Six years ago, Rosa was four times as old as her daughter. Ten years from now, she will be only twice as old as her daughter. How old are they now?
Answers
Answer:
- Rosa is 38 years old.
- Her daughter is 14 years old.
Step-by-step explanation:
Given :
- Six years ago, Rosa was four times as old as her daughter.
- Ten years from now, she will be only twice as old as her daughter.
To find :
- The present ages of Rosa and her daughter
Solution :
Let the present age of Rosa be x years
and present age of her daughter be y years
Six years ago,
Rosa's age = (x - 6) years
Her daughter's age = (y - 6) years
As given,
Rosa's age = 4(her daughter's age)
x - 6 = 4(y - 6)
x - 6 = 4y - 24
x - 4y = -24 + 6
x - 4y = -18
x = 4y - 18 ⇢ [eqn. 1]
After 10 years,
Rosa's age = (x + 10) years
Her daughter's age = (y + 10) years
As given,
Rosa's age = 2(her daughter's age)
x + 10 = 2(y + 10)
x + 10 = 2y + 20
x = 2y + 20 - 10
x = 2y + 10
Substitute x = 4y - 18 ( eqn. [1] )
4y - 18 = 2y + 10
4y - 2y = 10 + 18
2y = 28
y = 28/2
y = 14
Substitute y = 14 in eqn. [1]
x = 4y - 18
x = 4(14) - 18
x = 56 - 18
x = 38
Therefore, the present age of Rosa = 38 years
and the present age of her daughter = 14 years
Verification :
Six years ago,
Rosa's age = 38 - 6 = 32 years
Her daughter's age = 14 - 6 = 8 years
Condition : Rosa was four times as old as her daughter
32 = 4 × 8
32 = 32
LHS = RHS
After 10 years,
Rosa's age = 38 + 10 = 48 years
Her daughter's age = 14 + 10 = 24 years
Condition : Rosa will be twice as old as her daughter
48 = 2 × 24
48 = 48
LHS = RHS
Hence verified!