eq.1:

5y 620

y 124

Hence, the number of 25 paise coins 124.

Conclusion

This article has demonstrated several methods to solve a typical coin value problem. Using a system of equations allows for systematic and accurate solutions. Whether you use substitution, elimination, or manipulation of coefficients, the core concept remains the same - solving a system of linear equations. The discussed examples offer practical applications of algebra, enhancing problem-solving skills in mathematical settings.