Download mixture probles- a medley of methods

A Mixture of Problems
Presented by
Laurie Riggs, Ph.D., Cal Poly Pomona
Ed D’Souza, Ph.D.,Rialto USD
Melanie Janzen, Rialto USD
ALGEBRA STANDARDS ADDRESSED
This presentation scaffolds material covered in earlier years and takes students from 4 th/5th grade math standards to Algebra Standards.
It is hope that the visual methods will provide students with strategies that they can make the connection to some of the challenging
standards in Algebra. In no way does it state that the material HAS to be covered in this manner or suggest that one has to ignore the
normal algebra methods used to address percent problems, solving simultaneous equations and mixture problems. It is our hope that
by providing alternate methods to teach algebra standards, students will actually learn the Algebra and teachers will experience
success with their teaching.
Activity : Solving percent problems using percent converter
Weight puzzles
STANDARD
Algebra 1
Grade 7
Grade 6
Grade 5
Grade 4
5.0 Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one
variable and provide justification for each step.
NS1.6 Calculate the percentage of increases and decreases of a quantity.
NS1.7 Solve problems that involve discounts, markups, commissions, and profit and compute simple and compound interest.
NS1.3 Use proportions to solve problems (e.g., determine the value of N if 4/7 = N/ 21, find the length of a side of a polygon
similar to a known polygon). Use cross-multiplication as a method for solving such problems, understanding it as the
multiplication of both sides of an equation by a multiplicative inverse.
NS1.4 Calculate given percentages of quantities and solve problems involving discounts at sales, interest earned, and tips.
SDP 1.3 Use fractions and percentages to compare data sets of different sizes.
AF2.0 Students know how to manipulate equations:
2.2 Know and understand that equals multiplied by equals are equal.
Activity : Solving simultaneous equations using unknown weights
Balancing Scales and Teeter- Totter Method to Solve Solution Problems
STANDARD
9.0
Students
solve
a
system
of
two
linear
equations
in two variables algebraically and are able to interpret the
Algebra 1
answer graphically.
Grade 7
Grade 6
Grade 5
Grade 4
15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.
AF1.1 Use variables and appropriate operations to write an expression, an equation, an inequality, or a system of equations or
inequalities that represents a verbal description (e.g., three less than a number, half as large as area A).
AF 4.1 Solve two-step linear equations and inequalities in one variable over the rational numbers, interpret the solution or
solutions in the context from which they arose, and verify the reasonableness of the results.
AF1.1 Write and solve one-step linear equations in one variable.
AF 1.2 Write and evaluate an algebraic expression for a given situation, using up to three variables.
AF 1.2 Use a letter to represent an unknown number; write and evaluate simple algebraic expressions in one variable by
substitution.
AF 1.1 Use letters, boxes, or other symbols to stand for any number in simple expressions or equations (e.g., demonstrate an
understanding and the use of the concept of a variable).
AF2.0 Students know how to manipulate equations:
2.1 Know and understand that equal added to equals are equal
2.2 Know and understand that equals multiplied by equals are equal.
Steps needed for mastery of Standard 15
Solving Mixture Problems
Using the teeter-totter to find an unknown
Solving for more than 1 unknown using balance scales
Knowing and applying percents using proportions
2
WARM-UP
How many ways can you solve….
12.5% of 40
3
PERCENT PROBLEM TOOL
%
#
%
#
%
#
%
#
%
#
%
#
%
#
%
#
4
Percent Problems
1: Find 40% of 300
%
#
2: 30% of what number is 45 ?
%
#
3: 15 is what percent of 60?
%
#
4: A house is sold for a profit of 20%. If the actual cost of the house is
$200,000, determine its sale price?
%
#
5: Principal =$2,000 R = 8% T= 5 years, determine the interest in $.
%
#
5
Percent Problem Types
TYPE 1: Find 40% of 300
%
100
#
%
#
%
#
300
10
100
30
300
10
30
40
100
120
300
(Explanation: 100% = 300, so 10% =30 and 40% = 120)
TYPE 2: 30% of what number is 45 ?
%
30
#
100
45
%
10
100
#
15
%
10
100
#
15
150
(Explanation: 30% = 45, so 10% =15 and 100% = 150)
6
TYPE 3: 15 is what percent of 60?
%
100
#
60
%
10
20
30
100
#
6
12
18
60
%
10
20
25
#
6
12
15
30
100
18
60
(Explanation: 100% = 60, so 10% =6 and 20% = 12 and 30% = 18. This is greater than 15. But 15 lies half
way between 12 and 18. .So 15 is 25% of 60)
TYPE 4: A house is sold for a profit of 20%. If the actual cost of the house is
$200,000, determine its sale price?
