Learn how to solve word problems using equations. The first key point is deciding what to let “x” represent.
Steps for Solving Word Problems
- Decide what to let x represent (the unknown quantity)
- Express the relationships among the quantities as an equation
- Solve the equation
- Check whether the answer satisfies the conditions of the problem
Example 1 (Number of Items and Cost)
You buy a total of 20 apples costing 80 yen each and tangerines costing 50 yen each, for a total of 1,220 yen. How many apples did you buy?
Solution:
Let the number of apples be x. Then the number of tangerines is (20－x).
80x＋50(20－x)＝1220
80x＋1000－50x＝1220
30x＝220
x＝220÷30≒7.3 (Since this is not an integer, further checking is necessary.)
※Check the numerical values in the problem. Correctly: Recheck the conditions based on 30x＝220.
(In actual problems, the values are set so that the division comes out evenly.)
You buy a total of 20 apples costing 80 yen each and tangerines costing 50 yen each, for a total of 1,220 yen. How many apples did you buy?
Solution:
Let the number of apples be x. Then the number of tangerines is (20－x).
80x＋50(20－x)＝1220
80x＋1000－50x＝1220
30x＝220
x＝220÷30≒7.3 (Since this is not an integer, further checking is necessary.)
※Check the numerical values in the problem. Correctly: Recheck the conditions based on 30x＝220.
(In actual problems, the values are set so that the division comes out evenly.)
Example 2 (Speed)
It takes 8 minutes to walk from home to school at 60 meters per minute. How many minutes does it take to run at 90 meters per minute?
Solution:
Distance＝60×8＝480 m
Running time: 480÷90＝16/3≒5.3 minutes (approximately 5 minutes 20 seconds)
It takes 8 minutes to walk from home to school at 60 meters per minute. How many minutes does it take to run at 90 meters per minute?
Solution:
Distance＝60×8＝480 m
Running time: 480÷90＝16/3≒5.3 minutes (approximately 5 minutes 20 seconds)
Example 3 (Age)
The father's current age is three times the child's age. In 12 years, the father's age will be twice the child's age. How old is the father now?
Solution:
Let the child's current age be x. Then the father is 3x years old.
In 12 years: 3x＋12＝2(x＋12)
3x＋12＝2x＋24
x＝12 (child), father＝3×12＝36 years old
The father's current age is three times the child's age. In 12 years, the father's age will be twice the child's age. How old is the father now?
Solution:
Let the child's current age be x. Then the father is 3x years old.
In 12 years: 3x＋12＝2(x＋12)
3x＋12＝2x＋24
x＝12 (child), father＝3×12＝36 years old
Key Points
- First state what you are letting x represent.
- Use consistent units (m and km, minutes and hours, etc.).
- Check whether the answer satisfies the conditions of the problem (such as being a positive integer).
Practice Problems
- When 5 is subtracted from three times a certain number, the result is equal to the value obtained by adding 4 to twice the number. Find the number.
- Currently, an older brother is 16 years old and his younger brother is 12 years old. In how many years will the older brother's age be 1.2 times the younger brother's age?
Answers and Explanations
- Answer: 9
Explanation: Let the number be x. 3x－5＝2x＋4. x＝9. Check: 3×9－5＝22, 2×9＋4＝22 ✓ - Answer: In 4 years
Explanation: Let x be the number of years until the older brother's age is 1.2 times the younger brother's age. (16＋x)＝1.2×(12＋x). 16＋x＝14.4＋1.2x. 1.6＝0.2x. x＝8 years. Check: In 8 years, the older brother will be 24 and the younger brother will be 20; 24÷20＝1.2 ✓