This is the final lesson on using equations. We will study application problems involving properties of integers, ages, and consecutive integers. Develop the ability to read the conditions and choose your own variables.
Problems Involving Consecutive Integers
Tips for expressing consecutive integers with variables:
・Three consecutive integers: x、x+1、x+2 (or x−1、x、x+1)
・Three consecutive even integers: x、x+2、x+4 (or x−2、x、x+2)
・Three consecutive odd integers: x、x+2、x+4 (odd integers also differ by 2)
The sum of three consecutive integers is 78. Find the three integers.
Solution:
Let the middle number be x. The three numbers are x−1、x、x+1.
Equality: (x−1) + x + (x+1) = 78 → 3x = 78 → x = 26.
The three integers are 25、26、27.
Age Problems
Key points for age problems:
・The age difference does not change (everyone grows older by the same amount).
・“In ○ years” means adding ○ to everyone’s age; “○ years ago” means subtracting ○ from everyone’s age.
Currently, a father is 40 years old and his child is 12 years old. In how many years will the father be three times as old as the child?
Solution:
Suppose that in x years the father will be three times as old as the child.
Father’s age ＝ 40 + x; child’s age ＝ 12 + x.
Equality: 40 + x ＝ 3(12 + x)
40 + x = 36 + 3x → −2x = −4 → x = 2.
In 2 years.
Problems Involving Two-Digit Integers
A two-digit integer with a in the tens place and b in the ones place can be expressed as 10a + b.
Example: 73 = 10×7 + 3.
The number obtained by switching the digits is 10b + a.
Example: Switching the digits of 73 gives 37 = 10×3 + 7.
There is a two-digit integer. Its tens digit is 4, and the number obtained by switching the tens and ones digits is 27 greater than the original number. Find the original integer.
Solution:
Let the ones digit be x. Original number = 40 + x; switched number = 10x + 4.
Equality: 10x + 4 ＝ (40 + x) + 27 → 10x + 4 = 67 + x → 9x = 63 → x = 7.
The original integer is 47.
- Consecutive integers: x、x+1、x+2 (if the middle is x, then x−1, x, x+1)
- Consecutive even or odd integers: x、x+2、x+4 (the difference is 2)
- Age problems: the age difference does not change. In ○ years, add ○ to everyone’s age.
- Two-digit integers: tens digit a, ones digit b → 10a + b
Practice Problems
- The sum of three consecutive even integers is 66. Find the three even integers.
- Currently, a mother is 36 years old and her child is 8 years old. In how many years will the mother be three times as old as the child?
- A two-digit integer has digits whose sum is 9, and the number obtained by switching the tens and ones digits is 9 greater than the original number. Find the original integer.
Answers and Explanations
- Answer: 20、22、24
Explanation: Let the middle number be x. (x−2)+x+(x+2) = 66 → 3x = 66 → x = 22. Therefore, 20, 22, 24. - Answer: In 6 years
Explanation: In x years, 36+x = 3(8+x) → 36+x = 24+3x → 12 = 2x → x = 6. - Answer: 45
Explanation: Let the tens digit be a and the ones digit be 9−a. Original = 10a + (9−a) = 9a+9. Switched = 10(9−a)+a = 90−9a. Equality: 90−9a = (9a+9)+9 → 90−9a = 9a+18 → 72 = 18a → a = 4. Therefore, 45.