Number, Percent and Geometry Problems

Number Problems

To solve a word problem, it is helpful to remember some basic vocabulary:

Symbol	Terms	Examples
+	sum more than	The sum of two numbers is five more than one number
-	difference less than	the length and the width differ by three dollars less than the cost of a ticket
x	product	the product of two numbers
/	quotient	the quotient of two numbers
=	is	the length is twice the width

Examples
 
Convert to a math expression :

1.  Elizabeth is five years younger than the combined age of Sarah and Brian.
   
           E  =  S + B - 5
   

2. The product of the length and the width is 10 more than the height.
   
           lw  =  h + 10
   

3. The sum of two consecutive integers is 29.
   
           n + (n + 1)  =  29
   

4. The product of two consecutive even integers is 168.
   
           2n (2n + 2)  =  168

Percent Problems

The words "percent of" mean  "/100"

Example

20 percent of 50 means

         20 
                  50
        100

Example  

The price of a meal and a 15% tip was $11.50.  What was the price of the meal?

Solution  

Let x be the price of the meal.  
Then 

        x + 0.15x  =  11.50           Price of meal + 15/100 times the price of the meal

or 

        1.15x  =  11.50               Combining like terms

or 

                  11.50
        x =                   =  $10
                  1.15

Hence the price of the meal was $10.

Exercise:  

$800 is invested into an account paying 3% interest.  How much money should be invested into an account paying 4% interest so that the total interest earned is $30?

Geometry Problems

Below are some geometrical facts.

1. Area of a triangle = 1/2 bh      (b is the base and h is the height)
   
   

2. Area of a rectangle = bh      (b is the base and h is the height)
   
   

3. Sum of the angles of a triangle is 180.
   
   

4. The perimeter is the sum of the sides.
   
   

5. Isosceles means two sides (and two angles) are equal.
   

Example

Find the angles of a triangle if the smallest angle is 5 degrees less than the next smallest, which is 20 degrees less than the largest.

Solution:  

Let x be the measure of the smallest angle.  Then the middle angle has measure 

        5 + x

and the largest has measure 

        20 + (5 + x)

We have:  

        x + (5 + x) + (25+ x)  =  180         Sum of the angles of a triangle is 180

        3x + 30  =  180                              x + x + x = 3x,   5 + 25 = 30

        3x  =  150                                     Subtracting 30 on both sides

        x  =  150/3                                    Dividing by three

        x  =  50

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