Word Problems & Translation • Topic 2 of 4

Ages & Consecutive Integers

Age and consecutive-integer problems are translation with a time or sequence twist. For ages, define the present ages first, then adjust for time: "5 years ago" subtracts 5 from every person's current age, and "in 8 years" adds 8 to each — the mistake to avoid is adjusting only one person. For consecutive integers, model them as n, n + 1, n + 2, …; consecutive even or odd integers step by 2, as n, n + 2, n + 4. A useful GRE fact: the sum of an odd count of consecutive integers equals the middle term times the count, so three consecutive integers summing to 72 have middle term 24 (giving 23, 24, 25) — you can often skip the algebra entirely. Keep track of who or what each variable represents; these problems punish a variable you defined loosely more than they punish arithmetic.

✅ Solved examples

1. The sum of three consecutive integers is 72. Find them.
Let them be n, n + 1, n + 2. Sum = 3n + 3 = 72, so 3n = 69, n = 23. The integers are 23, 24, 25. (Middle term = 72/3 = 24 confirms it.)
2. Maria is 4 years older than Sam. In 6 years the sum of their ages will be 40. How old is Sam now?
Let Sam = s, Maria = s + 4. In 6 years: (s + 6) + (s + 4 + 6) = 40, so 2s + 16 = 40, 2s = 24, s = 12.
3. The sum of two consecutive even integers is 46. Find them.
Let them be n and n + 2. 2n + 2 = 46, so 2n = 44, n = 22. The integers are 22 and 24.
4. A father is three times as old as his son. In 12 years he will be twice as old. Find the son's current age.
Let son = s, father = 3s. In 12 years: 3s + 12 = 2(s + 12), so 3s + 12 = 2s + 24, giving s = 12.

✏️ Practice — try these, take hints as needed

1. The sum of three consecutive integers is 90. Find the largest.
Middle term = 90/3.
Middle is 30.
Largest is middle + 1.
31
2. Two consecutive odd integers sum to 56. Find them.
n and n + 2.
2n + 2 = 56.
n = 27.
27 and 29
3. Ravi is 5 years younger than Priya. Their ages sum to 33. How old is Ravi?
Priya = p, Ravi = p − 5.
p + (p − 5) = 33.
2p = 38.
Ravi is 14
4. A mother is 4 times as old as her daughter. In 20 years she will be twice as old. Find the daughter's age now.
Daughter = d, mother = 4d.
4d + 20 = 2(d + 20).
2d = 20.
10
5. The sum of four consecutive integers is 90. Find the smallest.
n + (n+1) + (n+2) + (n+3) = 90.
4n + 6 = 90.
4n = 84.
21

📝 Topic test — 8 questions

Auto-graded with full solutions; saved to your dashboard. Use the calculator and formula sheet (top-right) any time.

Loading questions…
← Translating Words to Algebra Interest & Growth →