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MAC1105-1
Solving Algebraic
Word Problems
Purpose
To assist the student in using a structured methodology to translate word problems into
mathematical structures that can then be solved.
Translating Phrases or Sentences into Algebraic Expressions or Equations
Phrase or Key Word
Algebraic Symbol
Added to, Sum of, More than, Increased by, Greater than,
Excess of, Combined with, Total
Less than, Difference of, Decreased by, Reduced,
difference between/ of, fewer than
Of, Product of, Times, Multiplied by, Increased/decreased
by a factor of
Divided by, per, ratio of, quotient of, percent
(divide by 100)
Addition (+)
Subtraction (-)
Multiplication (x)
Division(÷)
Equal to, Result is, Will be, Gives, Yields
Equal (=)
Examples
Algebraic Equation
Sixteen subtracted from five times a number (n) equals the
number plus 4.
5n-16=n+4
Twenty five is five less than six n diminished by two.
25=(6n-5) - 2
Twice the quantity of n+ two divided by six is fifty.
2(n+2)\6 = 50
A number is equal to 50 less 9 times the number
n = 50 – 9n
The quotient of 16*n divided by 100 equals 16.
16 n/ 100 = 16
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Common Types of Algebra Word Problems
1. Age Problems: usually compare the ages of people. They may involve a
single person, comparing his/her age in the past, present or future. They may
also compare the ages involving more than one person.
2. Average Problems: involve the computations for arithmetic mean or weighted
average of different quantities. Another common type is the average speed
computation.
3. Coin Problems: deals with items with denominated values.
4. Consecutive Integer Problems: deals with consecutive numbers, and the
number sequences may be even or odd, or some other simple number
sequences.
5. Digit Problems: involve the relationship and manipulation of digits in numbers
6. Distance Problems: involve the distance an object travels at a rate over a
period of time.
7. Integer Problems: involves numerical representations of word problems
8. Interest Problems: involves calculations of simple interest.
9. Mixture Problems: involves items or quantities of different values that are
mixed together.
10. Proportion Problems: involves proportional and inversely proportional
relationships of various quantities.
11. Ratio Problems: require you to relate quantities of different items in certain
known ratios, or work out the ratios given certain quantities.
12. Work Problems: involve different people doing work together at different
rates.
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Structured Approach to Solve the Problem
A freight train leaves a station travelling at 30 mph. A passenger train leaves 1
hours later travelling at 50 mph. At what time will the passenger train overtake
the freight train?
1. Identify the unknown. An unknown is a quantity that your problem requires
you to find out, or the quantity that is necessary to find out in order to obtain
the solution.
Unknown is “time”, represented by‘t’
2. Develop the equation needed to solve the problem. The critical piece that you
need in this problem is the fact that the trains are at the same point at the
same time moment they meet.
3. Let's denote the time it takes the passenger train to overtake the freight train
as t. The freight train had 1 more hours to travel. S, the distance d1 that the
freight train has travelled, is it speed 30, multiplied by (t+1). The passenger
train's situation is even more straightforward. It had t hours to travel, and
since it is traveling with speed 50, it would have travelled d2 = t*50 miles.
Since both trains are in the meeting point at the same time, these two
distances are equivalent. That is, d1 = d2. What is the equation for finding out
that distance?
d1 = d2, or 30*(t+1) = t*50
4. Given the equation “30*(t+1) = t*50”, we can simplify it as :
5. 30*t +30 = 50*t
30 = 20*t
t = 30/20 = 1.5 hours
Cross check:
We previously stated 30*t +30 = 50*t. Therefore, insert the value of‘t’
that was found to verify its accuracy:
30 * 1.5 + 30= 75 miles
50 * 1.5 = 75 miles
Since both sides are truly equal, 1.5 hours is the correct solution. This
solution can further written in English:
It will take the passenger train 1.5 hours to overtake the freight
train. At that point both trains will have traveled 75 miles.
