Algebra 1 > Notes > Yorkcounty Final > Unit 3 > Lesson 6 - Solving_practical_problems

Solving Practical Problems Lesson 6

What are Practical Problems?  Practical

Problems are commonly known as word problems.

 These

are problems that are written in common English. You have to read the question and try to figure out what it is you are supposed to find and how to solve it.

Where to start? 

First you need to identify what it is you are trying to find.



Key words that you want to look for are:      

What How long How many A number A product How much

Defining a variable  After

you have found what it is you are trying to find you need to label that thing.

 You

will call this your unknown variable, x. (you can call the variable whatever letter you choose)

 Then

you will need to change the English into math language.

Example 1  Wild

toads grow about 6 inches each year. If a toad is now 11 inches long, about how long will it take for the frog to become 59 inches long?

 First

we need to identify what it is we are trying to find.

Example 1  The

keyword in that passage was “about how long.”

 We

will make that our variable, x.

 “about  Our

how long” = x (in years)

variable x stands for how long, how many years.

Example 1 Now we need to write an equation so we can solve for x.  (wild toads grow 6 inches a year and they are now 11 inches tall) = 6x + 11.  How long will it take them to become 59 inches?  6x + 11 = 59 , now solve for x. 

Example 1  6x

+ 11 = 59 - we need to get x alone, what do we need to move first?

 The

( + 11 ). So we need to do the opposite of ( + 11 ) which is ( - 11, subtract 11) from both sides.  6x + 11 – 11 = 59 – 11 which produces  6x = 48 now we must get rid of that 6

Example 1  

6x = 48 what is keeping x from being alone? The 6 being multiplied to it. How do we get rid of that 6? Do the opposite of what it is doing. Divide by 6.

6 x 48 = 6 6

Which produces x = 8

Example 1  We

found that x = 8. So replace x with “about how long.” About how long = 8 years. We need to check that solution.  6x + 11 = 59  6(8) + 11 = 59  48 + 11 = 59  59 = 59  8 years is the correct solution. The frog will be 59 inches long in 8 years.

Example 2  Let

us try another example.  Thirteen more than four times a number is twenty nine. Find that number.  What

are we trying to find?  A number.  So

x = a number.

Example 2  Write

the passage from English to math language.

 (13



more) than (4 times a number) is (29).

13 +

 13

4x

=

29

+ 4x = 39 (remember: a number = x)

Example 2  

Now we have to solve for x. 13 + 4x = 29



First subtract 13 from both sides. 13 -13 + 4x = 29 – 13



Which produces



Now divide both sides by 4.



4x = 16

4 x 16 = 4 4 

Which produces x = 4.



Check your solution

Example 2  13

+ 4x = 29  13 + 4(4) = 29  13 + 16 = 29  29 = 29  So

Substitute x = 4

x = 4 is a correct solution.  The number is 4. (Answer the question in English)

Example 3  Try

A

this one.

mechanic charges a flat fee of 50 dollars to look at your car. He also charges 25 dollars an hour to service your car. How many hours did the mechanic service your car if your bill was 225 dollars.

Example 3  What

did you pick for your variable?  “How many hours” = h  Now

write your equation.



(Flat fee) + ( 25 dollars per hour) = (200 dollars)



50 + 25h = 200 (now solve for h)

Example 3 

50 + 25h = 225



50 – 50 + 25h = 225 – 50



25h = 175

(first subtract 50 from both sides)

( divide both sides by 25)

25h 175 = 25 25 

h=7



Check your answer.

Example 3  50

+ 25h = 225

+ 25(7) = 225  50 + 175 = 225  225 = 225

,h=7

 50

 The

h = 7 is a correct solution.

mechanic worked on the car for 7 hours.

Example 4  Try

A

this one

boat sold for 500 dollars. The selling price of 500 dollars is 100 dollars less than twice the cost of the boat. Find the cost of the boat.

Example 4 

Cost of the boat = c



(500 dollar selling price) = (100 dollars less than 2 times the cost)



500 = 2c – 100 Solve for c



500 + 100 = 2c – 100 + 100 (add 100 to both sides)



600 = 2c



300 = c



The cost of the boat was 300 dollars

Divide both sides by 2

Remember  First

read the whole passage and try to figure out what is it your are supposed to find.  Make that thing your variable.  Write the English into math language.  Solve for your variable.  Check your solution.