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2
WORD PROBLEMS
STEP
1.
LABEL UNKNOWNS
a. DRAW A PICTURE , if possible, or
b. MAKE A TABLE , or
c. USE “LET” STATEMENTS
NOTE: In order to determine what the unknowns are, simpl y read the
question. Usuall y, we let one of the unknowns be x, and label the
other unknowns in terms of x.
NOTE: The problem will give you two distinct pieces of information
about the unknowns. Use one of them to Label the Unknowns and
the other to Write the Equation.
2. WRITE
EQUATION
3.
SOLVE EQUATION
4.
CHECK SOLUTIONS
a. Answer all the questions (Go back to where you labeled the
unknowns).
b. Check to see that your solutions are feasible and do what
they are supposed to.
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MIXTURE
PERCENT
x
+
AMOUNT
PROBLEMS
PERCENT
x
=
AMOUNT
PERCENT
x
AMOUNT
THE EQUAT ION IS:
(PERCENT x AMOUNT) + (PERCENT x AMOUNT) = (PERCENT x AMOUNT)
EXAMPLE:
How many gallons on a 12% salt solution must be combined with a 42% salt solution
to obtain 30 gallons of an 18% solution?
12%
x
42%
+
30 – x
18%
=
30
.12x + .42(30 – x) = .18(30)
12x + 42(30 – x) = 18( 30)
12x + 1260 – 42x = 540
-30x = -720
x = -720/ -30
x = 24
24 gallons of the 12% salt solution must be used.
INVESTMENT PROBLEMS ARE A TYPE OF MIXTURE PROBLEMS.
EXAMPLE:
If you have t wice as much invested at 8% as at 5% and if your annual interest income
from these two invest ments is $315, how much is invested at each rate?
5%
x
.05x + .08(2x) = 315
5x + 8(2x) = 31500
5x + 16x = 31500
21x = 31500
x = 31500/21
x = 1500
$1500 at 5% and $3000 at 8%
8%
+
2x
=
315
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MOTION
D
PROBLEMS
R
T
A
B
THREE COLUMNS:
1.
G IVEN
2.
U KNOWN
3.
F ORMULA
You are gi ven the two distances, and the t wo rates, or the tw o ti mes.
Read the question to determine this column.
D = RT
R = D/T
T= D/R
THE EQUAT ION COMES FROM THE FORMULA COLUMN!!!
EXAMPLE:
The speed of a stream is 4 m/h. A boat travel s 36 miles downstream in the same ti me
it travels 12 miles upstream. Find the speed of the boat in still wat er.
THREE COLUMNS:
1.
G IVEN
You are gi ven the two DISTANCES , 36 miles downstream and 12
miles upstream.
2.
U KNOWN
You are asked to find the speed (RATE) of t he boat in still water.
Let this be x. Keep in mi nd that all entries i n the table appl y to
the boat in this stream, which is moving at a speed of 4 m/h.
Therefore, the rate of the boat going downst ream is x + 4 m/ h, and
the rate of the boat going upstream is x – 4 m/ h.
3.
F ORMULA
The remainin g column is the T IME column, and the for mula for
time is T = D/R; hence, the ti me going downstream is 36/(x+4)
and the ti me going upstream is 12/(x -4).
DOWNSTREAM
UPSTREAM
D
36
12
R
x + 4
x – 4
T
36/(x + 4)
12/(x – 4)
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Since the EQUAT ION come fr om the FORMULA COLUMN, we must read the problem
again and find a relati on between the two T IMES. It says the boat travels downstream
and upstream “in the SAME T IME,” hence
36
12
x4 x4
CLEAR the fractions by cross -multipl ying.
36(x – 4) = 12(x + 4)
Now CLEAR the parentheses by using the
distributive propert y.
36x – 144 = 12x + 48
Next, ISOLATE the “x -ter ms,”
36x – 12x = 48 + 144
…and combine li ke ter ms.
