The age difference never changes
Meera is 9 years old and her brother Atharv is 14. Every year both of them grow one year older. So Atharv always stays exactly 5 years older than Meera.
The gap between two ages is the age difference. Difference = age of the older one − age of the younger one.
The age difference never changes
In the app you can move this figure with your finger.
Open in the appThe difference stays the same when years go back or forward
| When | Meera | Atharv | Difference |
|---|---|---|---|
| 4 years ago | 5 | 10 | 5 |
| Today | 9 | 14 | 5 |
| After 6 years | 15 | 20 | 5 |
Age difference = age of the older one − age of the younger one.
- e.g. ages 17 and 11 : difference = 17 − 11 = 6 years.
Both ages go up (or down) by the same number of years, so the difference stays the same.
- e.g. today 17 and 11. After 8 years 25 and 19 : 25 − 19 = 6. The same difference.
If one age is given, the other age = add the difference (to find the older one) or subtract it (to find the younger one).
- e.g. difference 6. When the younger one is 20, the older one = 20 + 6 = 26 years old.
Meera is 9 and Atharv is 14. What is the difference in their ages?
- Difference = 14 − 9 = 5 years.
- After 6 years Meera will be 15 and Atharv 20. Even then 20 − 15 = 5. The difference did not change.
A mother is 35 and her daughter is 8. How old will the mother be when the daughter is 20?
- Difference = 35 − 8 = 27 years.
- When the daughter is 20, the mother = 20 + 27 = 47 years.
Today father is 45 and uncle is 38. What was the difference in their ages 10 years ago?
- Today's difference = 45 − 38 = 7.
- The difference never changes, so 10 years ago it was also 7. There is no need to take both ages back; it saves time.
Whether the question says "5 years ago" or "after 20 years", the difference between the two ages stays the same.
Adding years to (or taking them away from) only one person. Both must get the same years added or subtracted.
Memory clue : "Years go, the difference stays." As soon as you read the question, find the difference, write it in the margin and use only that from then on.
Spot the difference
| Point | Age difference | Age ratio (how many times) |
|---|---|---|
| When years pass | does not change | changes |
| Father 40, son 10 : today | 30 | 4 times |
| The same, after 10 years (50 and 20) | 30 | 2.5 times |
Practice questions
- 1Meera is 9 and Atharv is 14. What is the difference in their ages?
- 2Seema is 11 and her mother is 38. What will be the difference in their ages after 10 years?
- 3Two brothers are 8 and 13 today. What was the difference in their ages 4 years ago?
- 4A grandmother is 66 and her granddaughter is 12. How old will the grandmother be when the granddaughter is 20?
- 5The difference between the ages of an uncle and his nephew is 21 years. How old was the uncle when the nephew was 7?
- 6Ravi is 30 and Raju is 7 years younger than him. How old was Raju 9 years ago?
- 7A father is 27 years older than his son. How old was the son when the father was 40? And how old will the father be when the son is 31?
- 8Anu, Banu and Chanu are 10, 14 and 19. How old will Banu and Chanu be when Anu is 25?
Show answers (try them yourself first)
- 114 − 9 = 5 years.
- 2The difference does not change : 38 − 11 = 27 years.
- 313 − 8 = 5 years (the same 4 years ago too).
- 4Difference 66 − 12 = 54; when the granddaughter is 20 the grandmother = 20 + 54 = 74 years.
- 5When the nephew is 7 the uncle = 7 + 21 = 28 years.
- 6Raju is 30 − 7 = 23 today; 9 years ago 23 − 9 = 14 years.
- 7When the father is 40 the son = 40 − 27 = 13 years; when the son is 31 the father = 31 + 27 = 58 years.
- 8Anu went from 10 to 25, so 15 years pass; Banu 14 + 15 = 29 and Chanu 19 + 15 = 34 years.
Before, today and after — years forward and back
Today you are 10. After 3 years you will be 10 + 3 = 13. And 2 years ago you were 10 − 2 = 8. Going back means subtract, going forward means add.
"Before / years ago" means back, so subtract. "After / years later" means forward, so add. Today's age is the middle point of every question.
Before, today and after — years forward and back
In the app you can move this figure with your finger.
Open in the appThe three-column table — use it for every question
| Before | Today | After | |
|---|---|---|---|
| Age | today − years | today's age | today + years |
Special rule for the sum (many people) : The same number of years is added for every person. So the sum of all their ages changes by people × years.
