# Problem-Solving Models

## Key Questions

Jane spent $42 for shoes. This was$14 less than twice what she spent for a blouse. How much was the blouse?

First, identify what the question is asking for.

Jane spent $42 for shoes. This was$14 less than twice what she spent for a blouse . How much was the blouse?

Next, identify the numbers.

Jane spent $42 for shoes. This was$14 less than twice what she spent for a blouse . How much was the blouse?

Next, identify the key words. These include add, subtract, remove, spend, earn, less, more, times, twice, half, etc.

Jane spent $42 for shoes. This was$14 less than twice what she spent for a blouse . How much was the blouse?

Finally, convert everything into an equation.

$42 = 2 \cdot \text{blouse} - 14$

Now, solve the equation.

$56 = 2 \cdot b l o u s e$

$b l o u s e = 28$

The blouse was 28 dollars.

• If it is a solution of equations, then plug your solution into the equations you are working on to make sure that all equations are satisfied by your solution.

I hope that this was helpful.

#### Explanation:

To start, assign the variables to unknowns, known values to constants, and relate them by the relations between the variables and constants.

• If a mathematical problem is expressed in words, then first try to express all the relevant information in the problem in terms of symbols representing unknowns, in the form of equations, inequalities, etc.

Once you have a symbolic representation of the problem - typically in the form of a set of simultaneous equations to be solved, try eliminating variables one at a time by rearranging terms, using basic arithmetic operations and substitutions.Eventually this process will probably lead to values (or at least constraints) for all of the unknowns.

Convert these values or constraints back into the context and language of the problem to express the solution.

For example, suppose Andrew is half the age of Brian, Brian is three times older than Charles and the sum of their ages is 44 years. How old is Charles?

Use $A$ to represent Andrew's age, $B$ to represent Brian's age and $C$ to represent Charles's age.

Then expressing the information in the problem symbolically we have:

$A = \frac{1}{2} B$

$B = 3 C$

$A + B + C = 44$

and we want to determine $C$

From the first two equations, we have $A = \frac{1}{2} B = \frac{1}{2} \left(3 C\right) = \frac{3}{2} C$

Substituting $A = \frac{3}{2} C$ and $B = 3 C$ into the third equation we find:

$44 = A + B + C = \left(\frac{3}{2} C\right) + 3 C + C = \frac{11}{2} C$

Multiplying both ends by $\frac{2}{11}$ we find

$C = 8$

So Charles is 8 years old.