Learn how to solve word problems using simultaneous equations, as well as applied problems involving fractions and decimals.
How to Solve Word Problems
Example 1 (Cost)
Two pencils and three notebooks cost 650 yen, while four pencils and one notebook cost 550 yen. Find the price of one pencil and one notebook.
Solution:
Let the price of one pencil be x yen and the price of one notebook be y yen.
① 2x＋3y＝650
② 4x＋y＝550
②×3: 12x＋3y＝1650
③－①: 10x＝1000 → x＝100
Substitute into ②: 400＋y＝550 → y＝150
One pencil costs 100 yen, and one notebook costs 150 yen.
Two pencils and three notebooks cost 650 yen, while four pencils and one notebook cost 550 yen. Find the price of one pencil and one notebook.
Solution:
Let the price of one pencil be x yen and the price of one notebook be y yen.
① 2x＋3y＝650
② 4x＋y＝550
②×3: 12x＋3y＝1650
③－①: 10x＝1000 → x＝100
Substitute into ②: 400＋y＝550 → y＝150
One pencil costs 100 yen, and one notebook costs 150 yen.
Example 2 (Speed)
There is a mountain trail that takes 12 minutes uphill and 8 minutes downhill. The uphill speed is 40 m per minute, and the downhill speed is x m per minute. Find the length of the mountain trail.
Solution:
Length of the mountain trail＝uphill speed×uphill time＝40×12＝480m
Downhill speed: 480÷8＝60m/分
（Example of simultaneous equations: If the distance is y m, then y=40×12, y=x×8 → x=60）
There is a mountain trail that takes 12 minutes uphill and 8 minutes downhill. The uphill speed is 40 m per minute, and the downhill speed is x m per minute. Find the length of the mountain trail.
Solution:
Length of the mountain trail＝uphill speed×uphill time＝40×12＝480m
Downhill speed: 480÷8＝60m/分
（Example of simultaneous equations: If the distance is y m, then y=40×12, y=x×8 → x=60）
Key Points
- Two unknowns → Set up two equations.
- Identify the “number of items equation,” “cost equation,” and “distance equation.”
- Two-digit integer problems: Let x be the tens digit and y be the ones digit. (The number is 10x＋y.)
Practice Problems
- The total admission fee for 3 adults and 5 children was 3700 yen. For 5 adults and 3 children, it was 4300 yen. Find the admission fee for one adult and one child.
- There is a two-digit integer. The sum of its tens digit and ones digit is 11, and the number obtained by reversing the tens and ones digits is 27 greater than the original number. Find the original integer.
Answers and Explanations
- Answer: Adult: 650 yen, Child: 350 yen
Explanation: Let the adult admission fee be x yen and the child admission fee be y yen. ①3x＋5y＝3700, ②5x＋3y＝4300. ②×5－①×3: 25x＋15y－(9x＋15y)＝21500－11100. Since 16x＝10400, x＝650 yen. Substitute into ①: 1950＋5y＝3700, so y＝350 yen. Check: 3×650＋5×350＝3700, 5×650＋3×350＝4300. - Answer: The original integer is 47
Explanation: Let the tens digit be x and the ones digit be y. ①x＋y＝11, ②(10y＋x)＝(10x＋y)＋27→9y－9x＝27→y－x＝3. ①＋（y－x＝3）: 2y＝14→y＝7, x＝4. Original integer: 47. Check: 4＋7＝11 ✓, 74－47＝27 ✓