4. Rational Equations Word-Problems
Problems from the Multiplication–Division Operations
Following are some basic applications of rational equations.
All the multiplicative formulas of the form AB = C may be written
as A = .C
B
5. Rational Equations Word-Problems
Problems from the Multiplication–Division Operations
Following are some basic applications of rational equations.
All the multiplicative formulas of the form AB = C may be written
as A = . This divisional form leads to rational equations.C
B
6. Rational Equations Word-Problems
Problems from the Multiplication–Division Operations
Following are some basic applications of rational equations.
All the multiplicative formulas of the form AB = C may be written
as A = . This divisional form leads to rational equations.
The calculation of “per unit” is a good example:
C
B
7. Rational Equations Word-Problems
Problems from the Multiplication–Division Operations
Following are some basic applications of rational equations.
All the multiplicative formulas of the form AB = C may be written
as A = . This divisional form leads to rational equations.
Total amount
Number of units
The calculation of “per unit” is a good example:
Per unit amount =
C
B
8. Rational Equations Word-Problems
Problems from the Multiplication–Division Operations
Following are some basic applications of rational equations.
All the multiplicative formulas of the form AB = C may be written
as A = . This divisional form leads to rational equations.
Total amount
Number of units
Cost per Unit
The calculation of “per unit” is a good example:
Per unit amount =
C
B
9. Rational Equations Word-Problems
Problems from the Multiplication–Division Operations
Following are some basic applications of rational equations.
All the multiplicative formulas of the form AB = C may be written
as A = . This divisional form leads to rational equations.
Total amount
Number of units
Cost per Unit
The calculation of “per unit” is a good example:
Per unit amount =
C
B
Total cost
Number of units
Per unit cost =
10. Rational Equations Word-Problems
Problems from the Multiplication–Division Operations
Following are some basic applications of rational equations.
All the multiplicative formulas of the form AB = C may be written
as A = . This divisional form leads to rational equations.
Total amount
Number of units
For example, a group of 5 people rent a taxi that cost $20
so each person’s share is 20/5 or $4 per person.
Cost per Unit
The calculation of “per unit” is a good example:
Per unit amount =
C
B
Total cost
Number of units
Per unit cost =
11. Rational Equations Word-Problems
Problems from the Multiplication–Division Operations
Following are some basic applications of rational equations.
All the multiplicative formulas of the form AB = C may be written
as A = . This divisional form leads to rational equations.
Total amount
Number of units
For example, a group of 5 people rent a taxi that cost $20
so each person’s share is 20/5 or $4 per person.
However if one person backs out, then the cost goes up to
s = 20/4 = $5 per person.
Cost per Unit
The calculation of “per unit” is a good example:
Per unit amount =
C
B
Total cost
Number of units
Per unit cost =
12. Rational Equations Word-Problems
Problems from the Multiplication–Division Operations
Following are some basic applications of rational equations.
All the multiplicative formulas of the form AB = C may be written
as A = . This divisional form leads to rational equations.
Total amount
Number of units
For example, a group of 5 people rent a taxi that cost $20
so each person’s share is 20/5 or $4 per person.
However if one person backs out, then the cost goes up to
s = 20/4 = $5 per person. Each remaining person has to pay
$1 more.
Cost per Unit
The calculation of “per unit” is a good example:
Per unit amount =
C
B
Total cost
Number of units
Per unit cost =
13. Rational Equations Word-Problems
Problems from the Multiplication–Division Operations
Following are some basic applications of rational equations.
All the multiplicative formulas of the form AB = C may be written
as A = . This divisional form leads to rational equations.
Total amount
Number of units
For example, a group of 5 people rent a taxi that cost $20
so each person’s share is 20/5 or $4 per person.
However if one person backs out, then the cost goes up to
s = 20/4 = $5 per person. Each remaining person has to pay
$1 more. Let’s formulate this as a word problem and solve it
using rational equations.
Cost per Unit
The calculation of “per unit” is a good example:
Per unit amount =
C
B
Total cost
Number of units
Per unit cost =
15. Rational Equations Word-Problems
Example A. A taxi is rented by a group of x people for $20
and the cost is shared equally. If there is one less person in
the group then each of the remaining people has to pay $1
more. What is x?
A table is useful in organizing repeated calculations for
different inputs.
16. Rational Equations Word-Problems
Example A. A taxi is rented by a group of x people for $20
and the cost is shared equally. If there is one less person in
the group then each of the remaining people has to pay $1
more. What is x?
A table is useful in organizing repeated calculations for
different inputs.
Total Cost No. of People Cost per Person =
20
20
Total Cost
No. of people
17. Rational Equations Word-Problems
Example A. A taxi is rented by a group of x people for $20
and the cost is shared equally. If there is one less person in
the group then each of the remaining people has to pay $1
more. What is x?
A table is useful in organizing repeated calculations for
different inputs.
Total Cost No. of People Cost per Person =
20 x
20 (x – 1)
Total Cost
No. of people
18. Rational Equations Word-Problems
Example A. A taxi is rented by a group of x people for $20
and the cost is shared equally. If there is one less person in
the group then each of the remaining people has to pay $1
more. What is x?
A table is useful in organizing repeated calculations for
different inputs.
Total Cost No. of People Cost per Person =
20 x
20 (x – 1)
Total Cost
No. of people
20
x
20
(x – 1)
19. Rational Equations Word-Problems
Example A. A taxi is rented by a group of x people for $20
and the cost is shared equally. If there is one less person in
the group then each of the remaining people has to pay $1
more. What is x?
A table is useful in organizing repeated calculations for
different inputs.
Total Cost No. of People Cost per Person =
20 x
20 (x – 1)
Total Cost
No. of people
20
x
20
(x – 1)
Let’s compare the two different per/person costs.
20
(x – 1)
20
x
20. Rational Equations Word-Problems
Example A. A taxi is rented by a group of x people for $20
and the cost is shared equally. If there is one less person in
the group then each of the remaining people has to pay $1
more. What is x?