%
100
Price
$200,000
%
Price
%
Price
10
20
$20,000
$40,000
10
20
$20,000
$40,000
100
$200,000
100
$200,000
120%
$240,000
(Explanation: 100% =$200,000, so 10% =$20,000 and 20% = $40,000 and 120% = $200,000 + $40,000 or
$240,000)
7
TYPE 5: Simple Interest
Principal =$2,000 R = 8%
T= 5 years, determine the interest in $.
%
100
$
$
2,000
%1
8
100
20
160
2,000
(Explanation: 100% =$2,000, so 1% =$20 and 8% = $160 )
The amount of interest received in 1 year =$160. So in 5 years the interest received is $$160 × 5 =$800
8
Weight Puzzles
In each of the problems below the scale is balanced.
- Same shapes have same weights
- Different shapes have different weights
- A horizontal bar shows that there is balance
- There is only one correct solution in each problem
- Use the weights given in the problem to determine the other weights
1
Total weight = 40
=__________
= _________
Write a numerical equation showing that the scale is balanced
_______________________________________________________________________
2.
Total weight = 48
=_________
=_______
Write a numerical equation showing that the scale is balanced
_______________________________________________________________________
9
(3)
Total weight = 32 and
=10
=__________
=__________
Write a numerical equation showing that the scale is balanced
_______________________________________________________________________
(4)
Total weight = 48 and
=__________
=16
= _________
Write a numerical equation showing that the scale is balanced
_______________________________________________________________________
10
(5)
Total weight = 40 and
= 4
=__________
= _________
Write a numerical equation showing that the scale is balanced
_______________________________________________________________________
(6)
Total weight = 32
=__________
= _________
Write a numerical equation showing that the scale is balanced
_______________________________________________________________________
11
(7)
Total weight = 80
=__________
= _________
Write a numerical equation showing that the scale is balanced
_______________________________________________________________________
(8)
Total weight = 48
=__________
= _________
=________ and
=________
Write a numerical equation showing that the scale is balanced
_______________________________________________________________________
12
What is the Weight?
Solve using at least two different methods!
Set A
14
11
12
Set B
31
22
27
Set C
17
33
21
Set D
33
24
30
13
Using the strategy learned in the previous problem and apply it to solve for x and y in the
system of equations:
2x + y = 9
3x + y = 11
DONUTS AND COFFEE
The cost for two donuts and a coffee is $1.35 and the cost of four donuts and three cups of
coffee is $3.05. What is the cost of a single donut and a single cup of coffee?
14
Teeter Totter Problems
1. Determine where the yellow figure should sit on the teeter totter if the fulcrum (or pivot) is at the 50 cm mark
and the teeter totter is to be balanced.
10 cm
50 cm
100g
60 cm
? g
2. Determine where the yellow figure should sit on the teeter totter if the fulcrum (or pivot) is at the 35 cm mark
and the teeter totter is to be balanced.
15 cm
35 cm
100g
75 cm
? g
3. Determine the weight of the yellow figure if the fulcrum (or pivot) is at the 35 cm mark
and the teeter totter is to be balanced.
5 cm
100g
35 cm
75 cm
? g
15
Mixtures –Finding the Balance
1.
Coffee
Cost
$3/lb
$3.25/lb
$3.50/lb
Mixture
5lbs
x lbs
2.
Butterfat
%
2%
6.4%
8.4%
Mixture
1 liter
x liters
3.
Grape juice
%
15%
20%
100%
Mixture
800ml
x mL
16
MIXTURE PROBLEMS
THE JELLY BEAN PROBLEM
White jelly beans cost $0.25 a pound and blue jelly beans cost $0.85 a pound. How many pounds
of white jelly beans must be added to 25 pounds of blue jelly beans to arrive at a mixture worth
$0.45 a pound?
SALT SOLUTION
A mixture containing 6% salt is to be mixed with 2 ounces of a mixture which is 15% salt, in
obtaining a solution which is 12% salt. How much of the first solution must be used?
17
A SOLUTION
How many liters of a 70% acid solution must be added to 50 liters of a 40% acid solution to
produce a 50% acid solution?
COFFEE ANYONE?
Find the selling price per pound of a coffee mixture made from 8 pounds of coffee that sells
for $9.20 per pound and 12 pounds of coffee that costs $5.50 per pound.
18
A VEGETABLE MEDLEY
How many pounds of lima beans that cost $0.90 per pound must be mixed with 16 pounds of corn
that costs $0.50 per pound to make a mixture of vegetables that costs $0.65 per pound?
