A box has 210 coins of denominations one-rupee and fifty paise only. The ratio of their respective values is 13 : 11. The number of one-rupee coin is

Question:

A box has 210 coins of denominations one-rupee and fifty paise only. The ratio of their respective values is 13 : 11. The number of one-rupee coin is

A. 65
B. 66
C. 77
D. 78

Core Concept

This question is based on Ratio and Proportion, specifically relating to the total value of coins. The problem involves a mix of one-rupee coins and fifty paise coins. We need to use the given ratio of their values to find the number of one-rupee coins.

Key Concepts:

1. Ratio and Proportion: The question provides the ratio of the total values of one-rupee coins and fifty paise coins. We need to apply this ratio to find the required number of coins.
2. Value of Coins: The value of one-rupee coin is 1 rupee, and the value of fifty paise coin is 0.5 rupees (or 50 paise).

Step-by-Step Solution

Step 1: Define Variables

Let:

• $x$ be the number of one-rupee coins,
• $y$ be the number of fifty paise coins.

We know the following:

• The total number of coins is 210:

  $$x + y = 210$$

• The ratio of the total values of the one-rupee coins to the fifty paise coins is 13 : 11. The value of one-rupee coins is $1 \times x = x$ rupees, and the value of fifty paise coins is $0.5 \times y = 0.5y$ rupees. Therefore, the ratio is:

  $$\frac{x}{0.5y} = \frac{13}{11}$$

Step 2: Solve the Ratio Equation

Rewriting the equation for the ratio:

$$\frac{x}{0.5y} = \frac{13}{11} \quad \Rightarrow \quad \frac{x}{y} = \frac{13}{11} \times 0.5 = \frac{13}{22}$$

This implies:

$$x : y = 13 : 22$$

Step 3: Express $x$ and $y$ in Terms of a Common Variable

Let the common multiplier be $k$. Therefore, we can express $x$ and $y$ as:

$$x = 13k \quad \text{and} \quad y = 22k$$

Step 4: Substitute into the Total Number of Coins

We know that the total number of coins is 210, so:

$$x + y = 210 \quad \Rightarrow \quad 13k + 22k = 210$$ $$35k = 210$$ $$k = \frac{210}{35} = 6$$

Step 5: Calculate the Number of One-Rupee Coins

Now that we know $k = 6$, we can calculate $x$ (the number of one-rupee coins):

$$x = 13k = 13 \times 6 = 78$$

Correct Answer and Option

The correct answer is:
D. 78

Answer Reasoning

By applying the given ratio and using the total number of coins, we derived the number of one-rupee coins to be 78. The key to solving the problem is understanding how to set up a ratio involving total values and then translating that into a ratio of coin counts.

Explanation of Incorrect Options

• A. 65: This is incorrect because it does not match the result after solving the system of equations derived from the ratio and total number of coins.
• B. 66: This is incorrect because it is not the solution obtained after solving the equation for the number of one-rupee coins.
• C. 77: This is incorrect as the final number of one-rupee coins is 78, not 77.

Common Mistakes to Avoid

1. Ignoring the value of coins: Some students may incorrectly use the number of coins instead of considering their values in the ratio.
2. Incorrectly simplifying the ratio: It’s important to simplify ratios carefully. In this case, the simplification of the ratio was key to solving the problem.

Simplest Way to Solve (Shortcut or Insight)

A quicker way to solve this problem is by directly applying the ratio $x : y = 13 : 22$ and the total number of coins, then solving for $k$ in the equation $35k = 210$. This eliminates the need for extra steps and quickly gives the solution.

Visual Aid Suggestion

Here’s a simple table to visualize the breakdown of the calculation:

| Quantity               | One-Rupee Coins (x) | Fifty Paise Coins (y) | Total Coins (x + y) | Total Value (1x + 0.5y) |
|------------------------|---------------------|-----------------------|---------------------|-------------------------|
| Given Ratios           | 13 : 22             |                       | 210                 | 13 : 11                |
| Calculation of x and y | 13k                  | 22k                   | 210                 |                         |
| Result                 | 78                  | 132                   | 210                 |                         |

Explanation:

• Step 1: The table lists the ratios and the total number of coins.
• Step 2: It then shows how $x$ and $y$ are expressed as multiples of $k$.
• Step 3: The final calculation of $x = 78$ is shown clearly, confirming the solution.

This table is helpful in visualizing how the ratios and total coins work together to determine the number of one-rupee coins.

Application

Understanding ratios and proportions is crucial in various real-life scenarios, such as finance, business, and resource management. For example, when dividing profits or shares in a business partnership, it’s common to use ratios to ensure fair distribution based on initial investments or contributions. Similarly, this concept is applied in inventory management to divide different products based on their values or quantities.

Practice Questions

1. A box has 500 coins of denominations one-rupee and fifty paise only. If the ratio of the values is 5 : 4, then the number of one-rupee coins is:
   • A. 200
   • B. 250
   • C. 300
   • D. 350
     (Answer: B)
2. The ratio of the values of one-rupee coins to fifty paise coins in a box is 7 : 5. If the total number of coins is 240, find the number of one-rupee coins:
   • A. 112
   • B. 120
   • C. 126
   • D. 130
     (Answer: A)
3. A box has 120 coins of one-rupee and fifty paise. If the total value of the coins is 100 rupees, then the number of one-rupee coins is:
   • A. 40
   • B. 50
   • C. 60
   • D. 70
     (Answer: C)
4. In a box, the ratio of the number of one-rupee coins to fifty paise coins is 3 : 5. If the total value is 75 rupees, find the number of one-rupee coins:
   • A. 18
   • B. 20
   • C. 22
   • D. 24
     (Answer: B)
5. The total value of 120 coins consisting of one-rupee and fifty paise coins is 90 rupees. Find the number of one-rupee coins:
   • A. 50
   • B. 60
   • C. 70
   • D. 80
     (Answer: A)