Sum-and-Difference Problems ③ Solving Problems Skillfully with Line-Segment Diagrams
Line-segment diagrams are a powerful tool that lets you see sum-and-difference problems. Even when a problem is somewhat complicated, if you carefully draw a line-segment diagram, the solution method will naturally become clear.
How to Draw a Line-Segment Diagram
- ① Draw the larger number above and the smaller number below using horizontal lines.
- ② Align the left ends of the lines neatly.
- ③ Label the part where the upper and lower lengths differ as “difference.”
- ④ Label the whole amount as “total.”
Example (Somewhat Difficult)
Elder Brother has 300 yen more than twice Younger Brother's money, and together they have 2700 yen. How much money does each have?
Solution:
① Let Younger Brother's money be □ yen. Then Elder Brother＝2×□＋300.
② Since they have 2700 yen altogether,
□＋(2×□＋300)＝2700
③ Three □s＋300＝2700, so three □s＝2400
④ □＝2400÷3＝800
Answer: Younger Brother＝800 yen, Elder Brother＝2×800＋300＝1900 yen
Check: 800＋1900＝2700 ✓
Elder Brother has 300 yen more than twice Younger Brother's money, and together they have 2700 yen. How much money does each have?
Solution:
① Let Younger Brother's money be □ yen. Then Elder Brother＝2×□＋300.
② Since they have 2700 yen altogether,
□＋(2×□＋300)＝2700
③ Three □s＋300＝2700, so three □s＝2400
④ □＝2400÷3＝800
Answer: Younger Brother＝800 yen, Elder Brother＝2×800＋300＝1900 yen
Check: 800＋1900＝2700 ✓
Line-Segment Diagrams with a “Times” Condition
When “A is ○ times B” appears, represent “○ times” using the length of the line segment.
Younger Brother: |────────| □ yen
Elder Brother: |────────|────────|＋300|
←─── 2□ ──→←300→
←────────── 2700 ──────────────→
Key Points
- When the problem says “○ times plus △,” first draw the length for “○ times,” then add “△” afterward.
- Always align the left ends of the line segments.
- Even for difficult problems, a line-segment diagram makes it easier to create an equation.
- Most sum-and-difference problems can be solved with a line-segment diagram.
Practice Problems
- Elder Sister's savings are 200 yen less than three times Younger Sister's savings, and together they have 3400 yen. Find each sister's savings.
- The total of two numbers is 100, and the larger number is four times the smaller number. Find the two numbers.
- A red ribbon is 15cm longer than a blue ribbon, and the blue ribbon is 2/3 the length of the red ribbon. Find the length of the blue ribbon.
Answers and Explanations
- Answer: Elder Sister＝2450 yen, Younger Sister＝950 yen
Explanation: Let Younger Sister's savings be □ yen. Elder Sister＝3×□－200. □＋3×□－200＝3400. 4□＝3600. □＝900 yen (Younger Sister). Elder Sister＝3×900－200＝2700－200＝2500 yen. Check: 2500＋900＝3400✓. Note: Because Elder Sister has 200 yen less than three times Younger Sister's savings, Elder Sister＝3□－200. From 4□－200＝3400, □＝900, so Elder Sister＝2500 yen and Younger Sister＝900 yen. - Answer: Larger number＝80, smaller number＝20
Explanation: Let the smaller number be □. The larger number＝4×□. □＋4□＝100. 5□＝100. □＝20 (smaller number). Larger number＝4×20＝80. Check: 80＋20＝100✓, 80＝4×20✓ - Answer: Blue ribbon＝30cm
Explanation: Let the length of the blue ribbon be □cm. Red ribbon＝□＋15. Condition “blue is 2/3 of red”: □＝(2/3)×(□＋15). 3□＝2□＋30. □＝30cm (blue). Red＝30＋15＝45cm. Check: 30＝(2/3)×45＝30✓