A LITE PUNCH
Two hundred liters of a punch that contains 35% fruit juice is mixed with 300 L of another
punch. The resulting fruit punch is 20% fruit juice. Find the percent of fruit juice in the 300
liters of punch.
19
THE CEREAL PROBLEM
Ten grams of sugar are added to a 40-g serving of a breakfast cereal that is 30% sugar. What
is the percent concentration of sugar in the resulting mixture?
How many liters of water must be added to 50 L of 30% acid solution in order to produce
a 20% acid solution?
20
ADDITIONAL MIXTURE PROBLEMS
(Composed by Glenda Griffin)
1. Hal is doing a chemistry experiment that calls for a 30% solution of copper sulfate. Hal has 40 mL of
25% solution. How many milliliters of a 60% solution should Hal add to obtain the required 30%
solution?
2. How much whipping cream (9% butterfat) should be added to 1 gallon of milk (4% butterfat) to obtain a
6% butterfat mixture?
3. A chemist had 2.5 liters of a solution which is 70% acid. How much water should be added to obtain a
50% acid solution?
4. How much pure copper must be added to 50 kg of an alloy containing 12% copper to raise the copper
content to 21%?
5. A liter of cream has 9.2% butterfat. How much skim milk containing 2% butterfat should be added to
the cream to obtain a mixture with 6.4% butterfat?
6. How much coffee costing $3 a pound should be mixed with 5 pounds of coffee costing $3.50 a pound to
obtain a mixture costing $3.25 a pound?
7. Ground chuck sells for $1.75 a pound. How many pounds of ground round selling for $2.45 a pound
should be mixed with 20 pounds of ground chuck to obtain a mixture that sells for $2.05 a pound?
8. An advertisement for an orange drink claims that the drink contains 10% orange juice. How much pure
orange juice would have to be added to 5 quarts of the drink to obtain a mixture containing 40% orange
juice?
9. A pharmacist had 150 dL of a 25% solution of peroxide in water. How many deciliters of pure peroxide
should be added to obtain a 40% solution?
10. A health food store sells a mixture of raisins and roasted nuts. Raisins sell for $3.50/kg and nuts sell for
$4.75/kg. How many kilograms of each should be mixed to make 20 kg of this snack worth $4.00 kg?
11. An auto mechanic has 300 mL of battery acid solution that is 60% acid. He must add water to this
solution to dilute it so that it is only 45% acid. How much water should he add?
12. A chemist has 40 mL of a solution that is 50% acid. How much water should he add to make a solution
that is 10% acid?
13. If 800 mL of a juice drink is 15% grape juice, how much grape juice should be added to make a drink
that is 20% grape juice?
14. How many liters of water must be added to 50 L of a 30% acid solution in order to produce a 20% acid
solution?
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15. How many milliliters of water must be added to 60 mL of a 15% iodine solution in order to dilute it to a
10% iodine solution?
16. A spice mixture is 25% thyme. How many grams of thyme must be added to 12 g of the mixture to
increase the thyme content 40%?
17. A grocer mixes two kinds of nuts. One kind cost $5.00/kg and the other $5.80/kg. How many
kilograms of each type are needed to make 40 kg of a blend worth $5.50/kg?
18. Joanne makes a mixture of dried fruits by mixing dried apples costing $6.00/kg with dried apricots
costing $8.00/kg. How many kilograms of each are needed to make 20 kg of a mixture worth $7.20/kg?
Answers:
1.
6.7 mL of the 60% solution)
2.
0.67 gal
3.
1L
4.
5.7 kg
5.
0.64 L
6.
5 lbs
7.
15 lbs
8.
2.5 qts
9.
37.5 dL
10. 12 kg of raisins and 8 kg of nuts
11. 100 mL of water
12. 160 mL water
13. 50 mL juice
14. 25 L
15. 30 L
16. 3 g
17. (15 kg of $5.00/kg, 25 kg of $5.80/kg
18. 8 kg dried apples, 12 kg dried apricots
22
MIXTURE PROBLES- A MEDLEY OF METHODS
(Written by Ed D’Souza, Ph.D.)
Method 1 : Intuitive Method
White jelly beans cost $0.25 a pound and blue jelly beans cost $0.85 a pound.
How many pounds of white jelly beans must be added to 25 pounds of blue jelly
beans to arrive at a mixture worth $0.45 a pound?
With an intuitive approach, we would begin by reasoning that if the mixture contained the same
amount of white and blue jelly beans, it would cost an amount that is right in the middle. A
$0.60 difference exists between the two types. A middle price would be $0.25 + $0.30 (half the
difference), or $0.55. But the problem states that the new mixture must be worth $0.45.
Therefore, the mixture must contain more white jelly beans than blue. Also, it must be greater
than 25 pounds.
