Fix the mistake: Surplus-and-Deficiency Problems
Excess-and-Shortage Method ① "Left Over" and "Shortage" When Distributing Equal Amounts
The excess-and-shortage method is used for problems similar to difference-collection problems, but its basic situation is distributing the same number of items to each person. Learn to distinguish it carefully while keeping its difference from the difference-collection method in mind.
Basics of the Excess-and-Shortage Method
When the same number of items is distributed to each child,
- Giving ○ items to each person leaves an excess (overage)
- Giving △ items to each person creates a shortage
Find the number of children or items from this situation.
Example Problem
If 8 sweets are given to each child, 12 are left over; if 10 are given to each child, there are 6 too few. How many sweets are there in total?
Solution:
① Number of children＝（12＋6）÷（10－8）＝18÷2＝9 children
② Number of sweets＝8×9＋12＝72＋12＝84 sweets
Check: 10×9－6＝90－6＝84 ✓
If 8 sweets are given to each child, 12 are left over; if 10 are given to each child, there are 6 too few. How many sweets are there in total?
Solution:
① Number of children＝（12＋6）÷（10－8）＝18÷2＝9 children
② Number of sweets＝8×9＋12＝72＋12＝84 sweets
Check: 10×9－6＝90－6＝84 ✓
Let's Visualize It with a Diagram
8 each:|──────────────────|＋12 left over
10 each:|───────────────────────|－6 short
←─── Total amount of sweets ───→
The diagram shows that distributing 10 to each person requires "the number of people×2 items" more than distributing 8 to each. This difference is "12 left over＋6 short＝18 items," so the number of children is 18÷2＝9.
Key Points
- The formula is the same as for the difference-collection method: "（leftover＋shortage）÷（difference in the number distributed）＝number of people"
- First find the number of people, and then calculate the number of items (two stages).
- Always check using both conditions.
Practice Problems
- If 5 candies are given to each child, 3 are left over; if 7 are given to each child, there are 7 too few. How many children are there?
- If 4 apples are given to each child, 6 are left over; if 5 are given to each child, there are 2 too few. How many apples are there?
- If 9 stickers are given to each person, 4 are left over; if 12 are given to each person, there are 11 too few. How many stickers are there in total?
Answers and Explanations
- Answer:There are 5 children
Explanation: Giving 5 to each child leaves 3, while giving 7 creates a shortage of 7. Number of children＝（leftover＋shortage）÷（difference in distribution）＝（3＋7）÷（7－5）＝10÷2＝5. Check: candies＝5×5＋3＝28. 7×5－7＝28✓ - Answer:There are 26 apples
Explanation: Giving 4 apples to each child leaves 6, while giving 5 creates a shortage of 2. Number of children＝（6＋2）÷（5－4）＝8÷1＝8. Number of apples＝4×8＋6＝32＋6＝38. Check: 5×8－2＝38✓ - Answer:There are 49 stickers in total
Explanation: Giving 9 stickers to each person leaves 4, while giving 12 creates a shortage of 11. Number of people＝（4＋11）÷（12－9）＝15÷3＝5. Number of stickers＝9×5＋4＝45＋4＝49. Check: 12×5－11＝60－11＝49✓
Previous and next lessons in this course
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