Common Algebra Word Problems Made Easy!

(Consecutive Integer, Triangles, Mixed
Interest, D=RT, Mixtures, etc.)

Mr. Martin, 8^{th} Grade Algebra (?
2004-2007 Mark D. Martin)

__Some
Examples You Already Know How to Do__

- Name three consecutive integers that
equal 39.

- Second angle of a triangle is 20? more
than the first. The measure of the third angle is twice the measure of the
first angle. Find all three
angles.

- I invest $30,000 for one year. Part is
invested at 2% interest per annum and the rest is invested at 3% per
annum. I earn $800 after one
year. How much did I invest
at 2% and how much did I invest at 3%?

__General
Method to Solve__

First,
read the problem carefully. Decide what information you are given and what is
being asked. Draw a diagram if
applicable. Remember, there is more than one piece of information missing in
all these problems. After you have done this, do the following:

- Define
variable. Let n = (describe in English what n equals)__only one__

- Using the same variable, define the
other unknowns.

- Using the information given, write an
equation.

- Solve the equation for the variable.

- Find the other answers by going back
to the definitions you made.

__Solutions to
Examples__

__Problem 1:
Three consecutive integers equal 39____.__

**Step 1:** Define variable

Let
n = the first integer.

**Step 2: **Using the same variable, define the other
unknowns.

n
+ 1 = the second integer.

n
+ 2 = the third integer.

**Step 3:** Write equation that sum of integers equals
39.

n + (n + 1) + (n + 2) = 39

**Step 4:** Solve equation

n
+ (n + 1) + (n + 2) = 39

3n
+ 3 = 39

3n
= 36

n
= 12

**Step 5: **Find the other answers by going back to the
definitions you made.

n
+ 1 = the second integer. 12 + 1 =
13

n
+ 2 = the third integer. 12 + 2 =
14

**Answer
is 12, 13, 14**

__Problem 2
– Triangle –second angle 20? more than first, third angle twice the
first__

**Step 1:** Let n = the first angle.

**Step 2:** Let n + 20 = the second angle

Let 2n = the third angle

**Step 3:** n + (n + 20) + 2n = 180

**Step 4:** 4n + 20 = 180

4n = 160

n = 40

**Step 5:** second angle = n + 20 = 40 + 20 = 60

third angle = 2n = 2(40) = 80

**Answer:
40?, 60?, 80?**

__Problem 3
– invest $30,000 for one year, part at 2%, part at 3%, earn $800__

**Step 1:** Let x = amount invested at 2%

**Step 2:** $30,000 – x = amount invested at 3%

**Step 3: **Remember I = prt. Here, interest (I) ($800) will equal interest on amount (x)
at 2% (.02) and interest on amount ($30,000 –x ) at 3% (.03). t is one
year. Hence, x(.02)(1) + (30,000 – x)(.03)(1) = 800

**Step 4:[1]**

x(.02)(1)
+ (30,000 – x)(.03)(1) = 800

.02x
+ 900 - .03x = 800 (applying distributive property)

-.01x
+ 900 = 800 (combining like terms)

-.01x
= -100 (subtracting 800 from both sides of equation)

x
= $10,000 (dividing both sides by .01)

**Step
5:**

x = $10,000 = amount invested at 2%

$30,000 – x = $30,000 - $10,000 =
$20,000 = amount invested at 3%

__New
Types of Problems__

Solve
other types of problems the same way.

__Distance
= Rate x Time Problems__

__ __

__Example
1__: Two trains leave Los
Angeles at the same time. Train A
travels north. Train B travels south.
At the end of two hours they are 180 miles apart. Find the rate of both
trains if Train A is traveling 10 miles per hour slower than Train B.

**Preliminary
Steps:** On these types of
problems it is helpful to draw a diagram and a chart. Also, remember that
distance = rate x time. Rate is
the same as speed. The other steps are the same.

Drawing
and Chart

**Rate**
**Time**

**Distance**

**Train A**

x

2 hours

2x

**Train B**

x + 10

2 hours

2(x +10)

To San Francisco Los Angeles To San Diego |

**Step
1:** Let x = the rate of
train A

**Step
2: **x + 10 = the rate of
train B

**Step
3:** The total distance, 180
miles, equals the distance train A went plus the distance train B went. Distance = rate multiplied by
time or D = rt. t = 2 hours

2x
+ 2(x + 10) = 180

**Step
4:**

** **2x + 2(x + 10) = 180

2x
+ 2x + 20 = 180 (distributive property)

4x + 20 = 180 (combine like terms)

4x = 160 (subtract 20 from both sides)

x = 40 (Divide both sides by 4)

**Step
5:** Train A?s rate was 40
mph. Train B?s rate was x + 10 =
50 mph.

** **

[1] It is good form to include the units on all calculations and perform the dimensional analysis. This also helps insure you are doing the problem correctly. For sake of simplicity here, however, I have not included the units or dimensional analysis.