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Selected Examples
Coin Problem
Joan bought $5.61 worth of stamps. She bought the same number of 2- and 5cent stamp and double that number of 22-cent stamps. How many of each kind
of stamp did she buy?
1. Identify the unknown(s). An unknown is a quantity that your problem requires
you to find out, or the quantity that is necessary to find out in order to obtain
the solution.
x= number of 2-cent stamps
x= number of 5-cent stamps
2x= number of 22-cent stamps.
2x= value in cents of 2-cent stamps
5x=value in cents of 5-cent stamps.
22(2x) = value in cents of 22-cent stamps.
$5.61= 561 cents
2. Develop the equation needed to solve the problem.
2x + 5x + 22(2x) =561
3. Given the previous equation, we can simplify it as :
2x + 5x+ 44x = 561
51 x = 561;
x = 11 therefore:
x = 11 2-cent stamps
x = 11 5- cent stamps
2x = 22 22-cent stamps
Cross check:
We previously stated:
2x=
5x=
22(2x) =
value in cents of 2-cent stamps
value in cents of 5-cent stamps.
value in cents of 22-cent stamps.
With x=11:
2x=
22 cents
5x=
55 cents
44x= 484 cents
561 cents
Since the values add up to $5.61, we have the correct answer of x=11.
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Work Problem
Bill can mow the lawn in 2 hours, and his younger brother Tom can mow it in 3 hours. If
both brothers mow the lawn together, how many hours will it take them?
Model:
Persons
Bill
Tom
Together
Equation:
Hours to mow the lawn
2
3
x
ଵ
ଶ
ଵ
ଵ
ଷ
௫
൅ ൌ
Part done in one hour
1/2
1/3
1/x
Since the LCD= 6x, the equation be rewritten as 3x + 2x = 6, which reduces to
5x = 6
Cross Check
Verify 1/2 + 1/3 = 1/ 6/5
1/2 + 1/3 = 5/6 (after inverting the denominator and multiplying by 1)
Using LCD of 6, rewrite the equation to 3/6 + 2/6 = 5/6. This equation is
correct, so x does equal 6/5
Therefore, it will take 1 hour and 12 minutes to finish the job together.
Age Problem
Bill is 5 years older than his brother Jim. Five years ago, Bill was twice as old as Jim.
How old is each now?
Model:
Persons
Bill
Jim
Age Now
x+5
x
Age 5 years ago
x+5-5
x-5
Equation: x + 5 – 5 = 2 (x - 5)
x = 2x - 10
10 = 2x - x
10=x (Jim’s age now)
15=x+5 (Bill’s age now)
Cross Check
Is Bill 5 years older than Jim? Yes
Five years ago was Bill twice as old as Jim?
Bill was 10; Jim was 5, so the answer is ‘yes’. The answers are correct.
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Algebraic Word Problems Work Sheet
1. Bob's father is 3 times old as Bob. 4 years ago, he was 4 times older. How old is
Bob?
2. Jim invested $1000 in a bank savings account. The bank pays him 5% annually,
that is, for every dollar invested he would get $1.05 next year. How much would
Jim have 2 years from now?
3. Mr. Warren Buffett invested $20000. Part of it he put in the bank at 5% interest.
Part of it he invested in bonds which pay a 7% return. How much money did he
put in each vehicle, if his annual income from these investments was $1200?
4. The sum of two numbers is 5 times their difference. If one exceeds the other by
7, what are the numbers?
5. Find three consecutive even integers so that the largest is 2 times more than the
smallest.
6. A barge has speed over water of 5 miles per hour. A river flows downstream at
the speed of 1 miles per hour. How long will it take the barge to go from point A
to point B upstream, and then back, if the distance from A to B is 24 miles?
7. In a right triangle, one of the acute angles is 2 times as large as the other acute
angle. Find the measure of the two acute angles.
8. The Mandarin High School marching band sold gift wrap to earn money for a
band trip to Palatka, Florida .The gift wrap in solid colors sold for $4.00 per roll
,and the print gift wrap sold for $6.00 per roll .The total number of rolls sold was
480 ,and the total amount of money collected was $2340. How many rolls of
each of each kind of gift wrap were sold?