24x = 192
Now ISOLATE the x by di viding both sides by
its coefficient,
x = 192/24
…and si mplif y.
x = 8
The speed of the boat in still water is 8 m/h .
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WORK
PROBLEMS
ALTERNATE METHOD (Necessar y for some Work Problems after DSM 086):
TIME
ALONE
RATE
TIME
WORK ING
PART
COMPLETED
A
B
THE EQUAT ION COMES F ROM THE LAST COLUMN. THE TWO PARTS
COMPLETED ADD UP TO 1 COMPLETED JOB.
EXAMPLE:
In a certain post office, Alice can sort a stack of mail in 30 minut es; Bob can sort the
same stack in 40 mi nutes. If they wor k toget her, how fast can they sort the stack?
TIME
ALONE
RATE
TIME
WORK ING
PART
COMPLETED
Alice
30
1/30
x
x/30
Bob
40
1/40
x
x/40
x/30 + x/40 = 1
40x + 30x = 120
70x = 120
x = 120/70
x = 17.1 min
Working together, they can sort the stack of mail in about 17 min.
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WORK
Let
PROBLEMS
t 1 = ti me it t akes f or the first participant to do the j ob
t 2 = ti me it t akes f or the second participant to do the j ob
etc…
T = ti me it t akes f or all the participants to do the j ob w orking together
FORMULA:
1

1
t t
1
 
2
1
T
NOTE: If a participant is wor king against the others, then the fract ion 1/t for that
participant is negati ve instead of positi ve.
EXAMPLE:
In a certain post office, Alice can sort a stack of mail in 30 minut es; Bob can sort the
same stack in 40 mi nutes. If they wor k toget her, how fast can they sort the stack?
1
1 1


30 40 x
40x + 30x = 120
70x = 120
x = 120/70
x = 17.1 min
Working together, they can sort the stack of mail in about 17 min.
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MIXTURE
PROBLEMS
1. How many quarts of cr eam containing 36% butterfat must be added to 100 quarts
of mil k containing 3% butterfat to produce 4% milk?
2.
An automobile radiator contains 12 quarts of a 20% antifreeze and water
solution. How much must be replaced by 80% antifreeze to make a 30% solution?
3.
Al invested $10,000 in two savings accounts, one that paid 4% and the other
5%. If his total earnings in one year was $480, then how much was invested at each
rate?
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MOTION
PROBLEMS
1. One hour and 20 minutes after a hiker started down a mountain trail a ranger
started after him on horseback and overtook hi m in 40 minutes. Find the speed
of each if the ranger r ode 6 miles per hour faster than the man wal ked.
2. A lady drove to a cit y 30 miles away to shop and returned home in the evening.
She spent 15 minutes longer dri ving the retur n trip than in the going, and she
drover three -fourths as fast when returning as she did when going to the city.
How long did it take her to drive to the cit y?
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WORK
PROBLEMS
1. A boy could spade his garden in 6 hours, whi le his father could spade it in 4
hours. How long woul d it take the two wor ki ng together to spade t he garden?
2. It takes Ashley three t imes as long to paint a room as it does Aaron. Wor king
together, they can paint the room in 6 hours. How long would it take Aaron to
paint the room?
3. A laborator y instructor can prepare the solut ions for a certain chemical
demonstration in 1 hour, while his assistant requires 1 ½ hours to prepare the
same solutions. After t hey had wor ked together for 30 minutes, the instructor
had to meet a class and the assistant finished the wor k. How long did it take
hi m?
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WORD PROBLEMS
Age Problems
1. Rick is 8 years older than Steve. The sum of their ages is 46 years. How old is
each now? (Rick-27, Steve-19)
2. Charles is 4 years older than his wife Diana. The sum of their ages is 52. What
are their present ages? (Charles -28, Diana-24)
Age Problems – 2 time frames
1. Sue and Joan are sisters. Joan is 7 years older than Sue. In 3 years Joan’s age
will be twice Sue’s age. What are their present ages? (Sue -4, Joan-11)
2. Selma is now 3 times as old as Jo yce. Four years ago, Selma was 4 times as old
as Joyce was then. Find their present ages. (Selma -36, Joyce-12)
3. Mrs. Barry is 20 years older than Mrs. Cook. Sixteen years ago, Mrs. Barry was
3 times as old a Mrs. Cook was then. Find their present ages. (Mrs. Barry -46, Mrs.