Before, today and after — years forward and back
In the app you can move this figure with your finger.
Open in the app"Years ago / before" means subtract and "years later / after" means add.
- e.g. today 30 years : 7 years ago 30 − 7 = 23; after 7 years 30 + 7 = 37.
First find out which time the given age belongs to, and find today's age first.
- e.g. 5 years ago the age was 12. Today = 12 + 5 = 17 years.
The sum of the ages of many people changes by people × years (it goes up when we go forward and down when we go back).
- e.g. the sum of 4 people is 60 today. After 3 years = 60 + 4 × 3 = 72.
4 years ago Rohan was 15. How old will he be after 3 years?
- Today = 15 + 4 = 19 years.
- After 3 years = 19 + 3 = 22 years.
The sum of the ages of three sisters is 36 today. What will the sum be after 5 years?
- 5 years must be added for each of the three : 3 × 5 = 15.
- New sum = 36 + 15 = 51.
4 years ago the sum of the ages of three people was 51. What will the sum be after 3 years?
- Slow method : today's sum = 51 + 3 × 4 = 63. Then after 3 years = 63 + 3 × 3 = 72.
- Fast method : From the earlier time to the later time, 4 + 3 = 7 years pass in all. Sum = 51 + 3 × 7 = 72. There is no need to find today's sum.
If both "years ago" and "years later" appear, add the two numbers of years (the total gap). If both are "years later", subtract the smaller from the larger.
Forgetting the number of people in a sum. For 3 people, 5 years does not mean 5; add 3 × 5 = 15.
A child may not have been born "that many years ago". When going back, check that nobody's age falls below zero.
Memory clue : "Before subtract, after add; people × years."
Spot the difference
| Word in the question | Direction | Action |
|---|---|---|
| years ago / before | back | subtract |
| years later / after | forward | add |
| today / now | right now | keep as it is |
Practice questions
- 1Nisha is 13 today. How old will she be after 5 years?
- 2Father is 41 today. How old was he 6 years ago?
- 3Two friends are 10 and 12 today. What will be the sum of their ages after 3 years?
- 44 years ago Rohan was 15. How old will he be after 3 years?
- 5The sum of the ages of three sisters is 36 today. What will the sum be after 5 years?
- 6The sum of the ages of four children will be 70 after 6 years. What is the sum today?
- 75 years ago the sum of the ages of four people was 48. What will the sum be after 2 years?
- 8The sum of the ages of two siblings is 25 today. After how many years will the sum be 37?
Show answers (try them yourself first)
- 113 + 5 = 18 years.
- 241 − 6 = 35 years.
- 3Today's sum 22; add 2 × 3 = 6 : 28.
- 4Today 15 + 4 = 19; after 3 years 22 years.
- 536 + 3 × 5 = 51.
- 6Subtract 4 × 6 = 24 : 70 − 24 = 46.
- 7In all 5 + 2 = 7 years; 48 + 4 × 7 = 76.
- 8The sum must grow by 37 − 25 = 12; with two people it grows by 2 every year; 12 ÷ 2 = 6 years.
Given the sum and the difference, find both ages
The sum of the ages of a mother and her son is 50 and the difference is 28. How do we find the two ages? Take the difference 28 away from 50 and 22 is left. Now the two are "equal"; so split 22 into two equal parts, 11 each. That is the son. Add the difference back for the mother : 11 + 28 = 39.
Simple method (in words)
- Age of the older one = (sum + difference) ÷ 2
- Age of the younger one = (sum − difference) ÷ 2
Given the sum and the difference, find both ages
In the app you can move this figure with your finger.
Open in the appCheck : The two ages added should give the sum, and the younger subtracted from the older should give the difference. Mother 39 and son 11 : 39 + 11 = 50 and 39 − 11 = 28. Correct.
Age of the older one = (sum + difference) ÷ 2.
- e.g. sum 30, difference 6 : (30 + 6) ÷ 2 = 18.
Age of the younger one = (sum − difference) ÷ 2. Then add the two ages and check that the sum matches.
- e.g. (30 − 6) ÷ 2 = 12. Check : 18 + 12 = 30 and 18 − 12 = 6.
If the sum is given for another time and not for today, first find today's sum : today's sum = given sum − 2 × years (if it is "years later") or + 2 × years (if it is "years ago").
