In a company, initially the ratio of the foreign and domestic workers was 5 : 8. When some foreign and 12 domestic workers left the company, this ratio became 2 : 3. Later, when 10 more foreign workers were replaced by 10 new domestic workers, this ratio became 3 : 7. The initial number of domestic workers in the company was ___

Let $a, b, c, d$ be positive integers such that $a + b + c + d = 2023$. If $a: b = 2:5$ and $c:d=5:2$, then the maximum possible value of $a + c$ is

A fruit seller has oranges, apples, and bananas in the ratio $3:6:7$. If the number of oranges is a multiple of both 5 and 6, then the minimum number of fruits the seller has is:

The highest possible value of the ratio of a four-digit number and the sum of its four digits is

Ashok purchased pens and pencils in the ratio $2:3$ during his first visit and paid Rs. 86 to the shopkeeper. During his second visit, he purchased pens and pencils in the ratio $4:1$ and paid Rs. 112. The cost of a pen as well as a pencil in rupees is a positive integer. If Ashok purchased four pens during his second visit, then the amount he paid in rupees for the pens during the second visit is __________.

Ashok started a business with a certain investment. After a few months, Bharat joined him investing half the amount of Ashok's initial investment. At the end of the first year, the total profit was divided between them in the ratio $3:1$. Bharat joined Ashok after

$\dfrac{(a + b)}{(b + c)} = \dfrac{(c + d)}{(d + a)}$ which of the following statements is always true?