This document provides examples of solving problems by working backwards. The first example involves determining the number of seats on a school bus given information about the number of students that boarded at four stops. Working backwards from the information given, it is determined that there were 40 seats on the bus. The second example involves calculating how much money a student started with given the amount he has now and what he spent. Working backwards, it is determined he started with $56.07. The third example involves determining the number of plants in a garden before new plants were added, given the total number of plants after adding. Working backwards, it is determined there were 86 plants originally.

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