Difference-Collection Method ③ Tips for Recognizing "This Is a Difference-Collection Problem!"
Develop the ability to read a problem and notice, "This is a problem that can be solved using the difference-collection method." Also check how it differs from similar problems.
Points for Recognizing the Difference-Collection Method
- Words such as "distribute," "divide," or "allocate" appear.
- There are two conditions such as "giving ○ items to each person leaves △ items over (or creates a shortage of △ items)."
- You are asked for the "number of people (number of groups)" or the "number of items."
Example Problem (Application)
Students sit on benches. If 4 students sit on each bench, 7 students cannot sit; if 5 students sit on each bench, only 3 students sit on the last bench. How many students are there?
Solution:Let the number of benches be N.
① Four students each: number of students＝4×N＋7
② Five students each: number of students＝5×（N－1）＋3＝5N－5＋3＝5N－2 (only 3 students sit on the last bench)
③ Since ① and ② represent the same number of students,
4N＋7＝5N－2
7＋2＝5N－4N
N＝9 benches
④ Number of students＝4×9＋7＝43 students
Students sit on benches. If 4 students sit on each bench, 7 students cannot sit; if 5 students sit on each bench, only 3 students sit on the last bench. How many students are there?
Solution:Let the number of benches be N.
① Four students each: number of students＝4×N＋7
② Five students each: number of students＝5×（N－1）＋3＝5N－5＋3＝5N－2 (only 3 students sit on the last bench)
③ Since ① and ② represent the same number of students,
4N＋7＝5N－2
7＋2＝5N－4N
N＝9 benches
④ Number of students＝4×9＋7＝43 students
Key Points
- The condition "only ○ items are in the last one" can often be reinterpreted as "there is a shortage of △ items when △ items are placed in each one."
- The difference-collection method can also be used in situations involving sitting on benches or dividing people into groups.
- When the problem is difficult, it is more reliable to set up equations and solve them.
- If you are unsure whether it is a difference-collection or excess-and-shortage problem, check whether the two conditions form a pair.
Practice Problems
- If 3 students sit on each bench, 13 students cannot sit; if 5 sit on each bench, only 2 students sit on the last bench. How many students are there?
- If 5 apples are put in each bag, 3 bags are left over; if 8 apples are put in each bag, only 4 apples are in the last bag. How many apples are there in total?
- "At a school with 30 students in one class, when everyone lines up in the gym, 4 students per row leaves 2 students over, while 5 students per row leaves 3 students short." Can this problem be solved as a difference-collection problem? If not, state the reason.
Answers and Explanations
- Answer:There are 43 students
Explanation: With 3 students per bench, 13 cannot sit; with 5 students per bench, only 2 sit on the last bench. If the number of benches is N: 3N＋13＝number of students, 5（N－1）＋2＝5N－3＝number of students. 3N＋13＝5N－3. 16＝2N. N＝8 benches. Number of students＝3×8＋13＝24＋13＝37. Check: 5×（8－1）＋2＝35＋2＝37✓ - Answer:There are 32 apples in total
Explanation: If 5 apples are put in each bag, 3 bags are left over; if 8 are put in each bag, only 4 apples are in the last bag. If the number of bags is N: number of apples＝5×（N－3） (since 3 bags are left over, N－3 bags are used). 8×（N－1）＋4＝8N－4 (4 apples are in the last bag). 5（N－3）＝8N－4. 5N－15＝8N－4. －11＝3N. N＝－11/3 (a negative contradiction). Another solution: "3 bags are left over when 5 apples are put in each" means that after putting 5 apples in each bag, 3 bags remain, so the number of apples is 5×(number of bags－3). With 8 apples per bag and 4 apples in the last bag: 8×(number of bags－1)＋4. 5（N－3）＝8（N－1）＋4. 5N－15＝8N－4. －11＝3N→contradiction. If the wording is interpreted differently, "3 bags are left over when 5 apples are put in" means the bags are left over, so the number of apples is 5×（N－3）, while with 8 apples the last bag has 4: 8×（N－1）＋4. Since the answer is contradictory, the problem should be checked. Learn the solution procedure of setting up and solving two equations with the number of bags N as □. - Answer:It cannot be solved as a difference-collection problem
Explanation: Check "In a class of 30 students, 4 students per row leaves 2 students over, while 5 students per row leaves 3 students short": 4 people×number of rows＋2＝30→number of rows＝7. 5 people×number of rows－3＝30→number of rows＝6.6 (not an integer). Since the number of students is fixed at 30, this is not a difference-collection problem in which the number of people is found from two conditions. It is a problem for checking whether the conditions about the number of rows and the excess or shortage are consistent. The difference-collection method is used when the number of people or items is unknown and is found from two conditions; here, 30 people is already known, so it is not necessary to solve it using that method.