John gets Rs $57$ per hour of regular work and Rs $114$ per hour of overtime work. He works altogether $172$ hours and his income from overtime hours is $15\%$ of his income from regular hours. Then, for how many hours did he work overtime?

A donation box can receive only cheques of Rs. $100$, Rs. $250$, and Rs. $500$. On one good day, the donation box was found to contain exactly $100$ cheques amounting to a total sum of Rs. $15250$. Then, the maximum possible number of cheques of Rs. $500$ that the donation box may have contained, is

In a school with 1500 students, each student chooses any one of the streams out of science, arts, and commerce, by paying a fee of Rs 1100, Rs 1000, and Rs 800, respectively. The total fee paid by all the students is Rs 15,50,000. If the number of science students is not more than the number of arts students, then the maximum possible number of science students in the school is

If $a - 6b + 6c = 4$ and $6a + 3b - 3c = 50$, where $a$, $b$ and $c$ are real numbers, the value of $2a + 3b - 3c$ is

A basket of $2$ apples, $4$ oranges and $6$ mangoes costs the same as a basket of $1$ apple, $4$ oranges and $8$ mangoes, or a basket of $8$ oranges and $7$ mangoes. Then the number of mangoes in a basket of mangoes that has the same cost as the other baskets is

A shop wants to sell a certain quantity (in kg) of grains. It sells half the quantity and an additional 3 kg of these grains to the first customer. Then, it sells half of the remaining quantity and an additional 3 kg of these grains to the second customer. Finally, when the shop sells half of the remaining quantity and an additional 3 kg of these grains to the third customer, there are no grains left. The initial quantity, in kg, of grains is

For some constant real numbers $p, k$ and $a$ consider the following system of linear equations in $x$ and $y$:

$px - 4y = 2$

$3x + ky = a$

A necessary condition for the system to have no solution for $(x, y)$ is