Download applications: solving word problems using linear

APPLICATIONS: SOLVING WORD PROBLEMS USING LINEAR EQUATIONS
Introduction: Solving an algebraic problem consists of finding a number or set of numbers
(the unknown), when other numbers related to the unknown (the data), are given. The
relation between the data and the unknown will be the basis to set up an equation whose
solution is the answer to the problem.
Example: Certain number added to its half is 240. What is the number?
Referring to the unknown as x, the equation would be
x
1
x  240
2
The solution, x = 160, is the desired number. In fact, 160 + 160/2 = 240.
Steps for Solving Word problems: Solving word problems through algebraic equations
requires, in general, the following steps:
1.
2.
3.
4.
5.
Identification of unknowns
Selection of variables to represent the unknowns
Set up the equation
Solve the equation
Interpretation and verification of results.
Example: A mother is seven years older than twice the age of her son. If the mother is 47
years old, how old is her son?
Unknown: Son's age
Variable:
x
Equation:
2x + 7 = 47
Solution:
2x = 40
x = 20
Answer:
The son is 20 years old
Verification: 2(20)+7=40+7=47.
Number Problems: Statements expressing relationships between numbers. The goal is to
translate the words into algebraic expressions, set up the equation and solve it by known
means.
Key words:
Increase, more, more than
Decrease, less, less than
Consecutive integers
Consecutive odd or even integers
Times, times more, of (after a fraction)
Times less
Is
Meaning
Addition
Subtraction
Differ by 1
Differ by 2
Multiplication
Division
Equal
Examples: 1. One third of a number is 7 less than one-half of the number. Find the number.
Let the number be x. The corresponding equation is:
1
1
x  x7
3
2
Multiplying by 6 we get
2x  3x  42
or
x  42
Therefore the number is 42.
In fact, 42/3 = 14, and 42/2 - 7 = 21 - 7 = 14.
2. Find two consecutive even integers such that 4 times the larger is 8 less than 5 times the
smaller.
First even integer:
x
Second even integer:
x+2
Equation:
4(x + 2) = 5x - 8
x = 16
The numbers are 16 and 18.
In fact, 4(18) = 72 and 5(16) - 8 =80 - 8 = 72.
3. The sum of two numbers is 24. The smaller is three times less than the larger. Find the
numbers.
Larger number:
x
Smaller number:
x/3
Equation:
x + x/3 = 24
3x + x = 72
4x = 72
x = 18
The numbers are:
18 and 6
In fact, 18 + 6 = 24 and 18/3 = 6.
Percent Problems: Sometimes the relation between two numbers is expressed as a percent.
"Per-cent" means "per-hundred" or "over a hundred", and it is represented by the symbol %.
Thus, x% = x/100.
Examples: 1. What is 70% of 48?
Unknown: x
Equation:
x = (70/100)(48)
x = 33.6
2. 238 is what percent of 350?
Unknown: x
Equation:
Solution:
238 = (x/100)(350)
238 = 3.5x
x = 68
68%
When we deposit money in an account, the amount deposited id called the principal (P), and
the amount earned over a period of time is called the interest (I). The interest received at the
end of one year at a rate r is:
I=Pxr
As the next example shows, this formula is used to solve problems involving capital gain.
3. Two sums of money totaling $20,000 are invested at 5% and 6% respectively. Find the
two amounts if together in one year they earn $1,080.
First amount :
Second amount :
Equation:
Principal
x
20000-x
Rate
5%
6%
Interest
0.05x
0.06(20000-x)
0.05x + 0.06(20000 - x) = 1080
5x + 6(20000 - x) = 108000
5x - 120000 - 6x = 108000
x = 12000
Amount invested at 5% : $12,000
Amount invested at 6% : $8,000.
Notice that 12,000 + 8000 = 20,000, and also that 0.05(12000) + 0.06(8000) = 1080.
