The total cost of a dishwasher with a vacuum cleaner was Rs. 78750. The vacuum cleaner was sold at a profit of 37% and the dishwasher at a loss of 27%. If the sale price was the same for both items, then the cost price of the cheaper item was:

Let the cost price of the vacuum cleaner be V and that of the dishwasher be D.
Total cost price:

V+D=78750
The vacuum cleaner is sold at a 37% profit, so its selling price (SP
V
​
) is:

SP
V
​
=1.37V
The dishwasher is sold at a 27% loss, so its selling price (SP
D
​
) is:

SP
D
​
=(1−0.27)D=0.73D
Since the selling prices are identical:

1.37V=0.73D
D
V
​
=
1.37
0.73
​
=
137
73
​

This gives the ratio of their cost prices as V:D=73:137.
The cheaper item is the vacuum cleaner (V). Let's find its cost price:

V=
73+137
73
​
×78750
V=
210
73
​
×78750
V=73×375=27375
Thus, the cost price of the cheaper item is Rs. 27375.