Work Backwards

1

• Chapter 11
• Work Backwards

2

• After dinner, three friends leaving a restaurant
  noticed a bowl of mints on the front counter.
  Sean took one-fourth of the mints but returned
  four because he had a momentary pang of guilt.
  Faizah then took one-third of what was left but
  returned three for similar reasons. Eugene then
  took half the remainder but threw two green ones
  back.
• The bowl had only 17 mints left when the raid was
  over. How many mints were originally in the bowl?

3
Write down the story steps in order.

• Start.
• Sean took 1/3 of the mints.
• Sean returned 4.
• Faizah took 1/4 of what was left.
• Faizah returned 3.
• Eugene took half the remainder.
• Eugene threw 2 back.
• The bowl had 17 mints left.

4
Work backwards from the end to the beginning.

• Start.
• Sean took 1/3 of the mints.
• Sean returned 4.
• Faizah took 1/4 of what was left.
• Faizah returned 3.
• Eugene took half the remainder.
• Eugene threw 2 back.
• The bowl had 17 mints left.

17
5
Work backwards from the end to the beginning.

• Start.
• Sean took 1/3 of the mints.
• Sean returned 4.
• Faizah took 1/4 of what was left.
• Faizah returned 3.
• Eugene took half the remainder.
• Eugene threw 2 back.
• The bowl had 17 mints left.

15
17
6
Work backwards from the end to the beginning.

• Start.
• Sean took 1/3 of the mints.
• Sean returned 4.
• Faizah took 1/4 of what was left.
• Faizah returned 3.
• Eugene took half the remainder.
• Eugene threw 2 back.
• The bowl had 17 mints left.

30
15
17
7
Work backwards from the end to the beginning.

• Start.
• Sean took 1/3 of the mints.
• Sean returned 4.
• Faizah took 1/4 of what was left.
• Faizah returned 3.
• Eugene took half the remainder.
• Eugene threw 2 back.
• The bowl had 17 mints left.

27
30
15
17
8
Draw diagrams as needed.

• Start.
• Sean took 1/3 of the mints.
• Sean returned 4.
• Faizah took 1/4 of what was left.
• Faizah returned 3.
• Eugene took half the remainder.
• Eugene threw 2 back.
• The bowl had 17 mints left.

27
30
15
17
9
Work backwards from the end to the beginning.

• Start.
• Sean took 1/3 of the mints.
• Sean returned 4.
• Faizah took 1/4 of what was left.
• Faizah returned 3.
• Eugene took half the remainder.
• Eugene threw 2 back.
• The bowl had 17 mints left.

36
27
30
15
17
10
Work backwards from the end to the beginning.

• Start.
• Sean took 1/3 of the mints.
• Sean returned 4.
• Faizah took 1/4 of what was left.
• Faizah returned 3.
• Eugene took half the remainder.
• Eugene threw 2 back.
• The bowl had 17 mints left.

32
36
27
30
15
17
11
Draw another diagram if you need to.

• Start.
• Sean took 1/3 of the mints.
• Sean returned 4.
• Faizah took 1/4 of what was left.
• Faizah returned 3.
• Eugene took half the remainder.
• Eugene threw 2 back.
• The bowl had 17 mints left.

32
36
27
30
15
17
12
Finish at the beginning.

• Start.
• Sean took 1/3 of the mints.
• Sean returned 4.
• Faizah took 1/4 of what was left.
• Faizah returned 3.
• Eugene took half the remainder.
• Eugene threw 2 back.
• The bowl had 17 mints left.

48
32
36
27
30
15
17
13
Check from beginning to end.

• Start
• Sean took 1/3 of the mints.
• Sean returned 4.
• Faizah took 1/4 of what was left.
• Faizah returned 3.
• Eugene took half the remainder.
• Eugene threw 2 back.
• The bowl had 17 mints left.

48 16
32 4
36 9
27 3
30 15
15 2
17
14
Strategy Work backwards

• Write out the steps in order.

• Work from the end to the beginning.
• Draw diagrams as needed.
• Perhaps act it out.
• Use algebra or other tools to get from one step
  to the previous one.
• Check from the beginning to the end.