9. A plane takes 6 hours to fly from San Francisco to New York, and 5 hours to
return back. The wind velocity is 50 miles per hour, from New York to San
Francisco. What is the speed of the airplane.
10. The sum of two numbers is 41. The larger number is 1 more than 4 times the
smaller number. What are these numbers?
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Algebraic Word Problems Answers
1. Let Bob=x,
Bob’s father= 3x
Four years ago, Bob was x-4 and Bob’s father was 3x-4 At that time Bob’s father
was 4 times older. Therefore:
Bob’s father’s age 4 years ago)
3x-4=4*(x-4)
Bob’s age 4 years ago)
3x-4=4x-16
x= 12
(Bob’s age today)
3x= 36
(Bob’s father’s age today)
2. Beginning balance: $1,000.00
Interest at the end of year 1: .05*1000= $50.00
Balance at the end of Year 1: $1000.00 + $50.00= $1,050.00
Interest at the end of Year 2: .05*1,050= $52.50
Balance at the end of year 2: 1,050 + 52.50= $1,102.50
3. Let x = amount of money Mr. Buffett put in the bank
Let 20000-x = the amount of money in bonds.
Invest Return is [.05 * x] + [.07* (20000-x)]= 1200
.05x + 1400-.07x =1200
-.02x=200
x=$10,000 (Bank)
20,000-x=$10,000 (Bonds)
Cross Check: Income equals .05*10000 + .07*10,000=$1,200
4. Let n= first number
Let n+7= second number
Equation: (n) + (n+7)= 5((n+7)-n)
Simplify: 2n+7=35
2n=28
n=14
14 + 7=21
First number
Second number
14
21
Cross Check: Is the sum = 5 times their difference?
21+14=5(21-14)
35=5*7=35 Yes, the answers are correct.
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5. Let
n = First integer
(Smallest)
Let n+2 = Second integer
Let n+4 = Third integer
(Largest)
Equation: n+4=2n
n=4
Integers are therefore 4, 6, 8
6.
A
B
Current against you
24 miles
Going from Point A to Point B (upstream) =
24 miles / (5–1) mph = 6 Hours
Going from Point B to Point A (downstream) = 24 miles / (5+1) mph= 4 Hours
Total Time= 6 + 4 = 10 hours
Current with you
7. Let x= First acute (< 90 degrees) angle
Let 2x= Second acute angle
In a right triangle, one of the angles, by definition, is 90 degrees. All three internal
angles must add up to 180 degrees.
Equation: x + 2x +90 = 180
3x=90
x=30 degrees
First acute angle:
30
2x=60 degrees
Second acute angle:
60
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8. Let x= # rolls of solid
Let 480-x= #rolls of print
4x + 6(480 –x) = $2,340
4x + 2880 -6x = 2340
-2x = -540 such that x=270
Revenue= $4(x)
Revenue- $6(480-x)
Therefore, 270 rolls of solid and 210 rolls of print were sold.
Cross Check: 270 (4) + 210 (6) = 1080 + 1260 = $2,340
Correct!
9.
6 hours
5 hours
24 miles
MPH: Wind with you
Let v = Speed of airplane
Going from NY to SF (with the wind)=
5 hrs * (v + 50)= Distance
Going from Sf to NY (into the wind)=
6 hrs * (v - 50)= Distance
Equation: 5 (v + 50) = 6 (v -50)
MPH: Wind against you
5v + 250 = 6v -300
v= 550 mph
Cross Check: 5(550 + 50)= 6 (550 -50); 3000=3000
Answer is correct
10. Let n= smaller number; 1 + 4n is the larger number
Equation: n + (1 + 4n) = 41
5n +1 = 41
5n = 40; therefore n=8
Small Number: 8
Large Number: 33
Cross Check: 8 + (1+4(8))=41; 8 + 33 = 41
Yes; answer is correct!
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