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Age problems
Age problems pile “now”, “in 6 years” and “5 years ago” into one sentence, and that is the whole difficulty. Put the ages in a small table and every one becomes a one-line equation.
People in rows, times in columns
| Now | In 6 years | |
|---|---|---|
| Son | s | s + 6 |
| Nora | 3s | 3s + 6 |
Give the younger person a letter, write everyone else from it, and add the same number of years to everyone in a column. The sentence about the future becomes one equation using only the “in 6 years” column: 3s + 6 = 2(s + 6), so 3s + 6 = 2s + 12 and s = 6. The son is 6 and Nora is 18.
Everyone ages the same
Six years later, every person is exactly six years older. So the difference between two ages never changes, but “twice as old” or “3 times as old” does change over time.
How to solve them
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1
Set up the table before you write an equation
One row per person, one column per moment the problem mentions (now, in some years, some years ago). Fill each cell with an expression in your one letter.
Next step: One columnLeo is 14. In 4 years he will be twice as old as Max will be then.Now In 4 years Leo 14 18 Max m m + 4 -
2
Each equation uses one column only
A sentence about “in 4 years” compares the ages in the “in 4 years” column. Never set a “now” age equal to a future one: that is the most common mistake in this whole topic.
Next step: Years agoExampleFinish the Leo and Max problem.
- In 4 years: 18 = 2(m + 4).
- m + 4 = 9, so m = 5.
- Check: in 4 years Max is 9 and Leo is 18, twice as old.
Answer Max is 5 now
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3
Years ago: subtract from everyone
A column for the past works the same way, with the years taken away from every person.
Next step: The gapExampleFive years ago, Ivy was 3 times as old as Sam. Ivy is 20 now. How old is Sam now?
- Five years ago: Ivy was 20 − 5 = 15, and Sam was S − 5.
- 15 = 3(S − 5), so S − 5 = 5.
- S = 10.
Answer Sam is 10 now
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4
The age gap never changes
If one person is 9 years older today, they were 9 years older ten years ago and will be 9 years older in fifty years. Many questions fall in one step with this.
Next step: Sums of agesExampleTom is 9 years older than Ella. When Ella is 30, how old will Tom be?
- The gap is always 9.
- 30 + 9 = 39.
Answer 39
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5
Sums of ages: every person adds the years
If the sum of two ages is given, remember that each of the two people gets older. In 7 years, the sum grows by 7 + 7, not by 7.
Next step: Check the answerExampleTwo sisters’ ages add up to 50 now. What will they add up to in 7 years?
- Each sister gains 7 years.
- 50 + 7 + 7 = 64.
Answer 64
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6
Answer for the right person and the right time
Your letter is usually someone’s age now. Read the question again: it may ask for the other person, or for an age in the future or the past.
Which person?You found the son (6); it asks for Nora.
3 × 6 = 18
Which time?You found Max now (5); it asks for Max in 4 years.
5 + 4 = 9
Does it fit?Put the numbers back in the sentence.
In 6 years, 24 is twice 12: it fits
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Mixing two columns
In 5 years Rob will be twice as old as Kim, who is 7 now:
WrongR + 5 = 2 × 7
RightR + 5 = 2 × (7 + 5) → R = 19
Adding the years to one person
The sum of two ages is 40 now. In 3 years:
Wrong40 + 3 = 43
Right40 + 3 + 3 = 46
Answering for the wrong time
Asked for Sam’s age now; you found Sam 5 years ago:
Wrong5
Right5 + 5 = 10
“Times as old” turned around
Ann is 4 times as old as Ben, who is 5:
WrongAnn = 5 ÷ 4
RightAnn = 4 × 5 = 20
5 questions at test level
Pick an answer, then read why. No timer, no account.
Kara is 8 years older than her brother. Her brother is 15 now. How old will Kara be in 6 years?
Show the answer
C. Kara is 15 + 8 = 23 now, and in 6 years she will be 23 + 6 = 29. 23 is her age now, and 21 is her brother in 6 years. Kara will be 29.
A father is 4 times as old as his daughter. The sum of their ages is 45. How old is the father?
Show the answer
D. Let d = the daughter’s age; the father is 4d. d + 4d = 45, so 5d = 45 and d = 9. The father is 4 × 9 = 36. 9 is the daughter, and 11.25 divides 45 by 4. The father is 36.
In 3 years, Mia will be twice as old as her cousin will be then. Mia is 17 now. How old is her cousin now?
Show the answer
B. Use the “in 3 years” column: Mia will be 20, and 20 = 2(c + 3), so c + 3 = 10 and c = 7. 10 is the cousin in 3 years, and 8.5 halves Mia’s age now. Her cousin is 7.
Six years ago, Luis was 3 times as old as Pablo. Luis is 30 now. How old is Pablo now?
Show the answer
C. Six years ago Luis was 24, and 24 = 3(P − 6), so P − 6 = 8 and P = 14. 8 is Pablo six years ago, and 24 is Luis back then. Pablo is 14.
Rita is 3 times as old as Jon. In 10 years, Rita will be twice as old as Jon. How old is Rita now?
Show the answer
D. Now: Jon j, Rita 3j. In 10 years: 3j + 10 = 2(j + 10), so 3j + 10 = 2j + 20 and j = 10. Rita is 3 × 10 = 30. Check: in 10 years, 40 is twice 20. 10 is Jon. Rita is 30.
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