Markup id the amount added to the cost of an item to determine its selling price. It is
usually expressed as either, a percent of the cost or a percent of the selling price. Let us
consider the following example.
4. The cost of a radio is $80. What is the selling price if the markup is 20% of the selling
price?
5.
Selling price : x
Markup :
0.20x
Equation:
x = 80 + 0.20x
0.80x = 80
x = 100
The selling price is $100.
Mixture Problems: We will study here three examples involving mixtures and alloys.
1. How many liters of water need to be added to 6 liters of an 8% solution of salt and water
in order to obtain a 5% salted solution.
Amount of water in liters :
Concentration of salt :
Amount of salt in liters :
Equation :
Original
6
8%
0.08(6)
Added
x
0%
0
Final
x+6
5%
0.05(x + 6)
0.08(6) + 0 = 0.05(x + 6)
48 = 5(x + 6)
5x = 18
x = 3.6
The amount of water to be added is 3.6 liters.
2. If we mix 48 ounces of a 4% iodine solution with 40 ounces of a 15% iodine solution,
what is the percentage of iodine in the mixture?
First
Amount of solution in ounces: 48
Concentration of iodine:
4%
Amount of iodine in ounces: (4/100)(48)
Second
40
15%
(15/100)(40)
Mixture
88
x%
(x/100)(88)
Equation:
0.04(48) + 0.15(40) = 0.88x
4(48) + 15(40) = 88x
192 + 600 = 88x
x=9
The mixture is 9% iodine.
3. An engineer mixed a 48% aluminum alloy with a 72% aluminum alloy to make a 57%
aluminum alloy. If there are 20 pounds more of the 48% alloy than the 72% alloy, what
is the weight in pounds of the final alloy?
Weight of alloy in pounds:
Percent of aluminum:
Pounds of aluminum:
First
x + 20
48%
0.48(x + 20)
Second
x
72%
0.72x
Mixture
2x + 20
57%
0.57(2x + 20)
Equation:
0.48(x + 20) + 0.72x = 0.57(2x + 20)
48(x + 20) + 72x = 57(2x + 20)
48x + 960 + 72x = 114x + 1140
6x = 180
x = 30
The weight of the final alloy is 2(30) + 20 = 80 pounds.
Coin and Value problems: These problems involve relationships between different types
of coins and the corresponding values of various items. The following examples are a
typical sample of this category.
1. Maria has $4.45 in dimes and quarters. If she has 28 coins in all, how many are dimes
and how many are quarters?
Dimes
Quarters
Coin value in cents:
10
25
Number of coins:
x
28 - x
Total value in cents:
10x
25(28 - x)
Equation:
10x + 25(28 - x) = 445
10x + 700 - 25x = 445
-15x = -225
x = 17
Maria has 17 dimes and 28 - 17 = 11 quarters.
2. George bought three books for his History, Chemistry, and Mathematics classes. The
Chemistry book costs $5 more than the History book. The Mathematics book costs twice
as much as the Chemistry book. If the total bill for all three books was $145, what is the
price of each one?
History
Chemistry
Mathematics
Price of book in dollars:
x
x+5
2(x + 5)
Equation:
x + x+5 + 2(x + 5) = 145
4x = 130
x = 32.5
Price of History book $32.50
Price of Chemistry book $37.50
Price of mathematics book $75
Motion Problems: These problems are based on the relationship existing between the
distance, the velocity, and the time. The relationship is given by the formula
d=vxt
where d = distance, v = velocity, and t = time.
Two typical examples are shown next.
1. Two cars that are 375 miles apart from each other, and whose speeds differ by 5 miles
per hour, are moving toward each other. Under these circumstances they will meet in 3
hours. What is the speed of each car?
First car
Second car
Speed in miles per hour:
x
x+5
Time in hours:
3
3
Distance in miles:
3x
3(x + 5)
Equation:
3x + 3(x + 5) = 375
6x = 360
x = 60
The speed of the first car is 60 mph
The speed of the second car is 65 mph.