Cook-26)
4. Amy is 8 years older than her pet hamster George. In 2 years Amy’s age will be
3 times George’s age. What are their present ages? (Amy -10, George-2)
5. David is 3 years older than his brother John. Three years ago John’s age was 6
more than one-fourth of David’s age. What are their present ages? (David -15, John12)
Consecutive Integer
1. The sum of three consecutive integers is 300. What are the integers?
(99,100,101)
2. The sum of two consecutive integers is -34. What are the integers? ( -18, -16)
3. The sum of two consec utive odd integers is 1460. What are the integers? (729,
731)
4. The sum of three consecutive integers is -54. What are the integers? ( -19, -18, 17)
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5. Find three consecutive integers with the property that 3 times the first integer
plus the second integer mi nus 2 times the third integer is 9. (6, 7, 8)
6. Find three consecutive odd integers with the propert y that 3 times the first
integer plus twice the second integer is 5 more than twice the third integer. (3, 5,
7)
Perimeter
1. The perimeter of a rectangle is 21 cm. The length is 9.5 cm greater than the
width. Find the width and length of the rectangle. (width -.5 cm, length10cm)
2. The perimeter of a rectangle is 38 m. The width is 13 m less than the length.
Find the width and the length. (width -3m, length-16m)
3. The perimeter of a rectangle is 64 ft. The length of the rectangle is 3 ft less
than 4 times the width. What are the dimensions of the rectangle? (width -7
ft, length -25 fit)
4. The perimeter of a rectangle is 50 yd. The width is 1 yd more than one -third
of the length. What are the dimensions of the rectangle? (width -7 yd, length18yd)
5. The perimeter of a triangle is 17 cm. The second side is twice the length of
the first side, and the third side is 1 cm less than 3 times the length of the
first side. What a re the lengths of the sides of the triangle? (3cm, 6cm, 8cm)
6. The perimeter of a rectangle is 26 in. If 5 in. are added to the width of the
rectangle, a square with perimeter 36 in. is obtained. What are the
dimensions of the original rectangle? (width -4in, length-9in)
Unknown number
1. When 15 is subtracted from 11 times a certain number, the result is -37.
What is the number? ( -2)
2. If you double a number and then add 14, you get 5/6 of the original number.
What is the original number? ( -12)
3. If you subtrac t 2/3 of a number from the number itself, you get 7. What is
the number? (21)
4. If 5 times a number is decreased by 8, the result is three times the number
increased b y 4. Find the number. (6)
5. The difference between two numbers is 24. Find the numbers is t heir sum is
88. (32, 56)
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6. The larger of two numbers is 5 less than twice the smaller. If their sum is
70, find the numbers. (25, 45)
7. A 450 m wire is cut into 3 pieces. The second piece is 11 times as long as
the first. The third piece is 3 times as long a s the second. How long is each
piece? (10m, 110m, 330m)
8. Jack has $4.80 in quarters and dimes. He has a total of 30 coins. How many
quarters and how many dimes does he have? (18 dimes, 12 quarters)
9. Ray has $435 in fives, tens, and twenties. He has 2 more $10 bills than $5
bills. How many of each type of bill does he have? (17 fives, 19 tens, 8
twenties)
10. The difference of two numbers is 7. The sum of the numbers is -1. What are
the numbers? (-4, 3)
Motion Problems
1. Henry leaves town at noon on a bicycle t raveling west at a rate of 15 mph.
At 4:00 pm Betty leaves town in a car traveling in the same direction at a
rate of 45 mph. How many hours will it take Betty to catch Henry?
