Let's solve this problem step-by-step.
First, let's define the variables:
- Let [tex]\( CP \)[/tex] be the cost price of the watch.
- Let [tex]\( SP \)[/tex] be the selling price of the watch when Manish sold it at a 5% loss.
Step 1: Establish the relationship between SP and CP for the loss scenario.
If Manish sold the watch at a 5% loss, it means he sold it for 95% of its cost price. Mathematically, we can express it as:
[tex]\[ SP = 0.95 \times CP \tag{1} \][/tex]
Step 2: Establish the relationship between SP and CP for the gain scenario.
If Manish sold the watch for Rs 320 more, he would have sold it for a price at which he makes an 8% gain. This means he would sell it for 108% of its cost price. Mathematically, we can express it as:
[tex]\[ SP + 320 = 1.08 \times CP \tag{2} \][/tex]
Step 3: Substitute the expression for SP from Equation (1) into Equation (2).
[tex]\[ 0.95 \times CP + 320 = 1.08 \times CP \][/tex]
Step 4: Simplify the equation to find CP.
Subtract [tex]\( 0.95 \times CP \)[/tex] from both sides:
[tex]\[ 320 = 1.08 \times CP - 0.95 \times CP \][/tex]
[tex]\[ 320 = 0.13 \times CP \][/tex]
Now, solve for [tex]\( CP \)[/tex]:
[tex]\[ CP = \frac{320}{0.13} \][/tex]
[tex]\[ CP = 2461.5384615384614 \][/tex]
So, the cost price [tex]\( CP \)[/tex] is approximately Rs 2461.54.
Step 5: Find the selling price [tex]\( SP \)[/tex] using the cost price.
From Equation (1), we know:
[tex]\[ SP = 0.95 \times CP \][/tex]
[tex]\[ SP = 0.95 \times 2461.5384615384614 \][/tex]
[tex]\[ SP = 2338.461538461538 \][/tex]
So, the selling price [tex]\( SP \)[/tex] of the watch is approximately Rs 2338.46.
Therefore, Manish sold the watch for Rs 2338.46.