- e.g. the sum after 4 years is 40. Today = 40 − 2 × 4 = 32.
The sum of two ages is 20 and the difference is 4.
- Older = (20 + 4) ÷ 2 = 12. Younger = (20 − 4) ÷ 2 = 8.
The sum of the ages of a mother and her son is 50 and the difference is 28.
- Mother = (50 + 28) ÷ 2 = 39. Son = (50 − 28) ÷ 2 = 11.
After 5 years the sum of two ages will be 60. Today the difference in their ages is 8. What are the two ages today?
- 5 years were added for each of them; today's sum = 60 − 2 × 5 = 50.
- Older = (50 + 8) ÷ 2 = 29. Younger = (50 − 8) ÷ 2 = 21.
Add the difference and halve for the older one; subtract the difference and halve for the younger one. At the end add the two ages and check.
Forgetting to add the difference to the sum and divide by 2, or taking sum − difference as the younger age without halving it. Both must be divided by 2.
Forgetting "the sum is given after 5 years" and taking the same sum as today's. First find today's sum.
Memory clue : "Add - subtract - halve." Add for the older, subtract for the younger, halve both.
A 10-second exam trick : the answer comes in whole years only if the sum and the difference are both even or both odd. If one is even and the other odd, that option or question is wrong.
Spot the difference
| What is given | What to do |
|---|---|
| sum + difference | (sum ± difference) ÷ 2 |
| sum + "x times" | in the next part (parts method) |
| sum "after 5 years" | first today's sum = given − people × years |
Practice questions
- 1The sum of two ages is 20 and the difference is 4. What is the age of the older one?
- 2The sum of two ages is 30 and the difference is 10. What is the age of the younger one?
- 3The sum of two ages is 14 and the difference is 2. Tell both ages.
- 4The sum of the ages of a mother and her daughter is 52 and the difference is 26. What is the mother's age?
- 5The sum of the ages of two friends is 33 and the difference is 5. Tell both ages.
- 6The sum of the ages of a brother and a sister is 25. The brother is 3 years younger than the sister. What is the brother's age?
- 7Is it possible for the sum of two ages to be 41 and the difference 6? Why?
- 8After 5 years the sum of two ages will be 60 and today the difference is 8. What are the two ages today?
Show answers (try them yourself first)
- 1(20 + 4) ÷ 2 = 12 years.
- 2(30 − 10) ÷ 2 = 10 years.
- 3Older (14 + 2) ÷ 2 = 8, younger (14 − 2) ÷ 2 = 6.
- 4Mother (52 + 26) ÷ 2 = 39 years (daughter 13).
- 5Older (33 + 5) ÷ 2 = 19, younger (33 − 5) ÷ 2 = 14.
- 6Sister (25 + 3) ÷ 2 = 14, brother (25 − 3) ÷ 2 = 11 years.
- 7No. 41 is odd and 6 is even; (41 − 6) ÷ 2 = 17.5, which gives half a year.
- 8Today's sum 60 − 10 = 50; older (50 + 8) ÷ 2 = 29, younger (50 − 8) ÷ 2 = 21.
The ratio puzzle — the parts method
A father's age is 3 times his son's age. Think of the son's age as one box, so the father has three boxes. How many boxes is the difference? 3 − 1 = 2 boxes. If the difference is 24 years, then 2 boxes = 24, so 1 box = 12. The son is 12 and the father 36.
Simple method (in words)
- If the difference is given : age of the younger one = difference ÷ (times − 1)
- If the sum is given : age of the younger one = sum ÷ (times + 1)
- Age of the older one = age of the younger one × times
The ratio puzzle — the parts method
In the app you can move this figure with your finger.
Open in the appParts table (father 3 times)
| Younger | Older | Difference | |
|---|---|---|---|
| Parts | 1 | 3 | 2 |
| Years (difference 24) | 12 | 36 | 24 |
When "times" is given, draw parts : younger = 1 part, older = as many parts as the times.
- e.g. father is 4 times the son : son 1 part, father 4 parts.
If the difference is given, 1 part = difference ÷ (times − 1).
- e.g. difference 21, 4 times : 21 ÷ 3 = 7. Son 7, father 28.
If the sum is given, 1 part = sum ÷ (times + 1).
