A bag contains 50p, 25p, and 10p coins in the ratio 5:9:4, amounting to â¹206. Find the number of coins of each type.

Points to Remember:

The problem involves solving a system of equations based on a given ratio and total amount.
Understanding ratios and their implications is crucial.
Careful calculation and attention to detail are necessary to avoid errors.

Introduction:

This question is a mathematical word problem requiring the application of ratio and proportion principles to solve for unknown quantities. We are given that a bag contains 50p, 25p, and 10p coins in the ratio 5:9:4, with a total value of â¹206. Our task is to determine the number of coins of each denomination. We will use algebraic methods to solve this problem. Note that â¹1 = 100p.

Body:

1. Defining Variables and Setting up Equations:

Let’s represent the number of 50p coins as 5x, the number of 25p coins as 9x, and the number of 10p coins as 4x. This is based on the given ratio 5:9:4. The total value of the coins in pence is:

50(5x) + 25(9x) + 10(4x) = 20600 (Since â¹206 = 20600 pence)

2. Solving the Equation:

Simplifying the equation:

250x + 225x + 40x = 20600
515x = 20600
x = 20600 / 515
x = 40

3. Calculating the Number of Coins:

Now that we have the value of x, we can calculate the number of coins of each type:

Number of 50p coins = 5x = 5 * 40 = 200
Number of 25p coins = 9x = 9 * 40 = 360
Number of 10p coins = 4x = 4 * 40 = 160

4. Verification:

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Let’s verify our solution by calculating the total value:

(200 * 50) + (360 * 25) + (160 * 10) = 10000 + 9000 + 1600 = 20600 pence = â¹206

This confirms our calculations are correct.

Conclusion:

In conclusion, the bag contains 200 coins of 50p, 360 coins of 25p, and 160 coins of 10p. This solution was obtained by setting up and solving a linear equation based on the given ratio and total value. The verification step ensures the accuracy of our findings. This problem highlights the practical application of ratio and proportion in everyday scenarios. Understanding these mathematical concepts is crucial for problem-solving in various fields, fostering a more analytical and logical approach to challenges. This approach emphasizes accuracy and precision in calculations, essential for building a strong foundation in quantitative reasoning.

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