Q.A company must replace a machine costing ₹1,00,000 at the end of 5 years. It decides to build a sinking fund by depositing an equal sum at the end of each year into an account earning 12% per annum compounded annually. Find the annual deposit.

Step 1 — amount the fund must reach

The scrap value of the old machine (₹2,00,000) offsets the cost of the new machine (₹10,00,000), so the fund must provide:

(The present cost ₹8,00,000 is extra data not needed for the accumulation.)

Step 2 — annuity accumulation factor at 7% for 20 years

Step 3 — annual sinking-fund provision

$A=\frac{8,00,000}{40.9957}\approx ₹19,514.$ …

Step 1 — Identify the sinking-fund structure

Equal annual investments $A$ must grow to a target $FV = ₹5,00,000$ over $n = 10$ years at $i = 10\% = 0.1$.

Step 2 — Future value of annuity factor

Step 3 — Solve for the annual deposit

Why the other options are wrong: (B) ₹3,137.27 and (D) ₹3,000.32 are off by a factor of 10; (C) ₹31,312.71 is a rounding/arithmetic slip on the same factor. …

Step 1 — Recognise the sinking fund

We need equal annual deposits that grow to a target future value — that is a sinking fund, i.e. an ordinary annuity whose future value is known.

Step 2 — Substitute the values

Step 3 — Solve for the instalment

$R = \frac{15,00,000}{57.274999} = ₹26,189.44$ …

Step 1 — Recognise the sinking fund (FV of annuity)

Equal yearly deposits compounding to reach a fixed future target is a sinking-fund problem. The accumulated value of an ordinary annuity is

Step 2 — Substitute the data

$FV=20{,}00{,}000$, $i=0.14$, $n=12$, $(1.14)^{12}=4.8179$.

Step 3 — Solve for the annual saving

$P=\dfrac{FV}{27.2707}=\dfrac{20{,}00{,}000}{27.2707}\approx ₹73{,}339$ …

Step 1 — The sinking-fund relation

Each year an equal amount $P$ is set aside and earns 6% compounded annually. After 10 years the deposits accumulate to the future value of an ordinary annuity:

Step 2 — Substitute the known values

Here the target $FV = ₹20{,}00{,}000$ and the factor $A(10, 0.06) = 13.180785$ is supplied.

Step 3 — Compute the deposit

$P = ₹1{,}51{,}736.03$ …

Step 1 — Recall the sinking-fund relation

The future value of $n$ equal deposits $D$ is $D\times A(n,i)$, where $A(n,i)=\dfrac{(1+i)^n-1}{i}$. Set this equal to the required corpus:

Step 2 — Substitute the given factor

Step 3 — Solve for the annual provision

$D=\frac{10{,}00{,}000}{15.937425}=62{,}745.4\approx\text{₹}62{,}745.$ …

Step 1 — Set up the accumulation equation

The future value of an ordinary annuity is $FV = P\times \dfrac{(1+i)^n-1}{i} = P\times A(n,i)$.

Step 2 — Solve for the annual deposit

Step 1 — FVIFA

turns an annuity into its future value: $FV=A\times FVIFA$.

Step 2 — Sinking fund factor

To accumulate a target $FV$, the required deposit is

The multiplier $\frac{i}{(1+i)^n-1}$ is the sinking fund factor.

Step 3 — Recognise the reciprocal

Step 1 — Identify the sinking fund relation

A constant year-end deposit $A$ for 10 years at 10% accumulates to

Step 2 — Solve for A