Difference-Collection Method ① Finding the Number of People or Items from Excess and Shortage
The difference-collection method is used in problems where the number of people or items is found from the "left over" or "shortage" that occurs when the same number of identical items is distributed to everyone.
Common Situations for the Difference-Collection Method
Consider a situation in which oranges are distributed to children.
- Giving a items to each person leaves r items over
- Giving b items to each person (b is greater than a) leaves a shortage of s items
In this case, the total of the "extra items given to each person (b－a)" exactly equals the "leftover＋shortage (r＋s)".
Example Problem
If 5 oranges are given to each child, 8 are left over; if 7 are given to each child, there are 4 too few. How many children are there?
Solution:
① Difference per person: 7－5＝2 items
② Total of the leftover and shortage: 8＋4＝12 items
③ Number of children: 12÷2＝6 children
Check: The oranges are 5×6＋8＝38, or 7×6－4＝38 ✓
If 5 oranges are given to each child, 8 are left over; if 7 are given to each child, there are 4 too few. How many children are there?
Solution:
① Difference per person: 7－5＝2 items
② Total of the leftover and shortage: 8＋4＝12 items
③ Number of children: 12÷2＝6 children
Check: The oranges are 5×6＋8＝38, or 7×6－4＝38 ✓
Formula
Number of people ＝（leftover ＋ shortage）÷（larger distribution amount － smaller distribution amount）
Key Points
- The basic formula is to divide "leftover＋shortage" by the "difference in the number distributed."
- Substitute the number of people you find into the original conditions and always check your answer.
- Even if you are not asked for the total number of items, calculating it as a check helps you answer with confidence.
Practice Problems
- If 2 notebooks are given to each student, 5 are left over; if 3 are given to each student, 4 are too few. How many students are there?
- If 4 cookies are given to each child, 7 are left over; if 6 are given to each child, 5 are too few. How many cookies are there in total?
- If 8 sheets of origami paper are given to each person, 12 are left over; if 10 are given to each person, 2 are too few. How many sheets of origami paper are there?
Answers and Explanations
- Answer:There are 9 students
Explanation: Giving 2 notebooks to each student leaves 5, while giving 3 leaves a shortage of 4. Difference in distribution＝3－2＝1 notebook. Total of leftover and shortage＝5＋4＝9 notebooks. Number of students＝9÷1＝9. Check: notebooks＝2×9＋5＝23. 3×9－4＝23✓ - Answer:There are 27 cookies in total
Explanation: Giving 4 cookies to each child leaves 7, while giving 6 creates a shortage of 5. Difference in distribution＝6－4＝2 cookies. Total of leftover and shortage＝7＋5＝12 cookies. Number of children＝12÷2＝6. Number of cookies＝4×6＋7＝24＋7＝31. Check: 6×6－5＝31✓ - Answer:There are 76 sheets of origami paper
Explanation: Giving 8 sheets to each person leaves 12, while giving 10 creates a shortage of 2. Difference in distribution＝10－8＝2 sheets. Total of leftover and shortage＝12＋2＝14 sheets. Number of people＝14÷2＝7. Number of sheets＝8×7＋12＝56＋12＝68. Check: 10×7－2＝68✓