Read the table carefully and answer the question: | Schemes | Time (In years) | Rate of interest | |----|----------------|---------------------------------| | A | 30 | $X^2 - 3X - 10 = 0$ | | B | $Y^2 - Y - 6 = 0$ | 20\% | | C | 40 | ------ | Note: (i) If time rate of interest, then simple interest should be considered in any scheme. (ii) If time

Question

Read the table carefully and answer the question:
| Schemes | Time (In years) | Rate of interest | 
|----|----------------|---------------------------------| 
| A | 30 | $X^2 - 3X - 10 = 0$ | 
| B | $Y^2 - Y - 6 = 0$ | 20\% | 
| C | 40 | ------ |

Note: (i) If time > rate of interest, then simple interest should be considered in any scheme. (ii) If time < rate of interest, then compound interest should be considered in any scheme. (iii) Some values are missing; you have to calculate according to the questions. (iv) P and Q are two friends and they invested their sum in scheme A and scheme B respectively. Q invested Rs.15000 and sum invested and profit earned by A is same. (v) P also invested some money in scheme C on simple interest and interest earned by her is same as profit earned by scheme A. (vi) Take positive value of equations for time and rate of interest. (vii) Rate in scheme C is greater that rate of interest in scheme A. If P invested her sum fo

Rate at which P invested in scheme C follows the equation $z^2-10z+24=0$and the interest earned by her in scheme A is 160% of amount invested by Q in scheme B. Find the amount received by P from scheme C?

Accepted Answer

**Solution:**

1. **Find rate of interest for scheme A:**
   Given quadratic equation for rate in scheme A:
   $$X^2 - 3X - 10 = 0$$

Factorizing or using quadratic formula:
   $$X = \frac{3 \pm \sqrt{9 + 40}}{2} = \frac{3 \pm 7}{2}$$

Possible values: $$X = 5$$ or $$X = -2$$ (discard negative)

So, rate of interest for scheme A, $$r_A = 5\%$$.

2. **Determine interest type for scheme A:**
   Time $$t_A = 30$$ years, rate $$r_A = 5\%$$.

Since time > rate, simple interest applies.

3. **Sum invested and profit earned by scheme A are equal:**
   Let sum invested in scheme A be $$P_A$$.

Simple Interest (SI) = $$\frac{P_A 	imes r_A 	imes t_A}{100} = P_A$$

$$\Rightarrow \frac{P_A 	imes 5 	imes 30}{100} = P_A$$

$$\Rightarrow \frac{150 P_A}{100} = P_A$$

$$\Rightarrow 1.5 P_A = P_A$$

This is only possible if $$P_A = 0$$ or the equation is consistent only if profit equals principal.

Since profit = principal, this confirms the condition.

4. **Find profit earned by scheme A:**
   Profit = $$P_A$$ (same as principal)

5. **Interest earned by P in scheme C equals profit earned by scheme A:**
   Interest earned by P = Profit of scheme A = $$P_A$$

6. **Interest earned by P in scheme C is 160% of amount invested by Q in scheme B:**
   Amount invested by Q in scheme B = Rs. 15000

Interest earned by P = 160% of 15000 = $$1.6 	imes 15000 = 24000$$

So, $$P_A = 24000$$.

7. **Find sum invested by P in scheme C:**
   Rate of interest in scheme C satisfies:
   $$z^2 - 10z + 24 = 0$$

Solving quadratic:
   $$z = \frac{10 \pm \sqrt{100 - 96}}{2} = \frac{10 \pm 2}{2}$$

Possible rates: $$z = 6\%$$ or $$z = 4\%$$

Since rate in scheme C is greater than rate in scheme A (5%), choose $$z = 6\%$$.

8. **Determine interest type for scheme C:**
   Time $$t_C = 40$$ years, rate $$r_C = 6\%$$.

Since time > rate, simple interest applies.

9. **Calculate principal invested by P in scheme C:**
   Let principal be $$P_C$$.

Simple Interest formula:
   $$SI = \frac{P_C 	imes r_C 	imes t_C}{100} = 24000$$

Substitute values:
   $$\frac{P_C 	imes 6 	imes 40}{100} = 24000$$

$$\Rightarrow \frac{240 P_C}{100} = 24000$$

$$\Rightarrow 2.4 P_C = 24000$$

$$\Rightarrow P_C = \frac{24000}{2.4} = 10000$$

10. **Calculate amount received by P from scheme C:**
    Amount = Principal + Interest = $$10000 + 24000 = 34000$$

**Answer:** Rs. 34000

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