1. Section 9-6
Problem Solving Skills:
Work Backwards
2. Essential Question
• How do you solve real-world applications by
working backward?
• Where you’ll see this:
• Part-time jobs, ﬁnance, travel
3. Example 1
After four stops, every seat in a school bus was taken.
Half as many students got on at the second stop as at
the ﬁrst stop, and half as many got on at the third stop
as at the second stop. At the fourth stop, ﬁve students
got on, the same number as at the third stop. How
many passenger seats were there on that bus?
4. Example 1
After four stops, every seat in a school bus was taken.
Half as many students got on at the second stop as at
the ﬁrst stop, and half as many got on at the third stop
as at the second stop. At the fourth stop, ﬁve students
got on, the same number as at the third stop. How
many passenger seats were there on that bus?
4th stop: 3rd stop:
2nd stop: 1st stop:
5. Example 1
After four stops, every seat in a school bus was taken.
Half as many students got on at the second stop as at
the ﬁrst stop, and half as many got on at the third stop
as at the second stop. At the fourth stop, ﬁve students
got on, the same number as at the third stop. How
many passenger seats were there on that bus?
4th stop: 5 got on 3rd stop:
2nd stop: 1st stop:
6. Example 1
After four stops, every seat in a school bus was taken.
Half as many students got on at the second stop as at
the ﬁrst stop, and half as many got on at the third stop
as at the second stop. At the fourth stop, ﬁve students
got on, the same number as at the third stop. How
many passenger seats were there on that bus?
4th stop: 5 got on 3rd stop:
Same as third stop
2nd stop: 1st stop:
7. Example 1
After four stops, every seat in a school bus was taken.
Half as many students got on at the second stop as at
the ﬁrst stop, and half as many got on at the third stop
as at the second stop. At the fourth stop, ﬁve students
got on, the same number as at the third stop. How
many passenger seats were there on that bus?
4th stop: 5 got on 3rd stop: 5 got on
Same as third stop
2nd stop: 1st stop:
8. Example 1
After four stops, every seat in a school bus was taken.
Half as many students got on at the second stop as at
the ﬁrst stop, and half as many got on at the third stop
as at the second stop. At the fourth stop, ﬁve students
got on, the same number as at the third stop. How
many passenger seats were there on that bus?
4th stop: 5 got on 3rd stop: 5 got on
Same as third stop 1/2 the 2nd stop
2nd stop: 1st stop:
9. Example 1
After four stops, every seat in a school bus was taken.
Half as many students got on at the second stop as at
the ﬁrst stop, and half as many got on at the third stop
as at the second stop. At the fourth stop, ﬁve students
got on, the same number as at the third stop. How
many passenger seats were there on that bus?
4th stop: 5 got on 3rd stop: 5 got on
Same as third stop 1/2 the 2nd stop
2nd stop: 10 got on 1st stop:
10. Example 1
After four stops, every seat in a school bus was taken.
Half as many students got on at the second stop as at
the ﬁrst stop, and half as many got on at the third stop
as at the second stop. At the fourth stop, ﬁve students
got on, the same number as at the third stop. How
many passenger seats were there on that bus?
4th stop: 5 got on 3rd stop: 5 got on
Same as third stop 1/2 the 2nd stop
2nd stop: 10 got on 1st stop:
1/2 the 1st stop
11. Example 1
After four stops, every seat in a school bus was taken.
Half as many students got on at the second stop as at
the ﬁrst stop, and half as many got on at the third stop
as at the second stop. At the fourth stop, ﬁve students
got on, the same number as at the third stop. How
many passenger seats were there on that bus?
4th stop: 5 got on 3rd stop: 5 got on
Same as third stop 1/2 the 2nd stop
2nd stop: 10 got on 1st stop: 20 got on
1/2 the 1st stop
12. Example 1
After four stops, every seat in a school bus was taken.
Half as many students got on at the second stop as at
the ﬁrst stop, and half as many got on at the third stop
as at the second stop. At the fourth stop, ﬁve students
got on, the same number as at the third stop. How
many passenger seats were there on that bus?
