In a showroom, the ratio of number of Apple products to Samsung products 2:7 and the ratio of number of Xiaomi products to Vivo products is 5:11. If the ratio of number of Samsung products to Vivo products is 5:9, what is the minimum possible sum of all the four type of product?

We are given that the ratio of number of Apple products to Samsung products 2:7. Assume the number of Apple products = 2x and  number of Samsung products = 7x.

Similarly, we are given that the ratio of number of Xiaomi products to Vivo products is 5:11 Assume the number of Xiaomi products = 5y and number of Vivo products = 11y.

We are also told that the ratio of number of Samsung products to Vivo products is 5:9.

The ratio of  number of Samsung products to Vivo products = 7x:11y

We are told that 7x:11y = 5:9

For this to be true, x has to be a multiple of both 11 and 5. Similarly y has to be multiple of 7 and 9.

Since we are asked for the minimum value, x = 11*5 = 55 and y = 7*9 = 63

number of Apple products = 2x = 110

number of Samsung products = 7x = 385

number of Xiaomi products = 5y = 315

number of Vivo products = 11y = 693

Sum = 1503