Savings Calculation for Public Provident Fund

This problem involves distributing a total amount of savings between two investment schemes, Kisan Vikas Patra (KVP) and Public Provident Fund (PPF), based on a given ratio. We need to find the amount saved specifically in the Public Provident Fund.

Setting Up the Equations

Let's denote Manoj's savings in Kisan Vikas Patra as K and his savings in Public Provident Fund as P.

From the question, we are given two pieces of information:

1. One-third of his savings in Kisan Vikas Patra is equal to one-half of his savings in Public Provident Fund. This can be written as an equation:

   $$ \frac{1}{3} \times K = \frac{1}{2} \times P $$

2. His total savings amount to Rs. 1,50,000. This gives us the second equation:

   $$ K + P = 1,50,000 $$

Solving for Public Provident Fund Savings

Our goal is to find the value of P.

First, let's simplify the relationship between K and P from the first equation:

$$ \frac{K}{3} = \frac{P}{2} $$

To express K in terms of P, we can multiply both sides by 3:

$$ K = 3 \times \frac{P}{2} $$

$$ K = \frac{3P}{2} $$

Now, we can substitute this expression for K into the second equation (total savings):

$$ \left( \frac{3P}{2} \right) + P = 1,50,000 $$

To solve for P, we first combine the terms involving P. We need a common denominator:

$$ \frac{3P}{2} + \frac{2P}{2} = 1,50,000 $$

$$ \frac{3P + 2P}{2} = 1,50,000 $$

$$ \frac{5P}{2} = 1,50,000 $$

Next, we isolate P by multiplying both sides by 2 and dividing by 5:

$$ P = 1,50,000 \times \frac{2}{5} $$

$$ P = \frac{3,00,000}{5} $$

$$ P = 60,000 $$

Conclusion

Therefore, Manoj has saved Rs. 60,000 in his Public Provident Fund.

We can also find his savings in Kisan Vikas Patra:

$$ K = 1,50,000 - P $$

$$ K = 1,50,000 - 60,000 $$

$$ K = 90,000 $$

Let's check if the initial condition holds:

$$ \frac{1}{3} \times K = \frac{1}{3} \times 90,000 = 30,000 $$

$$ \frac{1}{2} \times P = \frac{1}{2} \times 60,000 = 30,000 $$

Since both values are equal (30,000), our calculation is correct.