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 kgFrom the ratio, we can express WI and WC as:
WI (2/5) × 30 kg 12 kg WC (3/5) × 30 kg 18 kgStep 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 kgIf we add 3 kg of iron to this mixture, we get:
Total Fe 6 kg 3 kg 9 kg Carbon 9 kgResulting in a 1:1 ratio of iron to carbon.
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