A table is useful in organizing repeated calculations for
different inputs.
Total Cost No. of People Cost per Person =
20 x
20 (x – 1)
Total Cost
No. of people
20
x
20
(x – 1)
Let’s compare the two different per/person costs.
cost more cost less
20
(x – 1)
20
x
21. Rational Equations Word-Problems
Example A. A taxi is rented by a group of x people for $20
and the cost is shared equally. If there is one less person in
the group then each of the remaining people has to pay $1
more. What is x?
A table is useful in organizing repeated calculations for
different inputs.
Total Cost No. of People Cost per Person =
20 x
20 (x – 1)
Total Cost
No. of people
20
x
20
(x – 1)
Let’s compare the two different per/person costs.
cost more cost less ($1 more)
20
(x – 1)
20
x
22. Rational Equations Word-Problems
Example A. A taxi is rented by a group of x people for $20
and the cost is shared equally. If there is one less person in
the group then each of the remaining people has to pay $1
more. What is x?
Total Cost No. of People Cost per Person =
20 x
20 (x – 1)
Total Cost
No. of people
20
x
20
(x – 1)
20
(x – 1)
20
x
Let’s compare the two different per/person costs.
– = 1
cost more cost less ($1 more)
A table is useful in organizing repeated calculations for
different inputs.
23. Rational Equations Word-Problems
Example A. A taxi is rented by a group of x people for $20
and the cost is shared equally. If there is one less person in
the group then each of the remaining people has to pay $1
more. What is x?
Total Cost No. of People Cost per Person =
20 x
20 (x – 1)
Total Cost
No. of people
20
x
20
(x – 1)
20
(x – 1)
Let’s compare the two different per/person costs.
– = 1[ ] x (x – 1) clear the denominators
by LCM
A table is useful in organizing repeated calculations for
different inputs.
20
x
24. Rational Equations Word-Problems
Example A. A taxi is rented by a group of x people for $20
and the cost is shared equally. If there is one less person in
the group then each of the remaining people has to pay $1
more. What is x?
Total Cost No. of People Cost per Person =
20 x
20 (x – 1)
Total Cost
No. of people
20
x
20
(x – 1)
20
(x – 1)
20
x
Let’s compare the two different per/person costs.
– = 1[ ] x (x – 1) clear the denominators
by LCM
x (x – 1) x (x – 1)
A table is useful in organizing repeated calculations for
different inputs.
25. Rational Equations Word-Problems
Example A. A taxi is rented by a group of x people for $20
and the cost is shared equally. If there is one less person in
the group then each of the remaining people has to pay $1
more. What is x?
Total Cost No. of People Cost per Person =
20 x
20 (x – 1)
Total Cost
No. of people
20
x
20
(x – 1)
20
(x – 1)
Let’s compare the two different per/person costs.
= 1[ ] x (x – 1) clear the denominators
by LCM
x (x – 1) x (x – 1)
20x – 20(x – 1) = x(x – 1)
A table is useful in organizing repeated calculations for
different inputs.
20
x
–
26. Rational Equations Word-Problems
Example A. A taxi is rented by a group of x people for $20
and the cost is shared equally. If there is one less person in
the group then each of the remaining people has to pay $1
more. What is x?
Total Cost No. of People Cost per Person =
20 x
20 (x – 1)
Total Cost
No. of people
20
x
20
(x – 1)
20
(x – 1)
Let’s compare the two different per/person costs.
= 1[ ] x (x – 1) clear the denominators
by LCM
x (x – 1) x (x – 1)
20x – 20(x – 1) = x(x – 1)
20x – 20x + 20 = x2 – x
A table is useful in organizing repeated calculations for
different inputs.
20
x
–
27. Rational Equations Word-Problems
Example A. A taxi is rented by a group of x people for $20
and the cost is shared equally. If there is one less person in
the group then each of the remaining people has to pay $1
more. What is x?
Total Cost No. of People Cost per Person =
20 x
20 (x – 1)
Total Cost
No. of people
20
x
20
(x – 1)
20
(x – 1)
Let’s compare the two different per/person costs.
= 1[ ] x (x – 1) clear the denominators
by LCM
x (x – 1) x (x – 1)
20x – 20(x – 1) = x(x – 1)
A table is useful in organizing repeated calculations for
different inputs.
0 = x2 – x – 20
set one side as 0
20
x
–
20x – 20x + 20 = x2 – x
33. Rational Equations Word-Problems
Let R = rate (mph), T = time (hours) and D = Distance (m)
then RT = D or that T = . .D
R
Rate–Time–Distance
0 = x2 – x – 20
0 = (x – 5)(x + 4)
x = 5 or x = –4.
Hence there are 5 people.
34. Rational Equations Word-Problems
Let R = rate (mph), T = time (hours) and D = Distance (m)
then RT = D or that T = . .D
R
We hiked a 6–mile trail from A to B at a rate of 3 mph so the
trip took 6/3 = 2 hours.
Rate–Time–Distance
0 = x2 – x – 20
0 = (x – 5)(x + 4)
x = 5 or x = –4.
Hence there are 5 people.
35. Rational Equations Word-Problems
Let R = rate (mph), T = time (hours) and D = Distance (m)
then RT = D or that T = . .D
R
We hiked a 6–mile trail from A to B at a rate of 3 mph so the
trip took 6/3 = 2 hours. Going back from B to A our rate was
2 mph so the return took 6/2 = 3 hours.
Rate–Time–Distance
0 = x2 – x – 20
0 = (x – 5)(x + 4)
x = 5 or x = –4.
Hence there are 5 people.
