Question

A large mixing tank currently contains $200$ gallons of water into which $10$ pounds of sugar has been mixed. A tap will open pouring $20$ gallons of water per minute into the tank at the same time sugar is poured into the tank at a rate of $2$ pounds per minute. Find the concentration (pounds per gallon) of sugar in the tank after $14$ minutes.Then,
A. the concentration is greater than at the beginning
B. the concentration is lesser than at the beginning
C. the concentration is equal to the concentration at the beginning
D. None of the above

Answer 1

Hint: If you need to answer these types of questions, then you need to be clear with the language. Unless and until you understand the question clearly, you will not be able to come up with a solution. In the question, we are given a word problem in which we are provided with a mixing tank containing a specific quantity of water and sugar. Then, sugar and water is poured into the tank at a certain rate and we have to find the concentration in pounds per gallon after a fixed interval of time. Concentration of sugar in the solution is calculated by dividing the amount of sugar by the volume of water in the tank.

Complete step by step answer:
So, we have $200$ gallons of water and $10$ pounds of sugar in the tank initially. Then, sugar is poured into the tank at a rate of $2$ pounds per minute and water is poured through a tap at a rate of $20$ gallons per minute.Now, we have to find the concentration of sugar in the tank after $14$ minutes and compare it with the concentration at the beginning.We know that the concentration of sugar is calculated by dividing the amount of sugar in the tank by the volume of water in the tank.

So, concentration in the beginning is $10$ pounds of sugar per $200$ gallons of water. Thus, simplifying the concentration, we get, $1$ pound of sugar per $20$ gallon of water. So, the initial concentration of sugar in the tank is $0.05$ pounds per gallon.Now, when we start pouring more water and sugar into the tank, the concentration of sugar in the tank changes.

Rate of sugar pouring into tank $ = 2$ pounds per minute
Rate of water pouring into tank $ = 20$ gallons per minute
Now, the time interval is $14$ minutes.
So, the quantity of sugar poured in the tank in $14$ minutes $ = 2 \times 14 = 28$ pounds.
Also, the quantity of water poured into the tank in $14$ minutes $ = 20 \times 14 = 280$ gallons.
So, cumulative quantity of water in the tank $ = 280 + 200 = 480$ gallons
Cumulative quantity of sugar $ = 10 + 28 = 38$ pounds.

Again calculating the concentration of sugar by computing the amount of sugar per gallon of water.Hence, the final concentration of sugar in the tank is $38$ pounds of sugar per $480$ gallons of water. Thus, the final concentration of sugar in the tank is $0.0791$ pounds per gallon. Hence, the final concentration of sugar in the tank is greater than the initial concentration of sugar in the beginning.

So, option A is correct.

Note:The above-mentioned method can be remembered as a complete process for solving these types of questions. Be careful with the language while attending these types of questions, because there are many questions which may confuse you with the tricky and typical language. Take care while simplifying the calculations to be sure of the final answer. We must know how to calculate the concentration of sugar in the tank in order to solve the problem. One should also take care of calculations while doing such problems. One must have attended several mixture problems in order to tackle the problem in a limited time frame.