Master Algebra Word Problems

A 400-gallon tank contains 250 gallons of liquid. Two pumps feed ingredients into the top of tank. A discharge pump at the base of the tank removes the ingredients in the tank. The rates at which the pipes feed and remove liquid are:

a. pump 15 gallons per minute

b. pump 210 gallons per minute

c. discharge pump22 gallons per minute

Throughout this note, t represents time in minutes.

Example 1: Every minute pump 1 operates, it feeds 5 gallons into the tank. So, in 7 minutes, pump 1 feeds5 times 7 gallons into the tank. If t represents the amount of time (in minutes) pump 1 feeds the tank. At the end of this time interval, pump 1 will have fed 5 times t gallons into the tank.

The crux: The amount of liquid flowing into the tank is the product of the amount of time the pump feeds the tank and the amount of liquid the pump feeds into the tank every minute.

Problem 1: Give a mathematical expression for the amount of gallons fed into the tank by pump 2 in t minutes. Also write the amount in gallons the discharge pump eliminates in t minutes.

Example 2: Assume the tank is empty. Find the amount of time pump 1 needs to fill the tank, if it is the only pump operating. Observe:

1. Pump 1 feeds 5 t gallons into the tank in t minutes.

2. The tank is full when the amount pumped in is 400 gallons.

The time, we seek, is the moment the tank is full. At that moment, the tank is full so, 5 t = 400.

Solving for t: t = 400 / 5 = 80 minutes ( 1 hour and 20 minutes)

The crux of the solution: The tank is full when the amount pumped into the tank is the capacity of the tank.

Problem 2: Assume the tank is empty. Find the time it takes for pump 2 to fill the tank. Also, how long will it take for the discharge pump to empty a full tank?

Example 3: The tank contains 250 gallons, and pump 1 is the only pump operating. How much time is needed to fill the tank.

The crux of the solution: The tank is full when the original content of the tank and the amount pumped into the tank is the capacity of the tank.

The original content is 250 gallons.

The amount pump 1 feeds into the tank 5 t.

Tank full: 250 + 5 t = 400

Here t is the time that passes until the tank fills.

Solving for t: 5 t = 400 - 250

5 t = 150

So, t = 150 / 5 = 30 minutes.

Problem 3: The tank originally contains 260 gallons. Pump 2 is the only operating pump. How much time is needed to fill the tank.

Example 4: The tank contains 250 gallons and only the discharge pump is turned on. How much time is needed to empty the tank?

The crux of the solution:If the discharge pump is the only operating pump, the amount discharged is the same as the original content. So the tank will be empty when the discharged amount is the original amount.

The discharge pump discharges 10 t gallons in t minutes.

The amount in the tank is 250 gallons.

So the tank is empty, when 10 t = 250.

Solve for t. t = 250 / 10 = 25 minutes.

Problem 4: The tank originally contains 150 gallons and only the discharge pump is operating. How much time is needed to empty the tank?

Example 5: When all pumps are operating, the tank is empting.

The crux:

Amount flowing into the tank from both feed pumps (5+10) is gallons per minute.

Amount flowing out of the tank is 22 gallons per minute.

So the amount flowing into the tank is smaller than the amount flowing out of the tank.

In fact, 7 gallons leaves the tank every minute the tank contains liquid.

Example 6: The tank contains 50 gallons. Pump 1 is turned on. Three minutes later, pump 2 is turned on. How long will it take to empty the tank if the discharge pump is turned on 5 minutes after pump 2 starts to operate?

The tank empties when all pumps are operating. How long will this take?

Let t be the amount of time pump 1 operates until the tank empties.

The amount of liquid in the tank when the discharge pump is turned on:

50 gallons - original content of the tank

5 t - amount fed into the tank by pump 1

10 ( t - 3) - amount fed into the tank by pump 2

The amount the discharge pump eliminates for the tank:

22 (t - 8) - discharged amount

The crux of the solution: The tank is empty when the discharge pump removes all liquid from the tank. It happens when the amount eliminated by the discharge pump = the contents in the tank prior to starting the discharge tank.

22 (t-8) = 50 + 5 t + 10 ( t - 3).

7 t = 50 - 30 + 176

7 t = 196

t = 196 / 7 minutes or 28 minutes.

Problem 6: The tank now contains 70 gallons. Pump 1 is turned on. After 2 minutes, pump 2 is turned on. How long will it take to empty the tank if the discharge pump is turned on after 7 minutes?

Something to think about.

Yesterday a huge rainstorm dumped 2 inches of water on a football field. The pump drains 28 gallons of water from the field every minute it operates. Before starting the drainage pump, the rain started again and it rained exactly 3 hours dumping 10 gallons onto the field each minute. After the rain stopped, the drainage pump was started. How much time is needed to drain the water from the field? The amount of water on the field, before the rain started again, is 5,984 gallons.

Something further to think about.

The drainage pump starts 90 minutes after the rain starts. How much time is needed to drain the water from the field?

Reference: The Use of Worked Examples as a Substitute for Problem Solving in Learning Algebra, Cognition and Instruction, Vol. 2, No. 1 (1985), pp. 59-89.