4th stop: 5 got on 3rd stop: 5 got on
Same as third stop 1/2 the 2nd stop
2nd stop: 10 got on 1st stop: 20 got on
1/2 the 1st stop 5+5+10+20
13. Example 1
After four stops, every seat in a school bus was taken.
Half as many students got on at the second stop as at
the ﬁrst stop, and half as many got on at the third stop
as at the second stop. At the fourth stop, ﬁve students
got on, the same number as at the third stop. How
many passenger seats were there on that bus?
4th stop: 5 got on 3rd stop: 5 got on
Same as third stop 1/2 the 2nd stop
2nd stop: 10 got on 1st stop: 20 got on
1/2 the 1st stop 5+5+10+20 =40
14. Example 1
After four stops, every seat in a school bus was taken.
Half as many students got on at the second stop as at
the ﬁrst stop, and half as many got on at the third stop
as at the second stop. At the fourth stop, ﬁve students
got on, the same number as at the third stop. How
many passenger seats were there on that bus?
4th stop: 5 got on 3rd stop: 5 got on
Same as third stop 1/2 the 2nd stop
2nd stop: 10 got on 1st stop: 20 got on
1/2 the 1st stop 5+5+10+20 =40 seats
15. Example 2
Matt Mitarnowski has $14 in his pocket now. During
the day, he spent $27.45 on fez hats, $8.62 on bubble
gum, and $6 on toys from vending machines. How
much money did he start out with?
16. Example 2
Matt Mitarnowski has $14 in his pocket now. During
the day, he spent $27.45 on fez hats, $8.62 on bubble
gum, and $6 on toys from vending machines. How
much money did he start out with?
Has $14, spent $27.45, $8.62, $6
17. Example 2
Matt Mitarnowski has $14 in his pocket now. During
the day, he spent $27.45 on fez hats, $8.62 on bubble
gum, and $6 on toys from vending machines. How
much money did he start out with?
Has $14, spent $27.45, $8.62, $6
14+27.45+8.62+6
18. Example 2
Matt Mitarnowski has $14 in his pocket now. During
the day, he spent $27.45 on fez hats, $8.62 on bubble
gum, and $6 on toys from vending machines. How
much money did he start out with?
Has $14, spent $27.45, $8.62, $6
14+27.45+8.62+6
=$56.07
19. Example 3
After Shecky planted 42 bubble gum trees, 32 cotton
candy bushes, and 25 graphing calculator plants in his
garden, he had 185 plants in all. How many plants
were in the garden before he added to them?
20. Example 3
After Shecky planted 42 bubble gum trees, 32 cotton
candy bushes, and 25 graphing calculator plants in his
garden, he had 185 plants in all. How many plants
were in the garden before he added to them?
In the end, he has 185 plants
21. Example 3
After Shecky planted 42 bubble gum trees, 32 cotton
candy bushes, and 25 graphing calculator plants in his
garden, he had 185 plants in all. How many plants
were in the garden before he added to them?
In the end, he has 185 plants
He just planted 42, 32, and 25 plants
22. Example 3
After Shecky planted 42 bubble gum trees, 32 cotton
candy bushes, and 25 graphing calculator plants in his
garden, he had 185 plants in all. How many plants
were in the garden before he added to them?
In the end, he has 185 plants
He just planted 42, 32, and 25 plants
185-42-32-25
23. Example 3
After Shecky planted 42 bubble gum trees, 32 cotton
candy bushes, and 25 graphing calculator plants in his
garden, he had 185 plants in all. How many plants
were in the garden before he added to them?
In the end, he has 185 plants
He just planted 42, 32, and 25 plants
185-42-32-25
= 86 plants
24. Homework
25. Homework
p. 400 #1-17
“Live up to the best that is in you: Live noble lives, as
you all may, in whatever condition you may ﬁnd
yourselves.” - Henry W. Longfellow
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