A person invested a total sum of ₹1900 in three different schemes of simple interest at 2%, 4%, and 5% per annum. At the end of one year, he got the same interest from all three schemes. What was the amount (in ₹) invested at 4%?
₹500
Let the amounts invested at 2%, 4%, and 5% be \(x\), \(y\), and \(z\) respectively, and let the common interest be \(k\).
Step 1 — equal interest in 1 year:
\(\dfrac{2x}{100} = \dfrac{4y}{100} = \dfrac{5z}{100} = k\)
So \(x = \dfrac{k}{2}\cdot 50 = 50k\,/\,1\)… Let's just write \(2x = 4y = 5z\), which gives:
\(x = \dfrac{2x}{2},\quad y = \dfrac{2x}{4} = \dfrac{x}{2},\quad z = \dfrac{2x}{5}\)
Step 2 — total invested:
\(x + \dfrac{x}{2} + \dfrac{2x}{5} = 1900\)
\(\dfrac{10x + 5x + 4x}{10} = 1900 \;\Longrightarrow\; \dfrac{19x}{10} = 1900 \;\Longrightarrow\; x = 1000\)
Step 3 — amount at 4%:
\(y = \dfrac{x}{2} = \dfrac{1000}{2} = 500\)
Hence the amount invested at 4% is ₹500 — option (2).
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