Content

• Steps
• Tips

Many people think that mathematics deals only with numbers, however, there are problems in the real world that can be solved using it. To introduce students to such problems, verbal problems are included in many algebra programs. However, many tasks can be very difficult if you don’t know how to break them down and find the underlying numbers. Problem solving is the art of converting words and sentences into mathematical expressions and then applying traditional algebraic techniques to solve them.

Steps

1. 1 To start solving a problem, write it down. Writing down a task makes it easier to break it down into its component parts. Let's say you need to solve the following problem:
   • “Zhenya went to a bookstore and bought a book for 1200 rubles. In the store, Zhenya found another interesting book, and bought it for an amount of 400 rubles. less than three times the cost of the first book. How much did the second book Zhenya bought cost? "
   • Write this text in a notebook so that important information can be emphasized and highlighted.
2. 2 To start looking for the answer, define the unknown. Each problem has at least one unknown quantity that is asked to find, usually at the end of the question.
   • The unknown is usually the answer to the problem.
   • In the problem from the example, if we carefully read the last sentence, then we will clearly see the question “How much did the second book bought by Zhenya cost?”.
   • Therefore, the unknown in this problem is the cost of the second book purchased.
3. 3 Come up with a variable to represent the unknown quantity. To make the calculation easier, you can set the unknown value to the variable "x".
   • Let "x" = the price of the second book purchased.
   • A variable can be thought of as a short designation for an unknown factor that can be used in mathematical equations.
   • We used the letter "x", but you can use any letter or combination of letters you like.
4. 4 Extract all numerical information from the problem. Numerical information is information described using numbers.
   • Take a look at the problem from the example. The first numerical value that we come across is 1200 rubles, which is the price of the first purchased book.
   • The next figure indicated is 400 rubles, which is the difference between the triple cost of the first book and the price of an unknown book.
5. 5 Determine the relationship between different numerical values. Now figure out how the different numbers relate to each other and translate those ratios into mathematical expressions.
   • For example, “… Zhenya found another interesting book and bought it for an amount of 400 rubles. less than three times the cost first book… ”.
   • You know that the price of the first book is 1200 rubles.
   • Substitute it in the condition as follows: “... Zhenya found another interesting book, and bought it for an amount of 400 rubles. less than three times 1200 RUB… ”.
6. 6 Continue to split the problem into parts, replacing the numbers with algebraic relations. Now you’ll turn the whole problem into an algebraic equation that can be solved.
   • But you know that the unknown is the price of the second book purchased, so you can substitute "x" for the price of the second book purchased.
   • From the previous step, you know that the second book you bought costs "400 rubles less than three times 1200 rubles."
   • If we use all this information, we get: "x = 400 rubles less than three times 1200 rubles."
7. 7 Turn words into math operations to complete the equation. Now you need to convert words that refer to mathematical operators, such as "greater", "less" and "times ...", into mathematical symbols.
   • Less means subtraction, so replace that word with the minus (-) operation.
   • Now we have "x = tripled 1200 rubles - 400 rubles."
   • Further, tripled 1200 rubles. equivalent (3 * 1200 rub.)
   • Our final equation looks like this: "x = (3 * 1200 rubles.) - 400 rubles.
   • The problem has been completely transformed from a verbal form to an algebraic expression.
8. 8 Solve the algebraic equation obtained in the last step. This algebraic equation is very simple and can be solved as follows:
   • x = (3 * 1200 rub.) - 400 rub.
   • x = 3600 rub. - 400 rubles.
   • x = 3200 rub.
   • We know that x = the price of the second purchased book, so the price of the second purchased book is 3200 rubles.

Tips

• Problems can contain more than one unknown and more than one variable.
• The number of variables is always equal to the number of unknowns.
• To gain experience in problem solving, you must train as much as possible.
• When solving a problem, you must carefully read each sentence and try to extract numerical information.
• The relationship between the unknown and various numerical parameters is given in the form of words like "more by ...", "less than", "more than", "by ... times", etc.