If the current age of a person be X, then

- age after n years = X + n
- age n years ago = X – n
- n times the age = nX
- If ages in the numerical are mentioned in ratio A : B, then A : B will be AX and BX

- Let John’s present age be x

John’s age before 5 years = (x - 5)

John’s age after 10 years = (x + 10)

We are given that, John’s age after 10 years (x + 10) is 5 times his age 5 years back (x – 5)

Therefore,

(x + 10) = 5 (x – 5)

Solving the equation, we get

x + 10 = 5x – 25

4x = 35

x = 8.75 years

- Let age of Rohan be y

Rahul is 15 years elder than Rohan = (y + 15). So Rahul’s age 5 years ago = (y + 15 – 5)

Rohan’s age before 5 years = (y – 5)

5 years ago,

(y + 15 – 5) = 3 (y – 5)

(y + 10) = (3y – 15)

2y = 25

y = 12.5

Rohan’s age = 12.5 years

Rahul’s age = (y + 15) = (12.5 + 15) = 27.5 years.

If sum of ages of x and y is A and ratio of their ages is p : q respectively, then u can determine age of y by using the formula shown below:

Age of y = [latex]\frac{Ratio \ of \ y}{Sum \ of \ ratios} \times sum \ of \ ages[/latex]

Age of y = [latex]\frac{q}{p + q} \times A[/latex]

- We are given that age ratio of Harry : Pitter = 5 : 6

Harry’s age = 5x and Peter’s age = 6x

One year ago, their age was 5x and 6x. Hence at present, Harry’s age = 5x +1 and Peter’s age = 6x +1

After 4 years,

Harry’s age = (5x +1) + 4 = (5x + 5)

Peter’s age = (6x +1) + 4 = (6x + 5)

After 4 years, this ratio becomes 6 : 7. Therefore,

[latex]\frac{Harry’s Age}{6}[/latex] = [latex]\frac{Peter’s Age}{7}[/latex]

[latex]\frac{(5X + 5)}{(6X + 5)}[/latex] = [latex]\frac{6}{7}[/latex]

7 (5x + 5) = 6 (6x + 5)

X = 5

Peter’s present age = (6x + 1) = (6 x 5 + 1) = 31 years

Harry’s present age = (5x + 1) = (5 x 5 + 1) = 26 years

Age of mother 10 years ago was 3 times the age of her son. After 10 years, mother’s age will be twice that of his son. Find the ratio of their present ages.

- We are given that, age of mother 10 years ago was 3 times the age of her son

So, let age of son be x and as mother’s age is 3 times the age of her son, let it be 3x, three years ago.

At present: Mother’s age will be (3x + 10) and son’s age will be (x + 10)

After 10 years: Mother’s age will be (3x + 10) +10 and son’s age will be (x + 10) + 10

Mother’s age is twice that of son

(3x + 10) +10 = 2 [(x + 10) + 10]

(3x + 20) = 2[x + 20]

Solving the equation, we get x = 20

We are asked to find the present ratio.

(3x + 10) : (x + 10) = 70 : 30 = 7 : 3

- Here, we have to calculate: How many years ago the ratio of their ages was 4 : 6

Let us assume x years ago

At present: Sharad is 60 years and Santosh is 80 years

x years ago: Sharad’s age = (60 – x) and Santosh’s age = (80 – x)

Ratio of their ages x years ago was 4 : 6

[latex]\frac{(60 – x)}{(80 – x)}[/latex] = [latex]\frac{4}{6}[/latex]

6(60 – x) = 4(80 – x)

360 – 6x = 320 – 4x

x = 20

Therefore, 20 years ago, the ratio of their ages was 4 : 6

- 1) At present: Ratio of their ages = 5 : 3. Therefore, 5 : 3 will be 5x and 3x.
Rohan’s age 4 years ago = 5x – 4
Rahul’s age after 4 years = 3x + 4

2) Ratio of Rohan’s age 4 years ago and Rahul’s age after 4 years is 1 : 1 Therefore, [latex]\frac{(5x – 4)}{(3x + 4)}[/latex] = [latex]\frac{1}{1}[/latex] Solving, we get x = 4

3) We are asked to find the ratio between Rohan’s age 4 years hence and Rahul’s age 4 years ago. Rohan’s age : (5x + 4) Rahul’s age: (3x – 4) Ratio of Rahul’s age and Rohan’s age [latex]\frac{(5x + 4)}{(3x – 4)}[/latex] = [latex]\frac{24}{8}[/latex] = [latex]\frac{3}{1}[/latex] = 3 : 1

- Let daughter’s age be x and father’s age be 3x.

Father’s age is 3 times more aged than his daughter, therefore father’s present age = x + 3x = 4x

After 5 years, father’s age is 3 times more than his daughter age.

(4x + 5) = 3 (x + 5)

(4x+5)=3 (x+5)

(4x + 5) = 3 (x + 5)

x = 10

After 5 years it was (4x + 5), then after further 5 years, father’s age = (4x +10) and daughter’s age = (x + 10)

[latex]\frac{(4x + 10)}{(x + 10)}[/latex] = ?

Substitute the value of x, we get

[latex]\frac{[(4 × 10) + 10]}{[10 + 10]}[/latex] = [latex]\frac{50}{20}[/latex] = 2.5

After further 5 years, father will be 2.5 times of daughter’s age.

- Let present age of brother be x and sister’s age be 34 – x.

Past Age (5 Yrs Ago) | Present Age | Future Age (After 6 Yrs) | |
---|---|---|---|

Brother | (x – 5) | x | (x + 6) = ? |

Sister | (34 – x) - 5 | (30 – x) |

We are given, 5 years ago sister’s age was 5 times the age of her brother.

