Multiple-Ratio Problems ③ Problems Combined with Age Problems
Problems combining multiple-ratio problems and age problems often appear in junior-high entrance examinations. They ask “how many years from now (or how many years ago) will the ratio become ○：○?”
Working Together with Age Problems
The key points of age problems are as follows.
- The difference between ages never changes (it is the same after any number of years and at any time in the past).
- After N years, everyone becomes N years older.
Example Problem
Now, the father is 40 years old and the child is 10 years old. How many years from now will the father’s age be three times the child’s age?
Solution:
① Father’s age after N years: 40＋N
② Child’s age after N years: 10＋N
③ Condition: 40＋N＝3×(10＋N)
40＋N＝30＋3N
10＝2N
N＝5 years later
Check: Father 45 years old, child 15 years old. 45＝3×15 ✓
Now, the father is 40 years old and the child is 10 years old. How many years from now will the father’s age be three times the child’s age?
Solution:
① Father’s age after N years: 40＋N
② Child’s age after N years: 10＋N
③ Condition: 40＋N＝3×(10＋N)
40＋N＝30＋3N
10＝2N
N＝5 years later
Check: Father 45 years old, child 15 years old. 45＝3×15 ✓
Example Problem 2 (Solving with a Ratio)
The current ratio of the older brother’s age to the younger brother’s age is 5：3, and the difference is 8 years. How many years from now will the ratio of their ages become 3：2?
Solution:
① Difference of 8 years and ratio 5：3 → Difference in the ratio is 5－3＝2 → One unit is 4 years
② Older brother＝20 years old, younger brother＝12 years old (now)
③ After N years: (20＋N)：(12＋N)＝3：2
2×(20＋N)＝3×(12＋N)
40＋2N＝36＋3N
N＝4 years later
Check: Older brother 24 years old, younger brother 16 years old. 24：16＝3：2 ✓
The current ratio of the older brother’s age to the younger brother’s age is 5：3, and the difference is 8 years. How many years from now will the ratio of their ages become 3：2?
Solution:
① Difference of 8 years and ratio 5：3 → Difference in the ratio is 5－3＝2 → One unit is 4 years
② Older brother＝20 years old, younger brother＝12 years old (now)
③ After N years: (20＋N)：(12＋N)＝3：2
2×(20＋N)＝3×(12＋N)
40＋2N＝36＋3N
N＝4 years later
Check: Older brother 24 years old, younger brother 16 years old. 24：16＝3：2 ✓
Key Points
- In age problems, use the fact that “the difference does not change” to find the current ages.
- For “N years later,” add N to each age and make a ratio equation.
- For “N years ago,” subtract N from each age (also check that the answer is greater than 0).
- When both multiple-ratio and age problems appear, the two facts “the difference is constant” and “the ratio changes” are used at the same time.
Practice Problems
- Now, the mother is 36 years old and her daughter is 6 years old. How many years from now will the mother’s age be four times her daughter’s age?
- The current ratio of the grandfather’s age to the grandchild’s age is 9：1. In 10 years, it will be 4：1. Find the grandfather’s current age.
- The current ratio of A’s age to B’s age is 7：4, and the difference is 12 years. Find the ratio of their ages in 5 years.
Answers and Explanations
- Answer: 4 years later
Explanation: Mother’s age after N years: 36＋N. Daughter’s age after N years: 6＋N. Condition: 36＋N＝4×(6＋N). 36＋N＝24＋4N. 12＝3N. N＝4 years later. Check: Mother 40 years old, daughter 10 years old→40＝4×10✓ - Answer: The grandfather is currently 54 years old
Explanation: Since the current ratio is 9：1, grandfather＝9k and grandchild＝k. In 10 years, (9k＋10)：(k＋10)＝4：1. From 9k＋10＝4(k＋10), 5k＝30 and k＝6. The grandfather is currently 54 years old, and the grandchild is 6 years old. In 10 years, they will be 64 and 16, and 64：16＝4：1. - Answer: The ratio of A’s age to B’s age in 5 years will be 11：7
Explanation: The current ratio is 7：4 and the difference is 12 years. The difference in the ratio is 3 units, so 1 unit＝12÷3＝4 years. A is currently 28 years old and B is 16 years old. In 5 years, they will be 33 and 21, so 33：21＝11：7.