A person allows a 10% discount for cash payment from the marked price of a toy and still he makes a 10% gain. What is the cost price of the toy which is marked ₹ 770 ?

The correct option is ₹ 630.

Let us understand the step-by-step solution to find the cost price of the toy.

Step 1: Understand the given data
Marked Price (MP) of the toy = ₹ 770
Discount percentage allowed = 10%
Gain percentage made = 10%

Step 2: Calculate the Selling Price (SP) after the discount
A discount of 10% is allowed on the Marked Price (MP). Therefore, the Selling Price (SP) is calculated as:
$$\mathrm{SP}=\mathrm{MP}-(10\%\text{ of }\mathrm{MP})$$
Substitute the value of MP = ₹ 770:
$$\mathrm{SP}=770-\left(\frac{10}{100}\times 770\right)$$
$$\mathrm{SP}=770-77$$
$$\mathrm{SP}=₹693$$

Step 3: Calculate the Cost Price (CP) using the gain percentage
The person still makes a gain of 10% on the cost price (CP).
The relationship between Selling Price (SP), Cost Price (CP), and Gain Percentage is given by:
$$\mathrm{SP}=\mathrm{CP}\times \left(1+\frac{\text{Gain \%}}{100}\right)$$
Substitute the values of SP = ₹ 693 and Gain % = 10 into the formula:
$$693=\mathrm{CP}\times \left(1+\frac{10}{100}\right)$$
$$693=\mathrm{CP}\times \left(\frac{110}{100}\right)$$
$$693=\mathrm{CP}\times 1.1$$
Now, solve for CP:
$$\mathrm{CP}=\frac{693}{1.1}$$
$$\mathrm{CP}=\frac{6930}{11}$$
$$\mathrm{CP}=₹630$$

Thus, the cost price of the toy is ₹ 630.