A Mathematical Puzzle: Unraveling the Coin Conundrum in a Small Office

The petty cash box of a small office contains P16.25. If there are twice as many 5-centavo coins as 25-centavo coins and as many 10-centavo coins as the 5-centavo coins and 25-centavo coins combined, how many coins of each type were there? This puzzle, regularly encountered in budget and inventory management, presents an interesting challenge to anyone who enjoys a mathematical problem. Let's break down the solution step-by-step.

Understanding the Problem

The problem involves a set of coin types: 5-centavo, 10-centavo, and 25-centavo coins. The total value of the coins is P16.25, which equals 1625 cents. We are given the relationships between the numbers of these coins: there are twice as many 5-centavo coins as 25-centavo coins and as many 10-centavo coins as there are total 5-centavo and 25-centavo coins combined.

Solving the Puzzle

To solve this problem, we will use algebraic expressions to represent the relationships between the numbers of the coins. Let's denote the number of 25-centavo coins by ( n ). This is the smallest number of any type of coin, so it forms the basis of our solution.

Step-by-Step Solution

Define the number of each type of coin: We start with n coins of 25 centavos.

Calculate the number of 5-centavo coins: According to the problem, there are twice as many 5-centavo coins as 25-centavo coins. Therefore, we have:

2n coins of 5 centavos.

Calculate the number of 10-centavo coins: The number of 10-centavo coins is three times the sum of the 5-centavo and 25-centavo coins. Therefore, we have:

3(n 2n) 3(3n) 9n coins of 10 centavos.

Calculating the Total Value

Next, we calculate the total value of the coins in terms of ( n ):

The value of the 25-centavo coins is: 25 cents × ( n ) 25n cents.

The value of the 10-centavo coins is: 10 cents × 9n 90n cents.

The value of the 5-centavo coins is: 5 cents × 2n 10n cents.

Adding these values together, we get the total value of the coins:

25n 90n 10n 125n cents.

Given that the total value is 1625 cents, we set up the equation:

125n 1625.

Solving for ( n ):

n 1625 / 125 13.

Calculate the Number of Each Coin

Using the value of ( n 13 ), we can now determine the number of each type of coin:

Number of 25-centavo coins: ( n 13 ).

Number of 5-centavo coins: ( 2n 2 times 13 26 ).

Number of 10-centavo coins: ( 9n 9 times 13 117 ).

Verification

Let's verify the solution by calculating the total value of the coins:

Value of 25-centavo coins: ( 25 times 13 325 ) cents.

Value of 10-centavo coins: ( 10 times 117 1170 ) cents.

Value of 5-centavo coins: ( 5 times 26 130 ) cents.

Total value: 325 1170 130 1625 cents P16.25.

Conclusion

The solution to the puzzle is 13 25-centavo coins, 26 5-centavo coins, and 117 10-centavo coins, which, when added together, sum up to P16.25.