Then we would reason that the price difference between the white jelly beans and the mixture is
$.20. The price difference between the blue jelly beans and the mixture is $.40. As a ratio, the
price differences compare 1:2. It makes sense that the amounts will follow this ratio. Since
more white jelly beans are needed, the
ratio of white to blue must be 2:1. Therefore, since the mix has 25 pounds of blue, it must have
50 pounds of white.
23
METHOD 2:Teeter-Totter Method
White jelly beans cost $0.25 a pound and blue jelly beans cost $0.85 a pound.
How many pounds of white jelly beans must be added to 25 pounds of blue jelly
beans to arrive at a mixture worth $0.45 a pound?
Using the tug-of-war approach, imagine the white jelly beans having a tug-of-war game with
the blue jelly beans. Both want the final price to be closer to their price.
1. Draw a "rope" (straight line) and label the three numbers you know.
W=25
M =45
B= 85
2. The left point on the rope is the price of the white jelly beans (25), the right
point is the price of the blue jelly beans (85), and the "fulcrum" will be
located at the final price (45), and closer to the left side. Try to approximate
where 45 would land between 25 and 85.
Price (cents) W= 25
M= 45
B= 85
3. Label what you know - the pull to the right is 25. Label what you don't know the pull to the
left is x.
4.
Amount (lbs)
Price(cents)
Amount (lbs)
20
W= 25
x
40
M=45
B= 85
25
5. Calculate the gap in price between the knot and the left side (20cents) and between the
knot and the right side (40 cents- see bullet 3.). Since the gap on the left is half as big, the left
side must be pulling twice as hard. Therefore, we must have 50 pounds of white jelly beans.
24
Students will eventually discover the rule: (gap) x (pull) = (gap) x (pull). Multiply the gap on
the left (20) by the pull to the left (the unknown). Set this quantity to the gap on the right (40)
times the pull to the right (25). Solving for the unknown yields 50 pounds.
(20)(x) = (40)(25)
x
= (40)(25) = 50lbs
20
So , 50 lbs of white jelly beans need to be added to the mixture.
3rd Method: Column Method
White jelly beans cost $0.25 a pound and blue jelly beans cost $0.85 a pound. How many
pounds of white jelly beans must be added to 25 pounds of blue jelly beans to arrive at a
mixture worth $0.45 a pound?
Use only one unknown
Cost Per Pound
Pounds: (Decide what
is x)
Total Cost
White Jellybeans
0.25
?x
Blue Jelly Beans
0.85
25
Mixture
0.45
x+25
0.25 x
(0.85)(25)
(x+25) 0.45
Equation: 0.25x + (0.85)(25) =(x+25)(0.45)
Multiply throughout by 100 (this eliminates decimals)
(100) (0.25x) + (100) (0.85)(25) = (x+25)(0.45)(100)
25 x + (85)(25) = 45(x+25)
25 x + 2125 = 45x + 1125
2125 – 1125 = 45x – 25 x
1000 = 20 x
x =50
So, 50 white jelly beans need to be added to the mixture.
25
Strategies for Solving Problems
Strategy
When to use it
How to do it
When doing a simple
computation problem
Use order of operations, collect like
terms or simplify, write down all
calculations and check your work.
Make a picture
When you need to visualize
something to solve a problem.
Try to make a drawing or sketch a
diagram that will help you understand
the situation the problem is based on.
Make a table or
chart
When you are working with
different groups of numbers
or and you need to find a
pattern.
Use a t-chart or data table this will
allow you to see and identify patterns.
Look for a Pattern
When you are given a series of
pictures, geometric figures or
numbers and you are asked “what
would come next?”
Make a chart and look similarities is ending
numbers, beginning numbers, perfect
squares, multiples, palindromes, transposed
numbers etc….
When you are given what
seems to be the answer or
the end result and need to
find the start
Substitute the information you are given
back into the problem and work backwards
until you have all components of the
problem solved.
When the problem describes
a physical action or process
Use people or manipulatives to
represent the situation
When it is quicker to try a
few numbers and then check
the results when you have a
problem to solve.
Guess the answer and then checking that
the guess fits the conditions of the
problem If is does not fit, revise your
guess /try another number. Make a record
of the numbers you tried. The crucial step
here is recognizing that each result from
previous guesses can provide information
for improving the next guess.
When you have a complex
problem and you want to
break it into smaller steps,
Solve a similar or easier problem to see how
it works, change large numbers into smaller
numbers, or reduce the number of items
given in the problem. then go back to the
original problem and solve it.
Compute and solve
Work Backward
Act It Out
Guess, Check, and
Revise
Use an easier
problem
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