Translating & SolvingWord Problems

Some Hints: We are going to cover several different types

of word problems in this unit.

One of the keys to success will be the ability to recognize word problems by type and apply an appropriate problem-solving strategy to them.

Motion problems lend themselves to the use of one strategy while mixture problems require a slightly different approach.

There are several basic principles that can be applied to most any word problem and we will use them as we move from one type to another.

How To Solve Word Problems• Read the problem several times carefully.

Looking for key ideas. Guess the answer. Write down your guess on scratch paper and see if it fits what the problem is asking. Guessing brings common sense into play immediately.

• Define “x”, as the answer to the question. For example, if the question is: “How many pizzas did Dawn order?” then let x equal the number of pizzas Dawn ordered.

• Label other unknown quantities in terms of x. For example, if you know from reading the problem that Barry ordered 3 more pizzas than Dawn, let x+3 equal the number of pizzas Barry ordered.

How To Solve Word Problems cont.

Form an equation. This is a good time to read the problem again to be sure that you are making use of all important information.

Solve the problem. This may require solving an equation or making a table or counting something.

Answer the question. Compare your answer to your original guess. This may keep you from entering an answer that may not be correct due to a mistake in setting up the problem. You may then be able to trace your mistake and come up with an answer that seems reasonable.

* Lets work some problems to see how these strategies work.

??

??

? ?3

MYSTERY NUMBERS

MYSTERY NUMBERS1) The sum of three consecutive integers is 39. Find the three integers.

What are the Integers?

What does consecutive mean?

{…-4, -3, -2, -1, 0, 1, 2, 3, 4, …}

Next to, such as 1 next to 2, -2 next to -1

What is the difference between two consecutive integers?

1 unit

1) The sum of three consecutive integers is 39. Find the three integers.

n+1 = 13,

n + 2 = 14

How do we do this problem algebraically?

Let n = first number

n = 12,

then

n + n+1 + n+2 =39

3n + 3 = 39

3n = 36

n = 12,

n + 1 = second number

n + 2 = third number

Consecutive Odd or Even

Odd Integers are {…-3, -1,1,3,5,…}The difference between each consecutive pair is 2 units. Thus we would set up three consecutiveodd integers as n, n+2, and n + 4.

Even Integers are { …-6, -4, -2, 0, 2,4, …}The difference between each consecutive pair is2 units also. Thus we would set up three consecutive even integers exactly like we did theodd: n , n + 2, n + 4

Finding mystery numbersA mystery number is a number that has been changed to a new number by using one of the four operations: addition, subtraction, multiplication, and division. For example:

If twice our number is 10 what is our mystery number?

In other words 2 * ? = 10

To find the answer divide

We use division, the operation that

undoes multiplication to solve this problem.

So

Likewise if the sum of a number and 10 is 25.

We would use subtraction to undo

addition.

Undoing operations So to guess a mystery number

we would use the opposite operation.

Operation Opposite operation

+ Addition - Subtraction

- Subtraction + Addition

Multiplication

Division

Division Multiplication

More mystery numbers

Sometimes it is harder to find a mystery number because more than one operation is used on the number. In this case we need to use our problem-solving process. Let’s see how it works on an example.

Fifty-two is 2 less than 6 times a number. Find the number.

See if you can guess the answer before you do italgebraically. Write your guess down.

Define x. x = the number we are looking for.

Label the other unknown quantities in terms of x.

6 times a number is 6x 2 less than 6 times a number is 6x - 2

Most of the words in a sentence will translate directly into an equation. But the words “less than” reverse things.

Next form an equation.

Fifty-two is 2 less than 6 times a number.

52 = 6 x - 2

Verbs like “is” are where you place the “=” sign

HANDOUT

Click the back button when you finish.

Solve the equation: 52 = 6x - 2 add 2 to both sides 52 + 2 = 6x -2 +2 -2 + 2 = 0 54 = 6x Divide by 6 on both sides

9 = x

??

??

? ?YourTurn

Problem 1: Find three consecutive integers that add to 180.

1. Define x Let x = the first integer.

2. Label other unknown quantities in terms of x Let x + 1= the second integer

Let x + 2 = the third integer3. Form an equation First Integer +Second Integer + Third Integer = 180 x + (x+1) + ( x+2) = 180Go to next slide.

(You may want to go to the end of this presentation to use the LiveMath program to guess the answer. But first read on.)

4. Solve the equation.

x + (x+1) + (x+2)= 180

3x + 3= 180 Add Like Terms

3x = 177 Subtract 3 from both sides.

x = 59 Divide by 3 on both sides.5. Answer the question.

x = 59

x + 1= 60

x + 2= 61

Check to see if your answer has a sum of 180.

59 + 60 + 61 = 180

180 = 180

How does this compare to your original guess?

2) The larger of two integers is 8 less than twice the smaller. When the smaller number is subtracted from the larger, the difference is 17. Find the two numbers.

Let x = smaller8 less than twice the smaller is written2x - 8 Note: when you see “less than” reverse

The order when you write them down.

2x – 8 is the larger number

Now subtract the smaller from the larger

2x - 8 - x = 17 then solve

x - 8 = 17x = 25 Next, find the larger by substituting the answerinto 2x – 82(25) - 8 = 50 - 8 = 42The two numbers are 25 and 42.