2. Michael drove 45 minutes at a certain speed. Then he increased the speed by 16 mph for
the rest of the trip. If the total distance traveled was 114 miles, and it took him 2 hours
and 15 minutes, how many miles did he drive at the higher speed?
Speed in mph:
Time in hours:
Distance in miles:
Equation:
First part
x
45/60 = 3/4
(3/4)x
Second part
x + 16
2 1/4 - 3/4 = 3/2
(3/2)(x + 16)
(3/4)x + (3/2)(x + 16) = 114
3x + 6(x + 6) = 456
9x = 360
x = 40
The distance traveled at the higher speed was (3/2)(40 + 16) = 84 miles.
Temperature Problems: Three common scales used to measure temperatures are the
Fahrenheit, the Celsius or Centigrade, and the Kelvin or Absolute. The relationship between
these scales is given by the equations:
F = (9/5) + 32
and
K = C + 273
Exercises:
1. At what temperature the Fahrenheit and the Centigrade scales coincide?
If x is the common temperature, the equation relation both scales becomes:
x = (9/5)x + 32
5x = 9x + 160
4x = -160
x = -40
Therefore the two scales coincide at -40 degrees.
2. Find the Kelvin or Absolute temperature corresponding to 86 Fahrenheit degrees.
First we will find the corresponding Centigrade reading. The equation is:
86 = (9/5)C + 32
(9/5)C = 54
C = 30
Thus the Kelvin reading is, K = 30 + 273 = 303 degrees.
Lever Problems: A uniform bar with negligible weight and balanced on a support
(fulcrum) with weights on both sides of it is called a lever. The distance from a weight to
the fulcrum is referred to as the arm of the weight.
For a lever to be balanced the following equation must be safisfied,
F1 x d1 = F2 x d2
Nutcrackers, scissors, and balances are simple machines to which the lever equation applies.
Examples:
1. Maria and Francisco together weight 75 lb. They balance a 10 feet teeterbore when
Francisco is 4 ft from the fulcrum. Find their respective weights.
Weight:
Arm of weight:
Equation:
Francisco
x
4
Maria
75 - x
6
4x = 6(75 - x)
10x = 450
x = 45
Francisco's weight is 45 lb, and Maria's weight is 75 - 45 = 30 lb.
2. A lever is balanced when a 90 lb weight is placed on one side, 8 ft away from the
fulcrum, and a 40 lb and 120 lb weights are placed 2 ft apart from each other on the side
of the fulcrum, with the 40 lb weight closer to it. How far from the fulcrum is the 40 lb
weight?
Equation:
40x + 120(x + 2) = 90(8)
60x = 480
x=3
The 40 lb weight is 3 ft away from the fulcrum.
Geometry Problems. Examples:
1. The length of a rectangle is 3 ft less than twice its width and its perimeter is 42 ft. Find
the dimensions of the rectangle.
Dimensions in feet:
Equation:
Width
x
Length
2x - 3
2x + 2(2x - 3) = 42
6x = 48
x=8
The width of the rectangle is 8 ft, and its length is 2(8) - 3 = 13 ft.
3. One of two complementary angles is 9o more than twice the other. Find the two angles.
Measure in degrees:
Equation:
Angle
x
Complementary
2x + 9
x + 2x+9 = 90
3x = 81
x = 27
One angle measures 27o and its complement measures 90 - 27 = 63o.
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EXERCISES
1. Find three consecutive odd integers such that, the product of the first and the third minus
the product of the first and the second, is 11 more than the third number.
2. The cost of a tape deck is $570. What is the selling price of the tape if the markup is
40% of the selling price?
3. The annual interest earned by $18,000 is $206 more than the interest earned by $16,000
invested at a rate 0.8% less than the one corresponding to the $18,000. What is the rate
of interest for each amount?
4. Find the Celsius reading corresponding to 23 Fahrenheit degrees.
5. If one of two supplementary angles is four times the other, what is the measure of each
angle?
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