Sum-and-Difference Problems ① Finding Two Numbers from Their Sum and Difference
Sum-and-difference problems are problems in which you find two numbers when you know their total and the difference between them. They are one of the most basic types of word problems in junior high school entrance examinations.
Formulas to Remember
If the larger of the two numbers is A and the smaller is B, they can be found as follows.
- Larger number A ＝ (total ＋ difference) ÷ 2
- Smaller number B ＝ (total － difference) ÷ 2
You do not have to memorize these formulas completely. If you understand why they work, you can create them yourself.
When the two equations A＋B＝total and A－B＝difference are added, there are two As, giving “total＋difference.” Therefore, A can be found by “(total＋difference)÷2.”
There are 20 apples and oranges altogether. There are 4 more apples than oranges. How many apples and oranges are there?
Solution:
Total＝20, difference＝4
① Apples (larger amount)＝(20＋4)÷2＝24÷2＝12
② Oranges (smaller amount)＝(20－4)÷2＝16÷2＝8
Check: 12＋8＝20, 12－8＝4 ✓
Visualize with a Line-Segment Diagram
Sum-and-difference problems become much easier to understand when you draw a line-segment diagram comparing lengths with horizontal lines.
Apples: |────────────────| 12
Oranges: |──────────────| 8
←── total=20 ──→
←difference 4→
Looking at the diagram, if you place one more orange line beside the first to make twice as much, you can see that “oranges×2＋difference＝total.” Therefore, oranges＝(total－difference)÷2.
- Instead of memorizing formulas, first develop the habit of drawing a line-segment diagram.
- Unless the total and difference are both even or both odd, the answer will not be an integer (a number that is not a decimal or fraction). In a well-formed problem, the answer will always be an integer.
- After finding the answer, check whether the original numbers give the correct total and difference.
Practice Problems
- When Elder Brother's and Younger Brother's money is combined, it totals 3600 yen. Elder Brother has 800 yen more than Younger Brother. How much money does each have?
- The total of two numbers is 84, and their difference is 12. Find the larger number.
- Class A and Class B have 65 students altogether. Class A has 7 more students than Class B. How many students are in Class A?
Answers and Explanations
- Answer: Elder Brother＝2200 yen, Younger Brother＝1400 yen
Explanation: Sum＝3600, difference＝800. Elder Brother＝(3600＋800)÷2＝4400÷2＝2200 yen. Younger Brother＝(3600－800)÷2＝2800÷2＝1400 yen. Check: 2200＋1400＝3600✓, 2200－1400＝800✓ - Answer: Larger number＝48
Explanation: Sum＝84, difference＝12. Larger number＝(84＋12)÷2＝96÷2＝48. Smaller number＝(84－12)÷2＝72÷2＝36. Check: 48＋36＝84✓, 48－36＝12✓ - Answer: Class A＝36 students
Explanation: Sum＝65, difference＝7. Class A＝(65＋7)÷2＝72÷2＝36 students. Class B＝(65－7)÷2＝58÷2＝29 students. Check: 36＋29＝65✓, 36－29＝7✓