An alloy contains only zinc and copper. One such alloy weighing 15 gm contains zinc and copper in the ratio of `2 : 3` by weight. If 10 gm of zinc is added then find what amount of copper has to removed from the alloy such that the final alloy has zinc and copper in the ratio of `4 : 1 `by weight.?

Answer

To solve the problem step by step, we will first analyze the given information and then perform the necessary calculations.

### Step 1: Determine the initial quantities of zinc and copper in the alloy.
The alloy weighs 15 grams and the ratio of zinc to copper is 2:3.

- Total parts = 2 + 3 = 5 parts
- Weight of 1 part = Total weight / Total parts = 15 gm / 5 = 3 gm

Now we can find the weight of zinc and copper:
- Weight of zinc = 2 parts = 2 * 3 gm = 6 gm
- Weight of copper = 3 parts = 3 * 3 gm = 9 gm

### Step 2: Add zinc to the alloy.
We are adding 10 gm of zinc to the initial 6 gm of zinc.

- New weight of zinc = 6 gm + 10 gm = 16 gm

### Step 3: Set up the equation for the new ratio.
Let x be the amount of copper to be removed. After removing x gm of copper, the weight of copper will be:

- New weight of copper = 9 gm - x gm

We want the final ratio of zinc to copper to be 4:1. Therefore, we can set up the equation:

\[ \frac{16}{9 - x} = \frac{4}{1} \]

### Step 4: Cross-multiply to solve for x.
Cross-multiplying gives us:

\[ 16 * 1 = 4 * (9 - x) \]

This simplifies to:

\[ 16 = 36 - 4x \]

### Step 5: Rearrange the equation to isolate x.
Rearranging the equation gives:

\[ 4x = 36 - 16 \]
\[ 4x = 20 \]
\[ x = \frac{20}{4} = 5 \]

### Conclusion: Amount of copper to be removed.
The amount of copper that needs to be removed is **5 grams**.