Try these problems:1. Five times an unknown number decreased by 7 is 43. Find

the number.    2. When 4 is subtracted from half of an unknown number the

result is 17. Find the number.3. The sum of 6 and twice an unknown number is 32. Find the

number.4. Fifty-two is 2 less than 6 times a number. Find the number.5. Eight less than 5 times a number is 57. Find the number.6. Eight less than 5 times a number is four more than 8 times

the number. Find the number.7. The sum of two numbers is 30. Three times the first plus

twice the second number is 72. Find the numbers.8. The sum of two numbers is 40. One number is one more

than twice the other. Find the numbers.9. The sum of two consecutive even integers is 106. Find the

integers.10. The sum of two consecutive odd integers is –40. Find the

integers.11. Find three consecutive integers if twice the largest is 16 less

than three times the smallest.Complete solutions follow this slide, so work these first before

moving to the next slide.

Complete solutions to Mystery Numbers: 1 – 11. 1.  Five times an unknown number decreased by 7 is 43. Find the number.      Read the problem several times. Guess the answer.             You might be able to get this one with just a couple of  guesses.        Define x, usually to answer the question.            Let x = the number.      Label other unknown quantities in terms of x.            This step is not necessary on this problem.        Form the equation.             5x – 7 = 43     Solve the equation.                  5x – 7 = 43 5x = 50 x = 10 Answer the question.  The number is 10. Check to see that 5 times this number decreased by 7 is 43.

2. When 4 is subtracted from half of an unknown number the result is 17. Find the number.a.       Read the problem several times. Guess.If we guess 40 then half of 40 is 20 and subtracting 4 from 20 leaves us with 16…pretty close. b.      Define x, usually to answer the question.Let x = the number. c.       Label other unknown quantities in terms of x.There are no other unknowns in this problem. d.      Form the equation.(1/2)x – 4 = 17e.       Solve the equation.(1/2)x – 4 + 4 = 17 + 4 (1/2)x = 21 (2)(1/2)x = 21(2) x = 42f.        Answer the question.Check to see that 42 behaves as advertised.

   

3. The sum of 6 and twice an unknown number is 32. Find the number.Let x = the number.      Form an equation.2x + 6 = 32 Solve the equation.2x + 6 = 32 2x = 26 x = 13 Answer the question. Check to see that the number 13 is correct.

4. Fifty-two is 2 less than 6 times a number. Find the number.  Let x = the number. Form an equation.52 = 6x – 2. Some people are tempted to try 52 = 2 – 6x. Be especially careful when translating subtraction. Solve the equation.52 = 6x – 2 54 = 6x 9 = x Answer the question. The mystery number is 9.    

5. Eight less than 5 times a number is 57. Find the number. Let x = the number 5x – 8 = 57   5x – 8 = 57 5x = 65 x = 13

Answer the question. To check, 5 times 13 is 65 and 8 less than 65 is 57.6.     Eight less than 5 times a number is four more than 8 times the number. Find the number.   Let x = the number.5x – 8 = 8x + 4 Solve the equation. 5x – 8 = 8x + 4 -3x = 12 x = -4 Answer the question. Is 5*(-4) – 8 = 4 + 8*(-4 ) -20 – 8 ? 4 – 32 -28 = -28 yes it checks

  7. The sum of two numbers is 30. Three times the first plus twice the second number is 72. Find the numbers.Let x = the first number. Label other unknown quantities in terms of x.Let 30 – x = the other number. Since the numbers have to add to 30, if we subtract one number (x) from 30 we will get an algebraic name for the other. Form an equation.3x + 2(30 – x) = 72 Solve the equation.3x + 60 – 2x = 72 x = 12 Answer the question. If x = 12 then 30 – x = 30 – 12 = 18. The two numbers are 18 and 12.

8. The sum of two numbers is 40. One number is one more than twice the other. Find the numbers.

Let x = one of the numbersLabel other unknown quantities in terms of x.

Let 40 – x = the other number. Form an equation.

x = 2(40 – x) + 1 Solve the equation.

x = 2(40 – x) +1 x = 80 – 2x + 1 3x = 81 x = 27 Answer the question. If x = 27 then 40 – x = 40 – 27 = 13.

9.   The sum of two consecutive even integers is 106. Find the integers.Let x = the numberLabel other unknown quantities in terms of x.Let x + 2 = the second even integer. Form an equation.x + (x + 2) = 106 Solve the equation.x + (x + 2) = 106 2x + 2 = 106 2x = 104 x = 52 Answer the question. If x = 52, then x + 2 = 54.

10.    The sum of two consecutive odd integers is – 40. Find the integers.Let x = the first odd integer. Label other unknown quantities in terms of x.Let x + 2 = the second odd integer. Form an equation.x + (x + 2) = - 40 Solve the equation.2x + 2 = - 40 2x = - 42 x = -21 Answer the question. If x = -21, then x + 2 = -19. Note that the two integers do add to – 40.

11.   Find three consecutive integers if twice the largest is 16 less than three times the smallest.Let x = the first integer Let x + 1 = the second integer. Let x + 2 = the third integer. Form an equation.2(x + 2) = 3x - 16 Solve the equation.2(x + 2) = 3x – 16 2x + 4 = 3x – 16 4 = x – 1620 = x Answer the question. If the first integer is 20 then the next two will be 21 and 22.