2. Jennie leaves school in a car traveling north at a rate of 45 mph. At the same
time Albert leaves school in a car traveling south at a rate of 55 mph. In how
many hours will Albert and Jennie be 250 miles apart? (2.5 hours)
3. A car travels from one town to another in 6 hours. On the return trip, the
speed is increased b y 10 mph and th e trip takes 5 hours. Find the rate on the
return trip. How far apart are the towns? (50 mph, 300 mi)
4. Carol has 8 hours to spend on a hike up a mountain and back again. She can
walk up the trail at an average of 2 mph and can walk down at an average of
3 mph. How long should she plan to spend on the uphill part of the hike? (4
4/5 hr)
5. Maria jogs to the country at a rate of 10 mph. She returns along the same
route at 6 mph. If the total trip took 1 hour 36 minutes, how far did she jog?
(6 mi)
6. Tom’s rowboat can travel 7 mi downstream in the same time that it can
travel 3 mi upstream. If the speed of the current is 1 mph, what is Tom’s
rowing speed in still water?
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Investment Problems
1. Phil has $20,000, part of which he invests at 8% interest and the rest at 6%.
If his total income was $1,460 from the two investments, how much did he
invest at each rate? ($13,000 at 8%, $7,000 at 6%)
2. Larr y made two investments totaling $21,000. One investment made him a
13% profit, but on the other investment he took 9% loss. If his net loss was
$196, how much was each investment? ($7,700 at 13%, $13,300 at 9%)
3. Jane had $30,000, part of which she invests at 9% interest and the rest at
7%. If her income from the 7% investment was $820 more than that from the
9% investment, how much did she invest at each rate? ($8,000 at 9%,
$22,000 at 7%)
4. An investment of $3,500 is made at an annual simple interest rate of 13.2%.
How much additional money must be invested at an annual simple interest
rate of 11.5% so that the total inter est earned is $1,037? ($5,000)
Mixture Problems
1. How many ounces of a gold alloy that costs $320 an ounce must be mixed
with 100 oz of an alloy that costs $100 an ounce to make a mixture that costs
$160 an ounce? (37.5 ounces)
2. How many pounds of peanuts that cost $2.25 per pound must be mixed with
40 lb of cashews that cost $6.00 per pound to make a mixture that costs
$3.50 per pound? (80 pounds)
3. A chemist mixes an 11% acid solution with a 4% acid solution. How many
milliliters of each solution should the chemist use to make a 700 ml solution
that is 6% acid? (200ml of 11%, 500 ml of 4%)
4. How many quarts of water must be added to 5 qt of an 80% antifreeze
solution to make 50% antifreeze solution? (3 qt)
5. A tea mixture was made from 30 lbs of tea costing ^ 6.00 per pound and 70
lbs of tea costing $3.20 per pound. Find the cost per pound of the tea
mixture. ($4.04 per pound)
Work Problems
1. Joyce can mow her yard in 2 hr. Her son Danny can mow the yard in 3 hr.
Working together, how long will it take them to mow the yard? (6/5 hr)
2. Jimmy can wash his car in 30 min. His 4 year old son Jimmy Jr. takes 2
hours to wash the car. Working together how long will it take them to wash
the car? (24 min)
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3. Working together, Larry, Joe, and Shirle y can wash the windows at t he Moss
Country Courthouse in 1 hr. Working alone, Larry can wash the windows in
2 hr and Joes can wash the windows in 3 hr. How long does it take Shirley to
wash the windows? (6 hrs)
4. Gary takes twice as long as Barbara to sort the Moss Country mail. Toge ther
they can sort the main in 8 hr. How long does it take each of them to sort the
mail alone? ( Barbara -12 hrs, Gar y-24 hrs)
5. A swimming pool can be filled by an inlet pipe in 12 hrs. The drain can
empty the pool in 36 hrs. How long will it take to fill the pool if the drain is
left open? (18 hrs)