problem solving
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12.6 – Rational Expressions. Problem Solving. The quotient of a number and 2 minus 1/3 is the quotient of a number and 6. Find the number. LCD = 6. 12.6 – Rational Expressions. Problem Solving. - PowerPoint PPT PresentationTRANSCRIPT
Problem Solving The quotient of a number and 2 minus 1/3 is the quotient of a number and 6. Find the number.
2
x
62
x
3 2x x 3 2x x
1x
LCD = 61
3
6
x
3
61
66
x
2 2x
12.6 – Rational Expressions
If one more than three times a number is divided by the number, the result is four thirds. Find the number.
3x
33 1
xx
x
3 3 1 4x x
9 3 4x x
5 3x 3
5x
LCD = 3x1
x 4
3
3
34
x
9 4 3x x
Problem Solving
12.6 – Rational Expressions
Mike and Ryan work at a recycling plant. Ryan can sort a batch of recyclables in 2 hours and Mike can short a batch in 3 hours. If they work together, how fast can they sort one batch?
Time to sort one batch (hours)
Fraction of the job completed in one hour
Ryan
Mike
Together
1
2
1
31
x
2
3
x
Problem Solving
12.6 – Rational Expressions
Time to sort one batch (hours)
Fraction of the job completed in one hour
Ryan
Mike
Together
1
2
1
31
x
2
3
x
1
2 61
2x
3x 5 6x 6
5x hrs.
LCD =1
3
1
x 6x
36
1x 1
6x
x
2x 61
15
Problem Solving
12.6 – Rational Expressions
Pippen and Merry assemble Ork action figures. It takes Merry 2 hours to assemble one figure while it takes Pippen 8 hours. How long will it take them to assemble one figure if they work together?
Time to Assemble one unit (hours)
Fraction of the job completed in one hour
Merry
Pippen
Together
2
8
x
1
21
8
1
x
Problem Solving
12.6 – Rational Expressions
Time to Assemble one unit (hours)
Fraction of the job completed in one hour
Merry
Pippen
Together
2
8
x
1
21
81
x
LCD:1
2 8
1
2x
4x 5 8x 8
5x
hrs.
1
8
1
x 8x
88
1x 1
8x
x
x 83
15
Problems about Numbers
12.6 – Rational Expressions
A sump pump can pump water out of a basement in twelve hours. If a second pump is added, the job would only take six and two-thirds hours. How long would it take the second pump to do the job alone?
1
12
1
x
26
3
1203
Time to pump one basement (hours)
Fraction of the job completed in one hour
1st pump
2nd pump
Together
x
12
20
3
Problem Solving
12.6 – Rational Expressions
1
12
1
x
26
3
120
3
Time to pump one basement (hours)
Fraction of the job completed in one hour
1st pump
2nd pump
Together
x
12
1
121 1
12 x
20
3
1
x 1
203
3
20
Problem Solving
12.6 – Rational Expressions
LCD:
601
12x
5x
60 4x hrs. 15x
1 1 3
12 20x 60x
160
xx
060
3
2x
60 9x
5x60 9x
Problem Solving
12.6 – Rational Expressions
Distance, Rate and Time Problems
If you drive at a constant speed of 65 miles per hour and you travel for 2 hours, how far did you drive?
d r t
65miles
hour 2 hours 130 miles
dt
r
dr
t
Problem Solving
12.6 – Rational Expressions
A car travels six hundred miles in the same time a motorcycle travels four hundred and fifty miles. If the car’s speed is fifteen miles per hour faster than the motorcycle’s, find the speed of both vehicles.
Rate Time Distance
Motor-cycle
Car
x
x + 15
450 mi
600 mi
t
t
r
dt
x
450
15
600
x
Problem Solving
12.6 – Rational Expressions
Rate Time Distance
Motor-cycle
Car
x
x + 15
450 mi
600 mi
t
t
r
dt
x
450
15
600
x
x
450
15
600
xLCD: x(x + 15)
15
600450
xx
x(x + 15) x(x + 15)
12.6 – Rational Expressions – Problem Solving
15
600450
xx
x(x + 15) x(x + 15)
xx 60045015
xx 60015450450
x15015450
x
150
15450
x45
45x mph
Motorcycle
6015 x mph
Car
12.6 – Rational Expressions – Problem Solving
A boat can travel twenty-two miles upstream in the same amount of time it can travel forty-two miles downstream. The speed of the current is five miles per hour. What is the speed of the boat in still water?
boat speedx
Rate Time Distance
UpStream
DownStream
x - 5
x + 5
22 mi
42 mi
t
t
r
dt
5
22
x
5
42
x
12.6 – Rational Expressions – Problem Solving
boat speedx
Rate Time Distance
UpStream
DownStream
x - 5
x + 5
22 mi
42 mi
t
t
r
dt
5
22
x
5
42
x
5
22
x
5
42
xLCD: (x – 5)(x + 5)
5
42
5
22
xx(x – 5)(x + 5) (x – 5)(x + 5)
12.6 – Rational Expressions – Problem Solving
542225 xx
2104211022 xx
xx 2242210110
x20
320
x1616 mph
Boat Speed
5
42
5
22
xx(x – 5)(x + 5) (x – 5)(x + 5)
x20320
12.6 – Rational Expressions – Problem Solving
12.7 – Rational Expressions
Simplifying Complex Fractions
LCD:
3759
3759
3 5
7 9
6337
63593 9
7 5
9 3
7 5
27
3527
35
63
Outers over Inners
3759
9 3
7 5
27
35
3 24 31 32 8
3 24 31 32 8
3 43 4
4
3 24 3
1 384 2
24 24
2
3 24 31
43
2 824
9 812 12
4 38 8
6 3 8 2
12 1 3 3
11278
18 16
12 9
2
211 8
12 7
1 8 2
12 3 7
2
21
LCD: 12, 8 LCD: 24
12.7 – Rational Expressions
Simplifying Complex Fractions
LCD: y1
2 1
xy
xy
1
2 1
y y
y
xy
xy
2 1
y x
x
yx
yx
12
1y–y
12.7 – Rational Expressions
Simplifying Complex Fractions
LCD: 6xy
56
3
yy xyx
2
2 2
5 6
2 6
x y
xy x y
56
3
6 6
6 6
yy x
xy xy
yy
xx xy
25 6x y2xy 3y x
xy
xy
y
3
65
6xy
6xy
12.7 – Rational Expressions
Simplifying Complex Fractions