36. Rational Equations Word-Problems
Let R = rate (mph), T = time (hours) and D = Distance (m)
then RT = D or that T = . .D
R
We hiked a 6–mile trail from A to B at a rate of 3 mph so the
trip took 6/3 = 2 hours. Going back from B to A our rate was
2 mph so the return took 6/2 = 3 hours. In a table,
Distance (m) Rate (mph) Time = D/R (hr)
Going
Return
Rate–Time–Distance
0 = x2 – x – 20
0 = (x – 5)(x + 4)
x = 5 or x = –4.
Hence there are 5 people.
37. Rational Equations Word-Problems
Let R = rate (mph), T = time (hours) and D = Distance (m)
then RT = D or that T = . .D
R
We hiked a 6–mile trail from A to B at a rate of 3 mph so the
trip took 6/3 = 2 hours. Going back from B to A our rate was
2 mph so the return took 6/2 = 3 hours. In a table,
Distance (m) Rate (mph) Time = D/R (hr)
Going 6 3 2
Return
Rate–Time–Distance
0 = x2 – x – 20
0 = (x – 5)(x + 4)
x = 5 or x = –4.
Hence there are 5 people.
38. Rational Equations Word-Problems
Let R = rate (mph), T = time (hours) and D = Distance (m)
then RT = D or that T = . .D
R
We hiked a 6–mile trail from A to B at a rate of 3 mph so the
trip took 6/3 = 2 hours. Going back from B to A our rate was
2 mph so the return took 6/2 = 3 hours. In a table,
Distance (m) Rate (mph) Time = D/R (hr)
Going 6 3 2
Return 6 2 3
Rate–Time–Distance
0 = x2 – x – 20
0 = (x – 5)(x + 4)
x = 5 or x = –4.
Hence there are 5 people.
39. Rational Equations Word-Problems
Let R = rate (mph), T = time (hours) and D = Distance (m)
then RT = D or that T = . .D
R
We hiked a 6–mile trail from A to B at a rate of 3 mph so the
trip took 6/3 = 2 hours. Going back from B to A our rate was
2 mph so the return took 6/2 = 3 hours. In a table,
Distance (m) Rate (mph) Time = D/R (hr)
Going 6 3 2
Return 6 2 3
Let’s turn this example into a rational–equation word problem.
0 = x2 – x – 20
0 = (x – 5)(x + 4)
x = 5 or x = –4.
Rate–Time–Distance
Hence there are 5 people.
40. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
41. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
The rate of the return is “1 mph faster then x” so it’s (x + 1).
42. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
D = Distance (m) R = Rate (mph) Time = D/R (hr)
Go
Return
The rate of the return is “1 mph faster then x” so it’s (x + 1).
Put these into the table
43. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
D = Distance (m) R = Rate (mph) Time = D/R (hr)
Go 6
Return 6
The rate of the return is “1 mph faster then x” so it’s (x + 1).
Put these into the table
44. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
D = Distance (m) R = Rate (mph) Time = D/R (hr)
Go 6 x
Return 6
The rate of the return is “1 mph faster then x” so it’s (x + 1).
Put these into the table
45. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
D = Distance (m) R = Rate (mph) Time = D/R (hr)
Go 6 x 6/x
Return 6
The rate of the return is “1 mph faster then x” so it’s (x + 1).
Put these into the table
46. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
D = Distance (m) R = Rate (mph) Time = D/R (hr)
Go 6 x 6/x
Return 6 x + 1
The rate of the return is “1 mph faster then x” so it’s (x + 1).
Put these into the table
47. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
D = Distance (m) R = Rate (mph) Time = D/R (hr)
Go 6 x 6/x
Return 6 x + 1 6/(x + 1)
The rate of the return is “1 mph faster then x” so it’s (x + 1).
Put these into the table
48. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
D = Distance (m) R = Rate (mph) Time = D/R (hr)
Go 6 x 6/x
Return 6 x + 1 6/(x + 1)
The rate of the return is “1 mph faster then x” so it’s (x + 1).
Put these into the table
Now we use the information about the times.
49. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
D = Distance (m) R = Rate (mph) Time = D/R (hr)
Go 6 x 6/x
Return 6 x + 1 6/(x + 1)
The rate of the return is “1 mph faster then x” so it’s (x + 1).
Put these into the table
6
(x + 1)
6
x
number of
hrs for going
Now we use the information about the times.
number of
hrs for returning
50. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
D = Distance (m) R = Rate (mph) Time = D/R (hr)
Go 6 x 6/x
Return 6 x + 1 6/(x + 1)
The rate of the return is “1 mph faster then x” so it’s (x + 1).
Put these into the table
6
(x + 1)
6
x
Now we use the information about the times.
(total trip 5 hrs)number of
hrs for going
number of
hrs for returning
51. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
D = Distance (m) R = Rate (mph) Time = D/R (hr)
Go 6 x 6/x
Return 6 x + 1 6/(x + 1)
The rate of the return is “1 mph faster then x” so it’s (x + 1).
Put these into the table
6
(x + 1)
6
x
Now we use the information about the times.
+ = 5
(total trip 5 hrs)number of
hrs for going
number of
hrs for returning
52. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
D = Distance (m) R = Rate (mph) Time = D/R (hr)
Go 6 x 6/x
Return 6 x + 1 6/(x + 1)
The rate of the return is “1 mph faster then x” so it’s (x + 1).
Put these into the table
6
(x + 1)
6
x
Now we use the information about the times.
+ = 5 use the cross
multiplication method
53. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
D = Distance (m) R = Rate (mph) Time = D/R (hr)
Go 6 x 6/x
Return 6 x + 1 6/(x + 1)
The rate of the return is “1 mph faster then x” so it’s (x + 1).
Put these into the table
6
(x + 1)
6
x
Now we use the information about the times.
+ = 5
6(x + 1) + 6x
x(x + 1)
= 5
use the cross
multiplication method
54. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
D = Distance (m) R = Rate (mph) Time = D/R (hr)
Go 6 x 6/x
Return 6 x + 1 6/(x + 1)
The rate of the return is “1 mph faster then x” so it’s (x + 1).
