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Answer :
To determine the amount of silver needed to make 10k white gold, let's go through the steps:
1. Understanding 10k Gold:
- 10k gold means the alloy consists of 10 parts gold out of 24. This implies the rest is composed of other metals (such as silver).
2. Amount of Pure Gold Used:
- The jeweler is using 112.5 grams of pure gold.
3. Calculate the Total Mass of 10k Gold:
- Since 10 parts out of 24 is pure gold, the proportion of pure gold is [tex]\(\frac{10}{24}\)[/tex].
- To find the total mass of 10k gold that can be made with 112.5 grams of pure gold, divide the mass of pure gold by the proportion of pure gold:
[tex]\[
\text{Total mass of 10k gold} = \frac{112.5 \text{ g}}{\frac{10}{24}} = 270 \text{ g}
\][/tex]
4. Calculate the Mass of Silver Needed:
- The other components of the alloy, primarily silver in this case, make up the difference between the total mass of the 10k gold and the mass of pure gold.
- Subtract the mass of pure gold from the total mass of 10k gold to find the mass of silver needed:
[tex]\[
\text{Mass of silver} = 270 \text{ g} - 112.5 \text{ g} = 157.5 \text{ g}
\][/tex]
Thus, the amount of silver that needs to be added is 157.5 grams. However, when selecting from the given options, the closest match is option D, which is 52 grams. There seems to be a discrepancy between the calculated result and the available options, suggesting a mismatch. Please ensure you've interpreted all question details correctly or check if there are any additional constraints or typographical errors in the options or question context.
1. Understanding 10k Gold:
- 10k gold means the alloy consists of 10 parts gold out of 24. This implies the rest is composed of other metals (such as silver).
2. Amount of Pure Gold Used:
- The jeweler is using 112.5 grams of pure gold.
3. Calculate the Total Mass of 10k Gold:
- Since 10 parts out of 24 is pure gold, the proportion of pure gold is [tex]\(\frac{10}{24}\)[/tex].
- To find the total mass of 10k gold that can be made with 112.5 grams of pure gold, divide the mass of pure gold by the proportion of pure gold:
[tex]\[
\text{Total mass of 10k gold} = \frac{112.5 \text{ g}}{\frac{10}{24}} = 270 \text{ g}
\][/tex]
4. Calculate the Mass of Silver Needed:
- The other components of the alloy, primarily silver in this case, make up the difference between the total mass of the 10k gold and the mass of pure gold.
- Subtract the mass of pure gold from the total mass of 10k gold to find the mass of silver needed:
[tex]\[
\text{Mass of silver} = 270 \text{ g} - 112.5 \text{ g} = 157.5 \text{ g}
\][/tex]
Thus, the amount of silver that needs to be added is 157.5 grams. However, when selecting from the given options, the closest match is option D, which is 52 grams. There seems to be a discrepancy between the calculated result and the available options, suggesting a mismatch. Please ensure you've interpreted all question details correctly or check if there are any additional constraints or typographical errors in the options or question context.
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