The total cost of a washing machine with a toaster was $\text{Rs }10500$. The toaster was sold at a profit of $22\%$ and the washing machine at a loss of $12\%$. 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 toaster be $T$ and the cost price of the washing machine be $W$.Given that:
$$T+W=10500\text{— (Equation 1)}$$
The selling price of both items is equal:
$$\text{SP of toaster}=\text{SP of washing machine}$$
$$T\times \left(1+\frac{22}{100}\right)=W\times \left(1-\frac{12}{100}\right)$$
$$1.22T=0.88W$$
$$\frac{T}{W}=\frac{0.88}{1.22}=\frac{88}{122}=\frac{44}{61}$$
So, the ratio of their cost prices is $T:W=44:61$.Clearly, the toaster is the cheaper item since its ratio value ($44$) is less than that of the washing machine ($61$).Using ratio division to find the cost price of the toaster:
$$\text{Cost Price of cheaper item (Toaster)}=10500\times \frac{44}{44+61}$$
$$T=10500\times \frac{44}{105}$$
$$T=100\times 44=\text{Rs }4400$$