Put these into the table
6
(x + 1)
6
x
Now we use the information about the times.
+ = 5
6(x + 1) + 6x
x(x + 1)
= 5
5
1
=x(x + 1)
12x + 6
use the cross
multiplication method
55. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
D = Distance (m) R = Rate (mph) Time = D/R (hr)
Go 6 x 6/x
Return 6 x + 1 6/(x + 1)
The rate of the return is “1 mph faster then x” so it’s (x + 1).
Put these into the table
6
(x + 1)
6
x
Now we use the information about the times.
+ = 5
6(x + 1) + 6x
x(x + 1)
= 5
5
1
=x(x + 1)
12x + 6
Cross again. You finish it.
use the cross
multiplication method
56. Rational Equations Word-Problems
Example B. We hiked 6 miles from A to B at a rate of x mph.
For the return trip our rate was 1 mph faster. It took us 5 hours
for the entire round trip. What is x?
D = Distance (m) R = Rate (mph) Time = D/R (hr)
Go 6 x 6/x
Return 6 x + 1 6/(x + 1)
The rate of the return is “1 mph faster then x” so it’s (x + 1).
Put these into the table
6
(x + 1)
6
x
Now we use the information about the times.
+ = 5
6(x + 1) + 6x
x(x + 1)
= 5
5
1
=x(x + 1)
12x + 6
Cross again. You finish it.
(Remember that x = 2 mph.)
use the cross
multiplication method
61. Rational Equations Word-Problems
Job–Rates
If we can eat 6 pizzas in 2 hours then our
6 pizzas
2 hours
pizza–eating rate = = 3 (piz/hr)
If we can eat 6 pizzas in 2 hours then our
pizza–eating rate =
62. Rational Equations Word-Problems
Job–Rates
If we can eat 6 pizzas in 2 hours then our
6 pizzas
2 hours
pizza–eating rate = = 3 (piz/hr)
If we can eat 6 pizzas in 2 hours then our
2 pizzas
6 hours
pizza–eating rate =
63. Rational Equations Word-Problems
Job–Rates
If we can eat 6 pizzas in 2 hours then our
6 pizzas
2 hours
pizza–eating rate = = 3 (piz/hr)
If we can eat 6 pizzas in 2 hours then our
2 pizzas
6 hours
pizza–eating rate = = (piz/hr)1
3
64. Rational Equations Word-Problems
Job–Rates
If we can eat 6 pizzas in 2 hours then our
6 pizzas
2 hours
pizza–eating rate = = 3 (piz/hr)
If we can eat 6 pizzas in 2 hours then our
2 pizzas
6 hours
pizza–eating rate = = (piz/hr)1
3
These types of rates (in per unit of time) are called job–rates.
65. Rational Equations Word-Problems
Job–Rates
If we can eat 6 pizzas in 2 hours then our
6 pizzas
2 hours
pizza–eating rate = = 3 (piz/hr)
If we can eat 6 pizzas in 2 hours then our
2 pizzas
6 hours
pizza–eating rate = = (piz/hr)1
3
These types of rates (in per unit of time) are called job–rates.
We note the following about job–rates.
66. Rational Equations Word-Problems
Job–Rates
If we can eat 6 pizzas in 2 hours then our
6 pizzas
2 hours
pizza–eating rate = = 3 (piz/hr)
If we can eat 6 pizzas in 2 hours then our
2 pizzas
6 hours
pizza–eating rate = = (piz/hr)1
3
These types of rates (in per unit of time) are called job–rates.
We note the following about job–rates.
* In all such problems, a complete job to be done may be
viewed as a pizza needing to be eaten and it’s set to be 1.
67. Rational Equations Word-Problems
Job–Rates
If we can eat 6 pizzas in 2 hours then our
6 pizzas
2 hours
pizza–eating rate = = 3 (piz/hr)
If we can eat 6 pizzas in 2 hours then our
2 pizzas
6 hours
pizza–eating rate = = (piz/hr)1
3
These types of rates (in per unit of time) are called job–rates.
We note the following about job–rates.
* In all such problems, a complete job to be done may be
viewed as a pizza needing to be eaten and it’s set to be 1.
Hence if it takes 3 hrs to paint a room, to mow a lawn, or to fill
a pool, then the job–rate for each is
1 job
3 hours
68. Rational Equations Word-Problems
Job–Rates
If we can eat 6 pizzas in 2 hours then our
6 pizzas
2 hours
pizza–eating rate = = 3 (piz/hr)
If we can eat 6 pizzas in 2 hours then our
2 pizzas
6 hours
pizza–eating rate = = (piz/hr)1
3
These types of rates (in per unit of time) are called job–rates.
We note the following about job–rates.
* In all such problems, a complete job to be done may be
viewed as a pizza needing to be eaten and it’s set to be 1.
Hence if it takes 3 hrs to paint a room, to mow a lawn, or to fill
a pool, then the job–rate for each is
1 job
3 hours
= (room), (lawn), or (pool) / hr1
3
70. Rational Equations Word-Problems
* The job–rate may be computed even if only a portion of the
job is completed. For example, if it took us 6 hrs to paint 2/3 of
the room then the job–rate is
2/3 room
6 hours
71. Rational Equations Word-Problems
* The job–rate may be computed even if only a portion of the
job is completed. For example, if it took us 6 hrs to paint 2/3 of
the room then the job–rate is
2/3 room
6 hours
=
2
3
1
6
*
72. Rational Equations Word-Problems
* The job–rate may be computed even if only a portion of the
job is completed. For example, if it took us 6 hrs to paint 2/3 of
the room then the job–rate is
2/3 room
6 hours
= (room/hr)1
9
2
3
1
6
* =
3
73. Rational Equations Word-Problems
* The job–rate may be computed even if only a portion of the
job is completed. For example, if it took us 6 hrs to paint 2/3 of
the room then the job–rate is
2/3 room
6 hours
= (room/hr)1
9
2
3
1
6
* =
* The reciprocal of a job–rate is the amount of time it would
take to complete one job.
