3-6 Equations & Problem Solving

The first step is to define the variables…There is no magic formula for doing this because each problem is a little different.

You need to identify the variable based on the problem.Usually you will need to identify one variable in terms of another. Ex.

The perimeter of a rectangle is 24 in. One side is 6 in. longer than the other side.The variables are side #1 and side #2.

Identify one of them as s,If Side #1 = s, then Side #2 = s + 6

There are 5 typical types of problems:Perimeter problems

Consecutive integer problemsCombined money problemAge/Comparison problems

Distance problems

Let’s do an example of each!

The perimeter of a rectangle is 24 in. One side is 6 in. longer than the other side. What are the dimensions?

Perimeter problems:These are usually pretty easy! You can always draw it…

The variables are side #1 and side #2Identify one of them as s,If Side #1 = s, then Side #2 = s + 6Since the perimeter is 24…add up all sidesAnd set them equal to 24.

s

s + 6

s + s + (s + 6) + (s + 6) = 24 4s + 12 = 24 4s = 12

s = 3 This isn’t the complete answer!The answer is 3 units by 9 units

The sum of 3 consecutive integers is 258, find them!

Consecutive integer problems:These are usually pretty easy as well!

The variables are integer #1, integer #2, and integer #3Identify one of them as x,

The next one would be 1 more, or x + 1The next one would be 2 more or x + 2

Remember that their sum is 258.

x + (x + 1) + (x + 2) = 258 3x + 3 = 258

3x = 255 x = 85 This isn’t the complete answer!The answer is 85, 86 & 87

Tony has twice as much money as Alicia. She has $16 less than Ralph. Together they have $200. How much money does each person have?

Combined money problems:

The variables are Tony’s $, Alicia’s $, and Ralph’s $.The key to these problems is picking the right variable!

So, let Alicia’s $ = a

That means Tony’s $ = 2a and Ralph’s $ = a + 16

And together they have a sum of 200…so the equation would be:

a + 2a + (a + 16) = 200

Look at the problem, and identify which one is somehow related to the other two. I know the relationship b/t Tony’s & Alicia’s

Alicia’s & Ralph’sSince Alicia is connected to both, make her $ the variable.

Age or comparison problems:These are very similar to the combined money problems.The key to these problems is picking the right variable!

State College has 620 students. There are 20 more women than men. How many men are there? How many women are there?If w = the number of women, then the number of men = w – 20Alternative methodIf m = the number of men, then the number of women = m + 20It’s a good idea to avoid subtraction if possible, but either method will work!The number of men plus the number of women = 620

m + (m + 20) = 620 2m + 20 = 620 2m = 600 m = 300

This isn’t the complete answer!There are 300 men and 320 women

Distance problems tend to be the toughest.

The problem is usually translating them from English into Algebra.

Start with the formula for speed (aka rate)

Rate =distance

timeThis formula can be manipulated…..

times

Start with the formula for speed (aka rate)

Rate =distance

timeThis formula can be manipulated…..

You end up with 3 different (similar) formulas: d = rt

r =

t =

dtdr

There are also three types of distance problems:

•Motion in opposite directions•Motion in the same direction•Roundtrip

Each type of problem is solved a little differently…depending on what information

you have.

Motion in opposite directions problems usually used1 + d2 = total d

Motion in the same direction usually use d1 = d2

Roundtrip problems usually use d1 = d2

No matter which type of problem it is, you should set up a chart like this:

distance timerate

Ex. Motion in opposite directions problems usually used1 + d2 = total d

Jane and Peter leave their home traveling in opposite directions on a straight road. Peter drives 15 mph faster than Jane. After 3 hours, they are 225 miles apart. Find Peter’s rate and Jane’s rate.

Jane Peter

225 miles

What should the variable be?Look at the question…find the rate…there’s the clue!

Ex. Jane and Peter leave their home traveling in opposite directions on a straight road. Peter drives 15 mph faster than Jane. After 3 hours, they are 225 miles apart. Find Peter’s rate and Jane’s rate.

If Jane’s rate = r, what is Peter’s rate?Peter’s rate = r + 15

The time and the distance are both given!t = 3 hours and d = 225 milesHow do I put this all together using a formula?

Think about the distance travelled…it’s their TOTAL distance…I have to add Jane’s distance plus Peter’s distance to get 225.

distance timerate

Jane

Peter

r

r + 15

3

3

Jane’s distance + Peter’s distance = 225 miles

3r + 3(r + 15) = 225 3r + 3r + 45 = 225 6r + 45 = 225 6r = 180 r = 30

Look back at how you defined the variables….Jane’s rate = 30 mph and Peter’s rate = 45 mph

Jane’s rate = rJane’s time = 3

So Jane’s distance = 3r

Peter’s rate = r + 15Peter’s time = 3

So Peter’s distance = 3(r + 15)

Peter’s distance = Peter’s rate * Peter’s timeJane’s distance = Jane’s rate * Jane’s time

Jane’s distance + Peter’s distance = 225 miles, so:

Motion in the same direction problems usually use d1 = d2

Ex. Motion in opposite directionsJane and Peter leave their home traveling in the same direction. Peter drives 55 mph, Jane drives 40mph. Jane leaves at noon, Peter leaves at 1 P.M. When will Peter catch up to Jane?

distance timerate

Jane

Peter

40

55

What should the variable be?Look at the question..”when”..there’s the clue…find the time!

NB! time = time spent traveling, NOT time of day!

Since she left 1 hour earlier, her time is

t + 1

t + 1

t

If Peter’s time = tWhat is Jane’s time?

Motion in the same direction problems usually use d1 = d2

Jane and Peter leave their home traveling in the same direction. Peter drives 55 mph, Jane drives 40mph. Jane leaves at noon, Peter leaves at 1 P.M. When will Peter catch up to Jane?

distance timerate

Jane

Peter

40

55

t + 1

t

Use the formula d = r * t

40(t+1)

55t1510 40(t+1) = 55t

40t + 40= 55t 40 = 15tt = 2

t = 2

1510

32

What does this mean? When Peter has been traveling 2 hours and Jane has been traveling 3 , they will have travelled the same distance…or to put it another way, Peter will have caught up to Jane. The time will be 3:40 P.M.

32 3

2

Jane’s d = Peter’s d

Ex. Round trip problemJane goes to the mall. She drives 35mph. On the way home, there is lots of traffic. She averages 15 mph. Her total travel time was 2 hours. How long did it take her to get to the mall?

distance timerate

there

back

35

15

What should the variable be?Look at the question…”how long”...there’s the clue…find the time!

If Jane’s time there = tWhat is her time home?Since her total time is 2 and it takes her t hours to get there, her time home is 2 – t

t

2 - t

Roundtrip problems usually use d1 = d2

You can’t avoid subtraction on this one.

Jane goes to the mall. She drives 35mph. On the way home, there is lots of traffic. She averages 15 mph. Her total travel time was 2 hours. How long did it take her to get to the mall?

distance timerate

there

back

35

15

t

2 - t

Use the formula d = r * t d there = d back 35t = 15(2-t)35t = 30 – 15t50t = 30 t =

53

35t

15(2-t)

What does this mean? It took Jane of an hour to get to the mall and it took her 1 hours to get home. If you wanted to convert this to minutes, just multiply by 60.

53

52