A child gets 1 random coin out of Re 1, Rs 2, and Rs 5 coins daily. He starts on 1st April and at the end of the month, he has coins in the ratio 5 ∶ 4 ∶ 6. Find the value of the coins with him
Rs 86
Understanding the Coin Collection Problem
This problem involves a child collecting coins daily for a month and determining the total value based on the ratio of the different denominations received.
The child receives one coin randomly each day from three possible denominations: Re 1, Rs 2, and Rs 5. The collection period starts on April 1st and ends at the end of April. The month of April has 30 days. Therefore, the child collects a total of 30 coins.
At the end of the month, the ratio of the number of Re 1, Rs 2, and Rs 5 coins is given as 5 ∶ 4 ∶ 6.
Calculating the Number of Each Coin Denomination
Let the number of Re 1, Rs 2, and Rs 5 coins be represented by $5k$, $4k$, and $6k$ respectively, based on the given ratio. The total number of coins is the sum of the number of coins of each denomination.
Total number of coins = Number of Re 1 coins + Number of Rs 2 coins + Number of Rs 5 coins
We know the total number of coins is 30 (the number of days in April). So, we can write the equation:
$$5k + 4k + 6k = 30$$
Combining the terms on the left side:
$$15k = 30$$
Now, we can solve for the value of $k$:
$$k = \frac{30}{15}$$
$$k = 2$$
Now that we have the value of $k$, we can find the actual number of coins for each denomination:
- Number of Re 1 coins = $5k = 5 \times 2 = 10$
- Number of Rs 2 coins = $4k = 4 \times 2 = 8$
- Number of Rs 5 coins = $6k = 6 \times 2 = 12$
Let's quickly check if the total number of coins is 30: $10 + 8 + 12 = 30$. This matches the number of days the child collected coins.
Calculating the Total Value of the Coins
To find the total value of the coins, we multiply the number of coins of each denomination by its respective value and sum the results.
- Value from Re 1 coins = Number of Re 1 coins $\times$ Value of Re 1 coin
- Value from Re 1 coins = $10 \times \text{Rs } 1 = \text{Rs } 10$
- Value from Rs 2 coins = Number of Rs 2 coins $\times$ Value of Rs 2 coin
- Value from Rs 2 coins = $8 \times \text{Rs } 2 = \text{Rs } 16$
- Value from Rs 5 coins = Number of Rs 5 coins $\times$ Value of Rs 5 coin
- Value from Rs 5 coins = $12 \times \text{Rs } 5 = \text{Rs } 60$
Total value of coins = Value from Re 1 coins + Value from Rs 2 coins + Value from Rs 5 coins
Total value = Rs $10$ + Rs $16$ + Rs $60$
Total value = Rs $86$
Thus, the total value of the coins with the child at the end of April is Rs 86.
Revision Table: Coin Collection Summary
| Denomination | Ratio Part | Value per Coin | Number of Coins (Ratio Part × 2) | Total Value (Number × Value) |
|---|---|---|---|---|
| Re 1 | 5 | Rs 1 | $5 \times 2 = 10$ | $10 \times 1 = 10$ |
| Rs 2 | 4 | Rs 2 | $4 \times 2 = 8$ | $8 \times 2 = 16$ |
| Rs 5 | 6 | Rs 5 | $6 \times 2 = 12$ | $12 \times 5 = 60$ |
| Total | 15 | 30 | $10 + 16 + 60 = 86$ |
Additional Information: Ratio and Proportion Concepts
This problem uses the concept of ratios and proportions. A ratio expresses the relationship between two or more quantities. In this case, the ratio 5 ∶ 4 ∶ 6 tells us the relative number of coins of each type.
When the total number of items is known and they are distributed according to a ratio, we can find the actual number of items for each part of the ratio. We sum the parts of the ratio ($5+4+6=15$) and set this total equal to the total number of items ($15k = 30$). Solving for the constant $k$ allows us to scale the ratio parts to the actual numbers.
Understanding ratios is important in various mathematical and real-life scenarios, such as dividing quantities proportionally, scaling recipes, or analyzing survey results.
Important Questions from Simple Ratios
Half of the villagers of a certain village have their own houses. One-fifth of the villagers cultivate paddy. One-third of the villagers are literate. Four-fifth of the villagers are under 25 years of age. Which one of the following statements is certainly correct ?
A bag contains Rs. 420 in the form of 5 rupee, 2 rupee, and 1 rupee coins. The number of coins of 5 rupee, 2 rupee, and 1 rupee is in the ratio of 2 : 3 : 5. What is the total number of coins in the bag?
If P/Q = 8/9 and Q - P = 12, then what is the value of P?
Find the proportion equivalent to M : N : O : P if M : N = 8 : 15, N : O = 5 : 9 and O : P = 18 : 19 ?
A number is 20 percent smaller than a second number. However, the second number now increases by 20 percent with no change in the first number. What is the ratio of the two numbers now?
