Sum-and-Difference Problems ② Problems with the Sum and Differences of Three Numbers
Even when there are three numbers, the way of thinking is simple. The key is to set the smallest number as a reference and express the other numbers as the reference number plus something.
How to Solve Three-Number Sum-and-Difference Problems
When there are three numbers A, B, and C and the differences in their sizes are known, use the smallest number as the reference.
- ① Let the smallest number be □.
- ② Write the other numbers in the form “□＋something.”
- ③ Make an equation for the total of the three numbers and find □.
Example
A, B, and C have 60 cards altogether. A has 6 more cards than B, and B has 3 more cards than C. How many cards does each person have?
Solution:
① Let the smallest number, C, be □.
② B＝□＋3, and A＝□＋3＋6＝□＋9.
③ Make an equation for the total.
□＋(□＋3)＋(□＋9)＝60
Three □s＋12＝60
Three □s＝48
□＝48÷3＝16
Answer: C＝16 cards, B＝19 cards, A＝25 cards
Check: 16＋19＋25＝60 ✓
A, B, and C have 60 cards altogether. A has 6 more cards than B, and B has 3 more cards than C. How many cards does each person have?
Solution:
① Let the smallest number, C, be □.
② B＝□＋3, and A＝□＋3＋6＝□＋9.
③ Make an equation for the total.
□＋(□＋3)＋(□＋9)＝60
Three □s＋12＝60
Three □s＝48
□＝48÷3＝16
Answer: C＝16 cards, B＝19 cards, A＝25 cards
Check: 16＋19＋25＝60 ✓
Another Method: Think “What If Everyone Had the Same Amount?”
If you temporarily imagine that A, B, and C all have the same number of cards as the smallest number, C, you can sometimes solve the problem more quickly.
- ① Imaginary total＝smallest number × number of people
- ② Total difference＝actual total－imaginary total
- ③ Combine the total difference with the extra amount each person has to find the answer
Key Points
- When there are three numbers, always let the smallest number be □.
- Draw three line segments side by side horizontally to make the relationships easy to see.
- After finding the answers, check both the total and each difference.
Practice Problems
- The total of three numbers is 90. The middle number is 4 greater than the smallest number, and the largest number is 10 greater than the smallest number. Find the smallest number.
- The total ages of Taro, Jiro, and Saburo is 33 years. Taro is 2 years older than Jiro, and Jiro is 3 years older than Saburo. How old is Saburo?
- There are 45 red, blue, and yellow balls altogether. There are 8 fewer blue balls than red balls, and 3 more yellow balls than red balls. How many red balls are there?
Answers and Explanations
- Answer: Smallest number＝76/3 (25 and 1/3)
Explanation: Let the smallest number be □. The second number is □＋4, and the largest is □＋10. □＋(□＋4)＋(□＋10)＝90. Since 3□＋14＝90, 3□＝76 and □＝76/3 (25 and 1/3). Check: 76/3＋88/3＋106/3＝270/3＝90. - Answer: Saburo＝25/3 years (8 and 1/3 years)
Explanation: Let Saburo's age be □ years. Jiro＝□＋3, and Taro＝□＋5. □＋(□＋3)＋(□＋5)＝33. Since 3□＋8＝33, 3□＝25 and □＝25/3 years (8 and 1/3 years). - Answer: No integer solution
Explanation: Let the number of red balls be □. Blue＝□－8, and yellow＝□＋3. □＋(□－8)＋(□＋3)＝45. Since 3□－5＝45, □＝50/3. However, the number of balls must be an integer, so there is no integer solution satisfying all the conditions in the problem.