From Word Problem to Equation

To be solved, word or story problems must be translated into equations with algebraic expressions that contain constants and variables. And how does one do the translation? Well, even in the most simple cases there may be more than one approach. As usual, to master the art one has to try. Let's return to our first example.

(Note: In the applet below, all underlied words and numbers can be clicked on. In fact you can see some changes clicking anywhere in the applet area. Click, click, click ... and see what happens.)

// foreground color // background color // information pieces start with '@', followed by // a single letter from {v,a,f,e}, followed by ": ". // v - variable - anything that may change on a click // variables must precede all other pieces // f - formulation - word problem with embedded variables. // Many formulations are possible for a single problem. // Only one is shown at a time // e - equation - equation is like a formulation with a // potential provision to, e.g., enclose negative numbers // into parentheses // a - answer - like an equation, but allowed to have a parsable // portion. Parsables are written in the reverse Polish notations // with variables and operations separated by a comma // d - directive // define color, skip line // // whatever it is, a variable can be of the three types: // I = integer - on click changes up (right of center) and down (left of center) // a = attribute - any clickable and changeable word // v = variable - like an attribute but without related attributes // all 'name's below are 1 letter from [A-Ba-b1-0] // I, name, initial value, min, max // a, name, related attribute, list of values // v, name, list of symbols // formulations // equations

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Here is a little different way to tackle the same problem. What in essence is the problem about? Forget for a moment about constants. Let's give names to the quantities involved:

// foreground color // background color // information pieces start with '@', followed by // a single letter from {v,a,f,e}, followed by ": ". // v - variable - anything that may change on a click // variables must precede all other pieces // f - formulation - word problem with embedded variables. // Many formulations are possible for a single problem. // Only one is shown at a time // e - equation - equation is like a formulation with a // potential provision to, e.g., enclose negative numbers // into parentheses // a - answer - like an equation, but allowed to have a parsable // portion. Parsables are written in the reverse Polish notations // with variables and operations separated by a comma // d - directive // define color, skip line // // whatever it is, a variable can be of the three types: // I = integer - on click changes up (right of center) and down (left of center) // a = attribute - any clickable and changeable word // v = variable - like an attribute but without related attributes // all 'name's below are 1 letter from [A-Ba-b1-0] // I, name, initial value, min, max // a, name, related attribute, list of values // v| name| list of symbols // formulations // equations

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

How does one solve two equations? In this case, the second equation is in fact the answer for one of the unknowns. We can simply substitute that value into the first equation:

// foreground color // background color // information pieces start with '@', followed by // a single letter from {v,a,f,e}, followed by ": ". // v - variable - anything that may change on a click // variables must precede all other pieces // f - formulation - word problem with embedded variables. // Many formulations are possible for a single problem. // Only one is shown at a time // e - equation - equation is like a formulation with a // potential provision to, e.g., enclose negative numbers // into parentheses // a - answer - like an equation, but allowed to have a parsable // portion. Parsables are written in the reverse Polish notations // with variables and operations separated by a comma // d - directive // define color, skip line // // whatever it is, a variable can be of the three types: // I = integer - on click changes up (right of center) and down (left of center) // a = attribute - any clickable and changeable word // v = variable - like an attribute but without related attributes // all 'name's below are 1 letter from [A-Ba-b1-0] // I, name, initial value, min, max // a, name, related attribute, list of values // v| name| list of symbols

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

Even in so simple a case, we can expand the initial step of naming variables:

// foreground color // background color // information pieces start with '@', followed by // a single letter from {v,a,f,e}, followed by ": ". // v - variable - anything that may change on a click // variables must precede all other pieces // f - formulation - word problem with embedded variables. // Many formulations are possible for a single problem. // Only one is shown at a time // e - equation - equation is like a formulation with a // potential provision to, e.g., enclose negative numbers // into parentheses // a - answer - like an equation, but allowed to have a parsable // portion. Parsables are written in the reverse Polish notations // with variables and operations separated by a comma // d - directive // define color, skip line // // whatever it is, a variable can be of the three types: // I = integer - on click changes up (right of center) and down (left of center) // a = attribute - any clickable and changeable word // v = variable - like an attribute but without related attributes // all 'name's below are 1 letter from [A-Ba-b1-0] // I, name, initial value, min, max // a, name, related attribute, list of values // v| name| list of symbols // formulations // equations

This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.


What if applet does not run?

We may learn a few things. First, there is much freedom in naming variables and putting equations together. Do not be surprised if a friend of yours solved the same problem differently. Second, when translating a word problem into the language of mathematics, it's quite normal to get more than 1 variable or more than 1 equation. When more than 1 equation results, the equations are called simultaneous and we talk of a system of simultaneous equations. To solve such a system is to obtain values for all variables involved. In other words, a system is solved when, for each of the variables that appear in the system, we get a simplest possible equation variable = constant which specifies a (constant) value for the variable on the left.

We'll tackle simultaneous equations on a separate page.

(There are many more word problems discussed and solved at this site. The math tutorial continues with a similar approach over several additional examples.)


Word Problems

  1. Problems of class a + x = b
  2. Problems of class a · x = b
  3. From Word Problem to Equation
  4. Problems of class x / a = b / c
  5. Problems of class x = a + b
  6. Problems of class x = a + b (II)
  7. Problems of class x = a × b
  8. Solving Two Simultaneous Linear Equations

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