In integer problems, express the place-value representation; in age problems, express the relationship between present and future ages.
Two-Digit Integer Problems
A two-digit integer with a in the tens place and b in the ones place is expressed as 10a+b.
Example ①
There is a two-digit integer. The sum of its digits is 11, and the number obtained by reversing the tens and ones digits is 27 greater than the original number. Find the original number.
Solution: Let the tens digit be x and the ones digit be y: { x+y=11、(10y+x)=(10x+y)+27. Simplifying equation 2: −9x+9y=27→−x+y=3. { x+y=11、−x+y=3. Add the equations: 2y=14→y=7,x=4。The original number is 47
There is a two-digit integer. The sum of its digits is 11, and the number obtained by reversing the tens and ones digits is 27 greater than the original number. Find the original number.
Solution: Let the tens digit be x and the ones digit be y: { x+y=11、(10y+x)=(10x+y)+27. Simplifying equation 2: −9x+9y=27→−x+y=3. { x+y=11、−x+y=3. Add the equations: 2y=14→y=7,x=4。The original number is 47
Age Problems
Example ②
At present, the father's age is four times the child's age. In 8 years, the father will be twice the child's age. Find their present ages.
Solution: Present: father x years old, child y years old. { x=4y、x+8=2(y+8)。Expand equation 2: x+8=2y+16→x=2y+8. { x=4y、x=2y+8. 4y=2y+8→y=4,x=16。Father: 16 years old; child: 4 years old
At present, the father's age is four times the child's age. In 8 years, the father will be twice the child's age. Find their present ages.
Solution: Present: father x years old, child y years old. { x=4y、x+8=2(y+8)。Expand equation 2: x+8=2y+16→x=2y+8. { x=4y、x=2y+8. 4y=2y+8→y=4,x=16。Father: 16 years old; child: 4 years old
Key Points
- Two-digit integer: tens digit a, ones digit b → 10a+b
- Age problems: n years later means (present age)+n
- Always check whether the solution is a natural number or positive integer
Practice Problem
- A two-digit integer has digits whose sum is 9. When the tens and ones digits are reversed, the resulting number is 27 less than the original. Find the original number.
Answer and Explanation
- Answer:63. { x+y=9、(10y+x)=(10x+y)−27→−9x+9y=−27→x−y=3. { x+y=9、x−y=3→x=6,y=3. The original number is 63.