Alloy Composition: Calculating the Required Amount of Iron to Achieve a 1:1 Ratio

Understanding Alloy Composition

Alloys are widely used in various industries due to their unique properties. A common problem in dealing with alloys is to determine the exact amount of a component, such as iron, to add to achieve a desired ratio between two components. Let's explore how to calculate the amount of iron needed to be added to an alloy to achieve a 1:1 ratio of iron to carbon.

Given Problem

Consider a particular type of alloy where the ratio of iron (Fe) to carbon (C) is 2:3. We want to find the amount of iron that should be added to 30 kg of this material to make the ratio of iron to carbon 1:1.

Step-by-Step Solution

Let's denote the amount of iron and carbon in 30 kg of the alloy by WI and WC, respectively.

Step 1: Determine the Initial Amounts

From the initial ratio 2:3, we can set up the following equations:

WI : WC 2 : 3 WI WC 30 kg

From the ratio, we can express WI and WC as:

WI (2/5) × 30 kg 12 kg WC (3/5) × 30 kg 18 kg

Step 2: Determine the Required Amount of Added Iron

Let's denote the amount of iron added by XI.

We need to find XI such that the new ratio of iron to carbon is 1:1:

(12 kg XI) : 18 kg 1 : 1

This simplifies to:

12 kg XI 18 kg

Therefore:

XI 18 kg - 12 kg 6 kg

Conclusion

To achieve a 1:1 ratio of iron to carbon in 30 kg of the alloy, we need to add 6 kg of iron.

Alternative Method

Alternatively, we can solve the problem by directly using the given compositions. If we start with 15 kg of alloy, the amounts of iron and carbon can be calculated as follows:

Fe (2/5) × 15 kg 6 kg C (3/5) × 15 kg 9 kg

If we add 3 kg of iron to this mixture, we get:

Total Fe 6 kg 3 kg 9 kg Carbon 9 kg

Resulting in a 1:1 ratio of iron to carbon.

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