3
74. Rational Equations Word-Problems
* The job–rate may be computed even if only a portion of the
job is completed. For example, if it took us 6 hrs to paint 2/3 of
the room then the job–rate is
2/3 room
6 hours
= (room/hr)1
9
2
3
1
6
* =
* The reciprocal of a job–rate is the amount of time it would
take to complete one job.
(lawn/hr),
1
9
Hence if the job–rate is
9
1
then it would take = 9 hrs to complete the entire lawn.
3
75. Rational Equations Word-Problems
* The job–rate may be computed even if only a portion of the
job is completed. For example, if it took us 6 hrs to paint 2/3 of
the room then the job–rate is
2/3 room
6 hours
= (room/hr)1
9
2
3
1
6
* =
* The reciprocal of a job–rate is the amount of time it would
take to complete one job.
(lawn/hr),
1
9
Hence if the job–rate is
9
1
then it would take = 9 hrs to complete the entire lawn.
Example C. We can paint 3/4 of a wall in 4 ½ hrs, what is the
job–rate? How long would it take to paint the entire wall?
3
76. Rational Equations Word-Problems
* The job–rate may be computed even if only a portion of the
job is completed. For example, if it took us 6 hrs to paint 2/3 of
the room then the job–rate is
2/3 room
6 hours
= (room/hr)1
9
2
3
1
6
* =
* The reciprocal of a job–rate is the amount of time it would
take to complete one job.
(lawn/hr),
1
9
Hence if the job–rate is
9
1
then it would take = 9 hrs to complete the entire lawn.
Example C. We can paint 3/4 of a wall in 4 ½ hrs, what is the
job–rate? How long would it take to paint the entire wall?
The job–rate is
3/4
4½
3
77. Rational Equations Word-Problems
* The job–rate may be computed even if only a portion of the
job is completed. For example, if it took us 6 hrs to paint 2/3 of
the room then the job–rate is
2/3 room
6 hours
= (room/hr)1
9
2
3
1
6
* =
* The reciprocal of a job–rate is the amount of time it would
take to complete one job.
(lawn/hr),
1
9
Hence if the job–rate is
9
1
then it would take = 9 hrs to complete the entire lawn.
Example C. We can paint 3/4 of a wall in 4 ½ hrs, what is the
job–rate? How long would it take to paint the entire wall?
The job–rate is
3/4
4½
=
3/4
9/2
=
3
4
2
9
*
3
78. Rational Equations Word-Problems
* The job–rate may be computed even if only a portion of the
job is completed. For example, if it took us 6 hrs to paint 2/3 of
the room then the job–rate is
2/3 room
6 hours
= (room/hr)1
9
2
3
1
6
* =
* The reciprocal of a job–rate is the amount of time it would
take to complete one job.
(lawn/hr),
1
9
Hence if the job–rate is
9
1
then it would take = 9 hrs to complete the entire lawn.
Example C. We can paint 3/4 of a wall in 4 ½ hrs, what is the
job–rate? How long would it take to paint the entire wall?
The job–rate is
3/4
4½
(wall / hr)=
1
6=
3/4
9/2
=
3
4
2
9
*
2 3
3
79. Rational Equations Word-Problems
* The job–rate may be computed even if only a portion of the
job is completed. For example, if it took us 6 hrs to paint 2/3 of
the room then the job–rate is
2/3 room
6 hours
= (room/hr)1
9
2
3
1
6
* =
* The reciprocal of a job–rate is the amount of time it would
take to complete one job.
(lawn/hr),
1
9
Hence if the job–rate is
9
1
then it would take = 9 hrs to complete the entire lawn.
Example C. We can paint 3/4 of a wall in 4 ½ hrs, what is the
job–rate? How long would it take to paint the entire wall?
The job–rate is
3/4
4½
(wall / hr)=
1
6=
3/4
9/2
=
3
4
2
9
*
2 3
The reciprocal of this job–rate is 6 (hr / wall) or it would
take 6 hours to paint the entire wall.
3
80. Rational Equations Word-Problems
Example D. Pipe A can fill a full tank of water in 3 hours,
pipe B can fill ¾ of the (same) tank of water in 1½ hr.
a. What is unit of the fill–rate and what is the rate of each pipe?
81. Rational Equations Word-Problems
Example D. Pipe A can fill a full tank of water in 3 hours,
pipe B can fill ¾ of the (same) tank of water in 1½ hr.
a. What is unit of the fill–rate and what is the rate of each pipe?
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A
B
82. Rational Equations Word-Problems
Example D. Pipe A can fill a full tank of water in 3 hours,
pipe B can fill ¾ of the (same) tank of water in 1½ hr.
a. What is unit of the fill–rate and what is the rate of each pipe?
The unit of the rate in question is tank/hr.
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A
B
83. Rational Equations Word-Problems
Example D. Pipe A can fill a full tank of water in 3 hours,
pipe B can fill ¾ of the (same) tank of water in 1½ hr.
a. What is unit of the fill–rate and what is the rate of each pipe?
The unit of the rate in question is tank/hr.
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A 1 3
B
84. Rational Equations Word-Problems
Example D. Pipe A can fill a full tank of water in 3 hours,
pipe B can fill ¾ of the (same) tank of water in 1½ hr.
a. What is unit of the fill–rate and what is the rate of each pipe?
The unit of the rate in question is tank/hr.
1
3
The rate of A = tank/hr.
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A 1 3 1/3
B
85. Rational Equations Word-Problems
Example D. Pipe A can fill a full tank of water in 3 hours,
pipe B can fill ¾ of the (same) tank of water in 1½ hr.
a. What is unit of the fill–rate and what is the rate of each pipe?
The unit of the rate in question is tank/hr.
1
3
The rate of A = tank/hr.