- e.g. sum 35, 4 times : 35 ÷ 5 = 7. Son 7, father 28.
The "times" is true only at that moment. When years pass, find the new ages and count the times again.
- e.g. 7 and 28 (4 times) ; after 7 years 14 and 35, which is 2.5 times. The difference is still 21.
The son is 7. The father is 5 times his age. How old is the father?
- 7 × 5 = 35 years.
The father is 4 times the son's age and the difference between them is 27.
- Parts : 4 − 1 = 3. Son = 27 ÷ 3 = 9. Father = 9 × 4 = 36.
Today a mother is 5 times her daughter's age and the sum of their ages is 42. How many times the daughter's age will the mother be after 7 years?
- Parts : 5 + 1 = 6. Daughter = 42 ÷ 6 = 7, mother = 35.
- After 7 years the daughter is 14 and the mother 42. 42 ÷ 14 = 3 times.
The "times" is true only at that moment. When years pass the times changes, but the difference does not.
Mixing up "3 times" and "3 more". 3 times means multiply, 3 more means add.
Taking 3 parts when the difference is given. Parts = times − 1 (2 parts if it is 3 times).
Memory clue : "For the difference, times-minus-one; for the sum, times-plus-one." That is, difference ÷ (times − 1) and sum ÷ (times + 1).
Spot the difference
| What is given | Number of parts | Age of the younger one |
|---|---|---|
| difference | times − 1 | difference ÷ (times − 1) |
| sum | times + 1 | sum ÷ (times + 1) |
Practice questions
- 1The son is 7 and the father is 5 times his age. How old is the father?
- 2A little girl is 6 and her elder sister is 3 times her age. What is the difference between them?
- 3The difference is 10 and the older one is 2 times the younger one. Tell both ages.
- 4The father is 4 times the son's age. The difference is 27. How old is the father?
- 5The sum of two ages is 40 and one is 3 times the other. Tell both ages.
- 6A grandfather is 6 times his grandson's age. The difference is 55. How old is the grandson?
- 7Today a mother is 5 times her daughter's age and the sum of their ages is 42. How many times the daughter's age will the mother be after 7 years?
- 8Today a father is 4 times his son's age. After 6 years he will be 3 times the son's age. How old is the son today? (Hint : the difference never changes. Today the difference is 3 parts; after 6 years the difference is 2 times the son's age, so the son is 1.5 parts. So 6 years = half a part.)
Show answers (try them yourself first)
- 17 × 5 = 35 years.
- 2Elder sister 6 × 3 = 18; difference 18 − 6 = 12 years.
- 3Younger = 10 ÷ (2 − 1) = 10; older = 20.
- 4Son 27 ÷ (4 − 1) = 9; father 9 × 4 = 36 years.
- 5Younger = 40 ÷ (3 + 1) = 10; older = 30.
- 6Grandson = 55 ÷ (6 − 1) = 11 years (grandfather 66).
- 7Daughter 42 ÷ 6 = 7, mother 35; after 7 years 14 and 42 : 3 times.
- 8Half a part = 6 years, so 1 part = 12. Son 12 and father 48. Check : after 6 years 18 and 54; 54 = 18 × 3.
10-second exam tricks (checking the options)
After buying something in a shop we quickly check whether the change is right. In the same way, in an age question, after choosing an answer from the options, put it back into the conditions of the question. If it is wrong, drop that option at once.
The 4-step exam frame
- Step 1 : Underline whose age is wanted today.
- Step 2 : Find the difference and write it aside (it does not change).
- Step 3 : Use "before = subtract, after = add" to find today's ages.
- Step 4 : Put the answer back into every condition of the question and check.
10-second exam tricks (checking the options)
In the app you can move this figure with your finger.
Open in the appThe two ages come out in whole numbers only if the sum and the difference are both even or both odd.
- e.g. sum 31, difference 6 : one odd, one even. So the ages will not be whole numbers. Such an option is wrong.
Change in the sum = total years that changed × people.
- e.g. the sum of 3 people was 33 four years ago. After 2 years : in all 4 + 2 = 6 years; 33 + 3 × 6 = 51.
Once you have the answer, put it back into every condition of the question (difference, sum, before-after).
- e.g. ages 18 and 12 : the sum 30 and the difference 6 both match.