Therefore,

(34 – x) – 5 = 5 (x – 5)

34 – x – 5 = 5x – 25

5x + x = 34 – 5 +25

6x = 54

x = 9

Future age (after 6 yrs) = (x + 6) = (9 + 6) = 15 years

- If the present age is [latex]x[/latex], then n times the age is n[latex]x[/latex]
- If the present age is [latex]x[/latex], then age of n years later = [latex]x[/latex] + n
- If the present age is [latex]x[/latex], then age of n years ago = [latex]x[/latex] - n
- If the ages are in the ratios of a : b, then a[latex]x[/latex] and b[latex]x[/latex]
- If the current age is [latex]x[/latex], then [latex]\frac{1}{n}[/latex] of the age is [latex]\frac{x}{n}[/latex]

- Let the Harry's present age be [latex]x[/latex]

After 25 years is [latex]x[/latex] + 25

3 years back is [latex]x[/latex] - 3

Therefore the equation is [latex]x[/latex] + 25 = 5([latex]x[/latex] - 3)

⇒[latex]x[/latex] + 25 = 5[latex]x[/latex] - 15

⇒25 + 15 = 5[latex]x[/latex] - [latex]x[/latex]

⇒40 = 4[latex]x[/latex]

⇒[latex]x[/latex] = [latex]\frac{40}{4}[/latex]

⇒[latex]x[/latex] = 10 years

Therefore, present age of Harry is 10 years

- Let Charlie's age be [latex]x[/latex]

Then James's age = [latex]\frac{120}{x} \quad[/latex](since [latex]x[/latex] +[latex]y[/latex] = 120, in order to eliminate [latex]y[/latex] variable replace [latex]y[/latex] = [latex]\frac{120}{x}[/latex])

Given that 2([latex]\frac{120}{x}[/latex]) - [latex]x[/latex] = 8

⇒[latex]\frac{120 - (x)^2}{x}[/latex] = 8

⇒240 - [latex](x)^2[/latex] = 8[latex]x[/latex]

⇒[latex](x)^2[/latex] + 8[latex]x[/latex] - 240 = 0

Now factorize the equation,

⇒[latex]x^2[/latex] + 20[latex]x[/latex] - 12[latex]x[/latex] - 240

⇒[latex]x(x + 20)[/latex] - 12[latex](x + 20)[/latex]

⇒[latex](x + 20)[/latex][latex](x - 12)[/latex]

⇒[latex]x = -20[/latex] and [latex]x = 12[/latex]

Hence [latex]x[/latex] = 12 years

Therefore, James's age is [latex]\frac{120}{12}[/latex] = 10 years.

- Given data

Henry's age is 30 years

George's age is 25 years

[latex]\frac{(30 - x)}{(25 -x)}[/latex] = [latex]\frac{3}{5}[/latex]

⇒[latex]5(30 - x)[/latex] = [latex]3(25 - x)[/latex]

⇒[latex]150 - 5x[/latex] = [latex]75 - 3x[/latex]

⇒[latex]150 - 75[/latex] = [latex]5x - 3x[/latex]

⇒[latex]75[/latex] = [latex]2x[/latex]

⇒[latex]x[/latex] = [latex]\frac{75}{2}[/latex]

⇒[latex]x[/latex] = [latex]37.5[/latex]

Therefore, 37.5 years ago was their ages 3 : 5.

- Let the present age of son be [latex]x[/latex]

Present age of father be [latex]y[/latex]

Given that [latex]x[/latex] = ([latex]\frac{2}{5}[/latex])[latex]y[/latex]

⇔[latex]y[/latex] = ([latex]\frac{5}{2}[/latex])[latex]x[/latex]

8 years later i.e. [latex]x + 8[/latex] = [latex]\frac{1}{2}[/latex] x [latex](y + 8)[/latex]

Substitute the value of [latex]y[/latex], and simplify

⇒[latex]x + 8[/latex] = [latex]\frac{1}{2}[/latex]([latex](\frac{5}{2})x + 8[/latex])

⇒[latex]2x + 16[/latex] = [latex]\frac{5x + 16}{2}[/latex]

⇒[latex]4x + 32[/latex] = [latex]5x + 16[/latex]

⇒[latex]5x - 4x[/latex] = [latex]32 - 16[/latex]

⇒[latex]x[/latex] = [latex]16[/latex]

Son's age is 16 years

Therefore, father's age is [latex]\frac{(5 \ * \ 16)}{2}[/latex] = 40 years

- Let Alice's age be [latex]x[/latex]

Amelie's age be [latex]y[/latex]

Given that [latex](x -1)[/latex] = [latex]4(y -1)[/latex]

⇒[latex](x -1)[/latex] = [latex](4y -4)[/latex]

⇒[latex](x -4y)[/latex] = [latex]-4 + 1[/latex]

⇒[latex](x -4y)[/latex] = [latex]-3[/latex] -------(i)

[latex]x + 6[/latex] = [latex](y + 6) + 9[/latex]

⇒[latex]x + 6[/latex] = [latex]y + 15[/latex]

⇒[latex]x - y[/latex] = [latex]15 - 6[/latex]

⇒[latex]x - y[/latex] = [latex]9[/latex] --------(ii)

By solving equations (i) and (ii)

[latex]y[/latex] = 4

[latex]x[/latex] = 13

Therefore, the ratio of ages of Alice and Amelie is x : y = 13 years : 4 years

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