At the end of 1986, R was half as old as his grandmother. The sum of the years…

Concept: When two people's ages are related by a ratio at a given reference year, and the sum of their birth years is known, set up a single linear equation in one unknown by writing each birth year as (reference year minus age at that reference year). Solving this equation gives the age at the reference year, which can then be projected forward or backward to any other year by adding or subtracting the elapsed number of years.

1. Let R's age at the end of 1986 be k years. Since R was half as old as his grandmother, the grandmother's age at the end of 1986 was 2k years.

2. R's birth year = 1986 minus k, and the grandmother's birth year = 1986 minus 2k.

3. The sum of their birth years is given as 3846, so (1986 minus k) + (1986 minus 2k) = 3846, which simplifies to 3972 minus 3k = 3846.

4. Solving, 3k = 3972 minus 3846 = 126, so k = 42. R was 42 years old at the end of 1986.

5. From the end of 1986 to the end of 1998 is 12 years, so R's age at the end of 1998 = 42 + 12 = 54.

Cross-check: R's birth year = 1986 minus 42 = 1944, and the grandmother's birth year = 1986 minus 84 = 1902. Their sum is 1944 + 1902 = 3846, matching the given data, and the grandmother's age (84) is indeed exactly twice R's age (42) at the end of 1986, confirming the value.