Difference-Collection Method ② When There Is Excess in Both Cases or Shortage in Both Cases
In patterns where "there is still an excess even when fewer are distributed" or "there is a shortage even when more are distributed," the "+" in the formula changes to "－." It is important to examine the situation carefully.
When There Is Excess in Both Cases
If giving a items to each person leaves r items over, and giving b items (b is greater than a) leaves s items over (r is greater than s):
Number of people ＝（r － s）÷（b － a）
Example Problem (Excess in Both Cases)
If 3 candies are given to each child, 15 are left over; if 5 are given to each child, 3 are left over. How many children are there?
Solution:
① Difference in the number distributed: 5－3＝2 items
② Difference in the excess: 15－3＝12 items
③ Number of children: 12÷2＝6 children
Check: The candies are 3×6＋15＝33, or 5×6＋3＝33 ✓
If 3 candies are given to each child, 15 are left over; if 5 are given to each child, 3 are left over. How many children are there?
Solution:
① Difference in the number distributed: 5－3＝2 items
② Difference in the excess: 15－3＝12 items
③ Number of children: 12÷2＝6 children
Check: The candies are 3×6＋15＝33, or 5×6＋3＝33 ✓
When There Is a Shortage in Both Cases
If giving a items to each person leaves a shortage of r items, and giving b items (b is greater than a) leaves a shortage of s items (s is greater than r):
Number of people ＝（s － r）÷（b － a）
Example Problem (Shortage in Both Cases)
If 4 oranges are given to each person, there are 6 too few; if 6 are given to each person, there are 18 too few. How many oranges are there?
Solution:
① Difference in the number distributed: 6－4＝2 items
② Difference in the shortage: 18－6＝12 items
③ Number of people: 12÷2＝6 people
④ Number of oranges: 4×6－6＝24－6＝18 oranges
If 4 oranges are given to each person, there are 6 too few; if 6 are given to each person, there are 18 too few. How many oranges are there?
Solution:
① Difference in the number distributed: 6－4＝2 items
② Difference in the shortage: 18－6＝12 items
③ Number of people: 12÷2＝6 people
④ Number of oranges: 4×6－6＝24－6＝18 oranges
Key Points
- It is easier to organize the problem by putting the excess and shortage patterns into a table.
- The calculation used changes depending on the case: "difference between excess amounts," "difference between shortage amounts," or "total of excess and shortage."
- In every pattern, the same point is to divide by the "difference in the number distributed."
Practice Problems
- If 2 notebooks are given to each person, 20 are left over; if 4 are given to each person, 6 are left over. How many notebooks are there?
- If 3 pencils are given to each person, there are 4 too few; if 5 are given to each person, there are 20 too few. How many pencils are there?
- If 7 items are given to each person, 12 are left over; if 9 are given to each person, 2 are left over. How many people are there?
Answers and Explanations
- Answer:There are 36 notebooks
Explanation: Giving 2 notebooks to each person leaves 20, while giving 4 leaves 6 (excess in both cases). Difference in distribution＝4－2＝2 notebooks. Difference in excess＝20－6＝14 notebooks. Number of people＝14÷2＝7. Number of notebooks＝2×7＋20＝14＋20＝34. Check: 4×7＋6＝28＋6＝34✓ - Answer:There are 11 pencils
Explanation: Giving 3 pencils to each person creates a shortage of 4, while giving 5 creates a shortage of 20 (shortage in both cases). Difference in distribution＝5－3＝2 pencils. Difference in shortage＝20－4＝16 pencils. Number of people＝16÷2＝8. Number of pencils＝3×8－4＝24－4＝20. Check: 5×8－20＝40－20＝20✓ - Answer:There are 5 people
Explanation: Giving 7 items to each person leaves 12, while giving 9 leaves 2 (excess in both cases). Difference in distribution＝9－7＝2 items. Difference in excess＝12－2＝10 items. Number of people＝10÷2＝5. Check: items＝7×5＋12＝35＋12＝47. 9×5＋2＝45＋2＝47✓