Math Problems. Easy points!.?

Question

1. What is the least positive integer that 180 should be multiplied by if the product is to be a perfect square? What is the least intiger we should multiply 180 by if the product to be a perfect cube?

2. A tank contains 50 liters of antifreeze. Ten liters are removed and replaced with water. The mixture is stirred. Ten liters of the mixture are removed and replaces with water. How many liters of water are now in the tank.

Answer 1

Answer 1:

The cube must be divisible by both 9 and 10.
The cube root of this cube has 3 as a prime factor. And 10 is also a factor of the cube root.

(3*10)^3 = 30^3 = 27000 is the lowest such cube.

27000/180 = 150.
We get a positive integer. So, no need to think of 60^3
The answer is 150.


Answer 2:
After first swap:

40 litres antifreeze, 10 litres water.
The mixture is 20% water and 80% antifreeze.

Now, ten litres of the mixture makes for 20% of the total volume of the mixture. So, 20% of 10 litres water and 20% of 40 litres antifreeze is removed. That's 2 litres water and 8 litres antifreeze out the window. And 10 more litres of water in.

Now, the new mixture has 8 litres more water than the original with 8 litres less antifreeze. So, the new mixture has 18 litres water and 32 litres antifreeze