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A 1 3 1/3
B 3/4 1 ½ = 3/2 1/2
86. Rational Equations Word-Problems
Example D. Pipe A can fill a full tank of water in 3 hours,
pipe B can fill ¾ of the (same) tank of water in 1½ hr.
a. What is unit of the fill–rate and what is the rate of each pipe?
The unit of the rate in question is tank/hr.
1
3
The rate of A = tank/hr.
The rate of B =
3/4
1½
=
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A 1 3 1/3
B 3/4 1 ½ = 3/2 1/2
87. Rational Equations Word-Problems
Example D. Pipe A can fill a full tank of water in 3 hours,
pipe B can fill ¾ of the (same) tank of water in 1½ hr.
a. What is unit of the fill–rate and what is the rate of each pipe?
The unit of the rate in question is tank/hr.
1
3
=
The rate of A = tank/hr.
The rate of B =
3/4
1½
=
3/4
3/2
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A 1 3 1/3
B 3/4 1 ½ = 3/2 1/2
88. Rational Equations Word-Problems
Example D. Pipe A can fill a full tank of water in 3 hours,
pipe B can fill ¾ of the (same) tank of water in 1½ hr.
a. What is unit of the fill–rate and what is the rate of each pipe?
The unit of the rate in question is tank/hr.
1
3
=
1
2
The rate of A = tank/hr.
The rate of B = tank/hr.
3/4
1½
=
3/4
3/2
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A 1 3 1/3
B 3/4 1 ½ = 3/2 1/2
89. Rational Equations Word-Problems
Example D. Pipe A can fill a full tank of water in 3 hours,
pipe B can fill ¾ of the (same) tank of water in 1½ hr.
a. What is unit of the fill–rate and what is the rate of each pipe?
The unit of the rate in question is tank/hr.
1
3
=
1
2
The rate of A = tank/hr.
The rate of B = tank/hr.
3/4
1½
=
3/4
3/2
b. How much time would it take for pipe B to fill the tank alone?
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A 1 3 1/3
B 3/4 1 ½ = 3/2 1/2
90. Rational Equations Word-Problems
Example D. Pipe A can fill a full tank of water in 3 hours,
pipe B can fill ¾ of the (same) tank of water in 1½ hr.
a. What is unit of the fill–rate and what is the rate of each pipe?
The unit of the rate in question is tank/hr.
1
3
=
1
2
The rate of A = tank/hr.
The rate of B = tank/hr.
3/4
1½
=
3/4
3/2
b. How much time would it take for pipe B to fill the full tank
alone?
We reciprocate the rate of B “½ (tank/hr)” and get 2 hr/tank
or that it would take 2hrs for pipe B to fill a full tank.
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A 1 3 1/3
B 3/4 1 ½ = 3/2 1/2
91. Rational Equations Word-Problems
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A 1 3 1/3
B 3/4 1 ½ = 3/2 1/2
c. What is the combined rate if both pipes are used and
how long would it take to fill the entire tank if both pipes are
used?
92. Rational Equations Word-Problems
The combined rate is the sum of the rates
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A 1 3 1/3
B 3/4 1 ½ = 3/2 1/2
c. What is the combined rate if both pipes are used and
how long would it take to fill the entire tank if both pipes are
used?
1
3 +
1
2
93. Rational Equations Word-Problems
5
6=
The combined rate is the sum of the rates
tank/hr.
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A 1 3 1/3
B 3/4 1 ½ = 3/2 1/2
c. What is the combined rate if both pipes are used and
how long would it take to fill the entire tank if both pipes are
used?
1
3 +
1
2
94. Rational Equations Word-Problems
5
6=
The combined rate is the sum of the rates
tank/hr.
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A 1 3 1/3
B 3/4 1 ½ = 3/2 1/2
The reciprocal of the combined rate is 6/5 therefore it would
take 1 1/5 hr if both pipes are used.
c. What is the combined rate if both pipes are used and
how long would it take to fill the entire tank if both pipes are
used?
1
3 +
1
2
95. Rational Equations Word-Problems
5
6=
The combined rate is the sum of the rates
tank/hr.
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A 1 3 1/3
B 3/4 1 ½ = 3/2 1/2
The reciprocal of the combined rate is 6/5 therefore it would
take 1 1/5 hr if both pipes are used.
c. What is the combined rate if both pipes are used and
how long would it take to fill the entire tank if both pipes are
used?
1
3 +
1
2
We note that in the above example, it takes pipe B one hour
less to fill the tank then pipe A does and that together they got
the job done in 6/5 hr.
96. Rational Equations Word-Problems
5
6=
The combined rate is the sum of the rates
tank/hr.
Pipes A = Amount (tank) T = Time (hr) Rate = A/T (tank/hr)
A 1 3 1/3
B 3/4 1 ½ = 3/2 1/2
The reciprocal of the combined rate is 6/5 therefore it would
take 1 1/5 hr if both pipes are used.
c. What is the combined rate if both pipes are used and
how long would it take to fill the entire tank if both pipes are
used?
1
3 +
1
2
We note that in the above example, it takes pipe B one hour
less to fill the tank then pipe A does and that together they got
the job done in 6/5 hr. Let’s treat this as a word problem.
97. Rational Equations Word-Problems
Example E. Pipe B takes one hour less to fill a full tank of
water then pipe A does and together they got the job done in
6/5 hr. How long will it take pipe A to fill the tank alone?
(What should x be?)
98. Rational Equations Word-Problems
Set x = number of hours for A to
fill the tank
Example E. Pipe B takes one hour less to fill a full tank of
water then pipe A does and together they got the job done in
6/5 hr. How long will it take pipe A to fill the tank alone?
(What should x be?)
99. Rational Equations Word-Problems
Set x = number of hours for A to
fill the tank
Example E. Pipe B takes one hour less to fill a full tank of
water then pipe A does and together they got the job done in
6/5 hr. How long will it take pipe A to fill the tank alone?
