In which of the following options is the amount of gold identical in the two coins? (Pure gold is 24 carat)
Understanding Gold Purity (Carat)
Pure gold is defined as 24 carat. The carat system indicates the proportion of pure gold in an alloy. A coin or piece of jewelry that is less than 24 carat contains a mixture of gold and other metals.
To find the amount of pure gold in a metal with a given carat, we use the ratio of its carat value to 24. The formula to calculate the amount of pure gold is:
\(\text{Amount of Pure Gold} = \text{Total Weight of Coin} \times \frac{\text{Carat Value}}{24}\)
Using this formula, we can calculate the amount of pure gold present in each coin described in the options.
Calculating Gold in Each Coin
Let's apply the formula to the coins mentioned in each option:
| Option | Coin 1 Details | Coin 1 Gold Amount | Coin 2 Details | Coin 2 Gold Amount | Identical Gold Amount? |
|---|---|---|---|---|---|
| 1 | 24g coin of 22 carat | \(24 \text{ g} \times \frac{22}{24} = 22 \text{ g}\) | 22g coin of 24 carat | \(22 \text{ g} \times \frac{24}{24} = 22 \text{ g}\) | Yes |
| 2 | 22g coin of 22 carat | \(22 \text{ g} \times \frac{22}{24} = 22 \text{ g} \times \frac{11}{12} \approx 20.17 \text{ g}\) | 24g coin of 24 carat | \(24 \text{ g} \times \frac{24}{24} = 24 \text{ g}\) | No |
| 3 | 22g coin of 22 carat | \(22 \text{ g} \times \frac{22}{24} \approx 20.17 \text{ g}\) | 22g coin of 24 carat | \(22 \text{ g} \times \frac{24}{24} = 22 \text{ g}\) | No |
| 4 | 24g coin of 22 carat | \(24 \text{ g} \times \frac{22}{24} = 22 \text{ g}\) | 24g coin of 24 carat | \(24 \text{ g} \times \frac{24}{24} = 24 \text{ g}\) | No |
Comparing the Results
From the calculations shown in the table:
- In Option 1, the 24g coin of 22 carat contains 22g of pure gold, and the 22g coin of 24 carat also contains 22g of pure gold. The amounts are identical.
- In Option 2, the amounts of pure gold are approximately 20.17g and 24g, which are not identical.
- In Option 3, the amounts of pure gold are approximately 20.17g and 22g, which are not identical.
- In Option 4, the amounts of pure gold are 22g and 24g, which are not identical.
Therefore, the amount of gold is identical only in the coins described in Option 1.
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