After 3 years the ages of two people will be 20 and 15. What is the younger one's age today? (1) 10 (2) 12 (3) 15 (4) 18
- Younger 15 − 3 = 12. The answer is option (2).
The sum of the ages of three people is 45 today. What will the sum be after 4 years? (1) 49 (2) 57 (3) 51 (4) 60
- Add 3 × 4 = 12 : 45 + 12 = 57. The answer is (2).
- 10-second check : three people, the sum grows by 12. 45 + 12 is exactly 57. 49 and 51 have grown too little, so drop them.
Today a father is 25 years older than his son. 5 years ago the sum of their ages was 45. How old is the son today? (1) 10 (2) 15 (3) 20 (4) 25
- Today's sum = 45 + 2 × 5 = 55.
- Son = (55 − 25) ÷ 2 = 15. The answer is (2).
- Check : son 15, father 40. 5 years ago 10 and 35, sum 45. Correct.
Work out the answer yourself before looking at the options; then match it with the options. If you look at the options first, you get confused.
Taking the sum of "5 years ago" as today's sum and working out the answer straight away. Always read which time is given in the question.
Choosing the ages of the wrong pair for the difference and the sum. Check both conditions (sum and difference).
Memory clue : "Difference, before-after, check."
Odd-even trick : the sum and the difference must be both even or both odd; otherwise drop the option.
Fast method : total years that changed × people.
Spot the difference
| Checking method | When to use it |
|---|---|
| Put the sum back | in options that give two ages |
| Put the difference back | when the condition says "x years older/younger" |
| Odd-even | when you suspect half a year in the options |
| Before/after check | after getting the answer, put it back into the original condition |
Practice questions
- 15 years ago Sunny was 8. How old is he today? (1) 3 (2) 13 (3) 10 (4) 12
- 2Today a teacher is 36 and a student is 12. What will be the difference in their ages after 6 years? (1) 24 (2) 30 (3) 18 (4) 48
- 3The sum of two ages is 18 and the difference is 4. What is the age of the older one? (1) 9 (2) 11 (3) 14 (4) 7
- 4The sum of the ages of a mother and her son is 45 and the mother is 4 times the son's age. How old is the son? (1) 9 (2) 11 (3) 15 (4) 5
- 53 years ago the sum of two ages was 33 and the difference is 7. How old is the older one today? (1) 20 (2) 22 (3) 23 (4) 26
- 6The sum of the ages of four people is 50 today. What will the sum be after 3 years? (1) 62 (2) 60 (3) 53 (4) 64
- 7After 6 years a father will be 3 times his son's age and the difference between them is 30. How old is the son today? (1) 9 (2) 10 (3) 12 (4) 15
- 8The sum of two ages today is 40 and the difference is 12. What are the ages of the older and the younger one, in that order? (1) 27, 13 (2) 26, 14 (3) 24, 16 (4) 28, 12
Show answers (try them yourself first)
- 1Option (2) : 8 + 5 = 13.
- 2Option (1) : the difference 36 − 12 = 24 does not change.
- 3Option (2) : (18 + 4) ÷ 2 = 11.
- 4Option (1) : 45 ÷ (4 + 1) = 9; mother 36.
- 5Option (3) : today's sum 33 + 2 × 3 = 39; older (39 + 7) ÷ 2 = 23, younger 16.
- 6Option (1) : 50 + 4 × 3 = 62.
- 7Option (1) : after 6 years the son is 30 ÷ (3 − 1) = 15; today 15 − 6 = 9 (father today 39; after 6 years 45 = 15 × 3).
- 8Option (2) : 26 + 14 = 40 and 26 − 14 = 12. The other options give a difference of 14, 8 or 16.
Quick revision of the whole lesson
| Concept | Memory clue |
|---|---|
| Age difference | Years go, the difference stays |
| Before / after | Before subtract, after add; people × years |
| Sum + difference | Add - subtract - halve : (sum ± difference) ÷ 2 |
| Times + difference / sum | parts of times − 1 (difference) and times + 1 (sum) |
| Checking the options | Difference, before-after, check; odd-even |
🎯 The most important points for the exam
- The difference in ages never changes; the times changes.
- "Years ago" = subtract, "years later" = add.
- The sum of the ages of many people changes by people × years.
- The two ages are whole numbers only if the sum and the difference are both even or both odd.
- Once you have the answer, put it back into every condition of the question and check.
"First the difference, then before-after; check at the end."