(What should x be?)
So the number of hours for B to
fill the tank is (x – 1)
100. Rational Equations Word-Problems
Set x = number of hours for A to
fill the tank
Example E. Pipe B takes one hour less to fill a full tank of
water then pipe A does and together they got the job done in
6/5 hr. How long will it take pipe A to fill the tank alone?
(What should x be?)
So the number of hours for B to
fill the tank is (x – 1)
It takes 6/5 hr to use both. .
101. Rational Equations Word-Problems
Set x = number of hours for A to
fill the tank
Pipes Time
(hr)
Rate
(tank/hr)
Example E. Pipe B takes one hour less to fill a full tank of
water then pipe A does and together they got the job done in
6/5 hr. How long will it take pipe A to fill the tank alone?
(What should x be?)
So the number of hours for B to
fill the tank is (x – 1)
It takes 6/5 hr to use both.
Put these into a table.
102. Rational Equations Word-Problems
Set x = number of hours for A to
fill the tank
Pipes Time
(hr)
Rate
(tank/hr)
A
B
A & B
Example E. Pipe B takes one hour less to fill a full tank of
water then pipe A does and together they got the job done in
6/5 hr. How long will it take pipe A to fill the tank alone?
(What should x be?)
So the number of hours for B to
fill the tank is (x – 1)
It takes 6/5 hr to use both.
Put these into a table.
103. Rational Equations Word-Problems
Set x = number of hours for A to
fill the tank
Pipes Time
(hr)
Rate
(tank/hr)
A x
B (x – 1)
A & B 6/5
Example E. Pipe B takes one hour less to fill a full tank of
water then pipe A does and together they got the job done in
6/5 hr. How long will it take pipe A to fill the tank alone?
(What should x be?)
So the number of hours for B to
fill the tank is (x – 1)
It takes 6/5 hr to use both.
Put these into a table.
104. Rational Equations Word-Problems
Set x = number of hours for A to
fill the tank
Pipes Time
(hr)
Rate
(tank/hr)
A x 1/x
B (x – 1) 1/(x – 1)
A & B 6/5 5/6
Example E. Pipe B takes one hour less to fill a full tank of
water then pipe A does and together they got the job done in
6/5 hr. How long will it take pipe A to fill the tank alone?
(What should x be?)
So the number of hours for B to
fill the tank is (x – 1)
It takes 6/5 hr to use both.
Put these into a table.
105. Rational Equations Word-Problems
Set x = number of hours for A to
fill the tank
Pipes Time
(hr)
Rate
(tank/hr)
A x 1/x
B (x – 1) 1/(x – 1)
A & B 6/5 5/6
Example E. Pipe B takes one hour less to fill a full tank of
water then pipe A does and together they got the job done in
6/5 hr. How long will it take pipe A to fill the tank alone?
(What should x be?)
So the number of hours for B to
fill the tank is (x – 1)
It takes 6/5 hr to use both. .
Put these into a table. The rates for A and B add to the
combined rate 5/6 so we get a rational equation in x.
106. Rational Equations Word-Problems
Set x = number of hours for A to
fill the tank
Pipes Time
(hr)
Rate
(tank/hr)
A x 1/x
B (x – 1) 1/(x – 1)
A & B 6/5 5/6
Example E. Pipe B takes one hour less to fill a full tank of
water then pipe A does and together they got the job done in
6/5 hr. How long will it take pipe A to fill the tank alone?
(What should x be?)
So the number of hours for B to
fill the tank is (x – 1)
It takes 6/5 hr to use both. .
5
6=
1
x
+ 1
(x – 1)
Put these into a table. The rates for A and B add to the
combined rate 5/6 so we get a rational equation in x.
107. Rational Equations Word-Problems
5
6=
1
x
+ 1
(x – 1)
clear the denominatorsx (x – 1)][
Set x = number of hours for A to
fill the tank
Pipes Time
(hr)
Rate
(tank/hr)
A x 1/x
B (x – 1) 1/(x – 1)
A & B 6/5 5/6
Example E. Pipe B takes one hour less to fill a full tank of
water then pipe A does and together they got the job done in
6/5 hr. How long will it take pipe A to fill the tank alone?
(What should x be?)
So the number of hours for B to
fill the tank is (x – 1)
It takes 6/5 hr to use both. .
Put these into a table. The rates for A and B add to the
combined rate 5/6 so we get a rational equation in x.
108. Rational Equations Word-Problems
5
6=
1
x
+ 1
(x – 1)
clear the denominators6x (x – 1)
6x
][
Set x = number of hours for A to
fill the tank
Pipes Time
(hr)
Rate
(tank/hr)
A x 1/x
B (x – 1) 1/(x – 1)
A & B 6/5 5/6
Example E. Pipe B takes one hour less to fill a full tank of
water then pipe A does and together they got the job done in
6/5 hr. How long will it take pipe A to fill the tank alone?
(What should x be?)
So the number of hours for B to
fill the tank is (x – 1)
Put these into a table. The rates for A and B add to the
combined rate 5/6 so we get a rational equation in x.
It takes 6/5 hr to use both. .
6(x – 1) x (x – 1)
109. Rational Equations Word-Problems
5
6=
1
x
+ 1
(x – 1)
clear the denominators6x (x – 1)
6x
][
Set x = number of hours for A to
fill the tank
Pipes Time
(hr)
Rate
(tank/hr)
A x 1/x
B (x – 1) 1/(x – 1)
A & B 6/5 5/6
Example E. Pipe B takes one hour less to fill a full tank of
water then pipe A does and together they got the job done in
6/5 hr. How long will it take pipe A to fill the tank alone?
(What should x be?)
So the number of hours for B to
fill the tank is (x – 1)
Put these into a table. The rates for A and B add to the
combined rate 5/6 so we get a rational equation in x.
It takes 6/5 hr to use both. .
