Algebraic Operations: Simple Word Problems

Mathematics, Preparatory Course

Problem 1. A student has a digital video disks in his collection. He is going to buy b more DVDs. How many DVDs will he have in all?

To solve Problem 1, we need the operation of addition of two quantities, a and b. The solution is represented by the following formula,

c = a + b.

The above formula holds true if to change the words "student", "buy", "DVD", etc. by any appropriate expressions. The kernel of the problem is the addition of two items.
However, each mathematical operation has its range of application. The operation of addition is defined for objects of the same (or similar) type. For instance, one can not sum up minutes with meters.

Problem 2. A student had c digital video disks in his collection. Then he gave b DVDs to his friend. How many DVDs did he have left for himself?

The solution is expressed in terms of the following subtraction formula,

a = c – b.

Note that the student will have the initial number of DVDs, if he get back b DVDs:

c – b + b = c.

It means that the addition of a number cancels the subtraction ot the number. If two operations cancel each other, then they are called the mutually inverse ones, and so the operations of addition and subtraction are the mutually inverse operations.
Two statements,

a = c – b and c = a + b,

are equivalent since one of them implies the other.

Problem 3. A student has a digital video disks in his collection. Each DVD contains b clips. How many clips does the collection contain?

The solution is given by the following multiplication formula,

c = a b.

Problem 4. During b days there were c events. How many events were during a day at the average?

The solution is expressed in terms of the following division formula,

a = c / b.

Note that the multiplication of a by b gives the initial value c.