6(x – 1) x (x – 1)
6x – 6 + 6x = 5x2 – 5x
110. Rational Equations Word-Problems
5
6=
1
x
+ 1
(x – 1)
clear the denominators6x (x – 1)
6x
][
Set x = number of hours for A to
fill the tank
Pipes Time
(hr)
Rate
(tank/hr)
A x 1/x
B (x – 1) 1/(x – 1)
A & B 6/5 5/6
Example E. Pipe B takes one hour less to fill a full tank of
water then pipe A does and together they got the job done in
6/5 hr. How long will it take pipe A to fill the tank alone?
(What should x be?)
So the number of hours for B to
fill the tank is (x – 1)
Put these into a table. The rates for A and B add to the
combined rate 5/6 so we get a rational equation in x.
It takes 6/5 hr to use both. .
6(x – 1) x (x – 1)
6x – 6 + 6x = 5x2 – 5x
0 = 5x2 – 17x + 6
(Question: Why is the other solution no good?)
you finish this...
111. Rational Equations Word-Problems
Ex. For problems 1 – 9, fill in the table, set up an equation
and solve for x. (If you can guess the answer first, go ahead.
But set up and solve the problem algebraically to confirm it.)
2. A group of x people are to share 6 slices of pizza equally.
If there are three less people in the group then each person
would get one more slice. What is x?
3. A group of x people are to share 6 slices of pizza equally.
If one more person joins in to share the pizza then each
person would get three less slices. What is x?
1. A group of x people are to share 6 slices of pizza equally.
If there is one less person in the group then each person
would get one more slice. What is x?
Total Cost No. of People Cost per Person = Total Cost
No. of people
112. Rational Equations Word-Problems
5. A group of x people are to share 12 slices of pizza equally.
If two more people want to partake then each person would
get one less slice. What is x?
6. A group of x people are to share 12 slices of pizza equally.
If four people in the group failed to show up then each person
would get four more slices. What is x?
7. A group of x people are to share the cost of $24 to rent a
van equally. If three more people want to partake then each
person would pay $4 less. What is x?
8. A group of x people are to share the cost of $24 to rent a
van equally. If four more people join in the group then each
person would pay $1 less. What is x?
9. A group of x people are to share the cost of $24 to rent a
van equally. If there are two less people in the group then
each person would pay $1 more. What is x?
113. Rational Equations Word-Problems
Distance (m) Rate (mph) Time = D/R (hr)
1st trip
2nd trip
10. We hiked 6 miles from A to B at a rate of x mph. For the
return trip our rate was 1 mph slower. It took us 5 hours for the
entire round trip. What is x?
HW C. For problems 10 – 16, fill in the table, set up an
equation and solve for x.
11. We hiked 6 miles from A to B at a rate of x mph. For the
return trip our rate was 3 mph faster. It took us 3 hours for the
entire round trip. What is x?
12. We hiked 6 miles from A to B at a rate of x mph. For the
return trip our rate was 1 mph faster. It took us 9 hours for the
entire round trip. What is x?
114. Rational Equations Word-Problems
13. We hiked 3 miles from A to B at a rate of x mph. Then we
hiked 2 miles from B to C at a rate that was 1 mph slower.
It took us 2 hours for the entire round trip. What is x?
14. We hiked 6 miles from A to B at a rate of x mph. Then we
hiked 3 miles from B to C at a rate that was 1 mph faster.
It took us 3 hours for the entire round trip. What is x?
15. We ran 8 miles from A to B. Then we ran 5 miles from B to
C at a rate that was 1 mph faster. It took one more hour to get
from A to B than from B to C. What was the rate we ran from A
to B?
16. We piloted a boat 12 miles upstream from A to B. Then we
piloted it downstream from B back to A at a rate that was
4 mph faster. It took one more hour to go upstream from A to B
than going downstream from B to C. What was the rate of the
boat going up stream from A to B?
115. Rational Equations Word-Problems
17. Its takes 3 hrs to complete 2/3 of a job.
What is the job–rate?
How long would it take to complete the whole job?
How long would it take to complete ½ of the job?
For problems 17–22, don’t set up equations. Find the answers
directly and leave the answer in fractional hours.
18. Its takes 4 hrs to complete 3/5 of a job.
What is the job–rate?
How long would it take to complete the whole job?
How long would it take to complete 1/3 of the job?
Time Rate
A
B
A & B
19. Its takes 4 hrs for A to complete
3/5 of a job and 3 hr for B to do 2/3 of
the job. Complete the table.
What is their combined job–rate?
How long would it take for A & B to
complete the whole job together?
116. Rational Equations Word-Problems
Time Rate
A
B
A & B
20. Its takes 1/2 hr for A to complete
2/5 of a job and 3/4 hr for B to do 1/6
of the job. Complete the table.
What is their combined job–rate?
How long would it take for A & B to
complete the whole job together?
Time Rate
A
B
A & B
21. It takes 3 hrs for pipe A to fill a
pool and 5 hrs for pipe B to drain
the pool. Complete the table.
If we use both pipes to fill and drain
simultaneously, what is their
combined job–rate?
Will the pool be filled eventually?
If so, how long would it take for
the pool to fill?
117. Rational Equations Word-Problems
Time Rate
leak
pump
both
22. A leak in our boat will fill the boat
in 6 hrs at which time the boat sinks.
We have a pump that can empty a
filled boat in 9 hrs. How much time
do we have before the boat sinks?
For 23 –26, given the information, how many hours it would
take for A or B to do the job alone.
23. It takes A one more hour than B to do the job alone.
It takes them 2/3 hr to do the job together.
24. It takes A two more hours than B to do the job alone. It
takes them 3/4 hr to do the job together.
25. It takes B two hour less than A to do the job alone.
It takes them 4/3 hr to do the job together.
26. It takes A three more hours than B to do the job alone.
It takes them 2 hr to do the job together.
