Question

today Rebecca who is 34 years old and her daughter who is 8 years old celebrate their birthdays how many years will pass before Rebecca's age is twice a daughter's age

Answer 1

We will have the following:

We are trying to find when Rebeca's age is twice her daughters, so the ratio between ages is 2, and a value "x" will represent the change in time, so we will have to solve for x when:

`(34+x)/(8+x)=2\Rightarrow34+x=16+2x\Rightarrow x=18`

Now, we corroborate:

`(34+18)/(8+18)=(52)/(26)=2`

So, there will have to pass 18 years until Rebeca's age is twice her Daugter's.

***Explanation***

We are told that Rebecca's age is 34. [We will name Rebeca's age "R"]

We are told that the Daughter's age is 8. [We will name the Daughter's age "D"]

Now, we are asked to find after how much time Rebecca's age will be twice her Daughter's, in other words:

`R=2D`

Now, we know that if time passes, it will be the same for both. In other words, if we add time which we will call "x" we will add it to both Rebecca and her Daughter, that is:

`\begin{cases}R+x \\ \\ D+x\end{cases}`

So, we replace in the first expression, that is:

`R=2D\Rightarrow(R+x)=2(D+x)`

Now, since we already know their original age R = 34 and D = 8, we replace in those values and solve for x, that is:

`(R+x)=2(D+x)\Rightarrow(34+x)=2(8+x)`
`\Rightarrow34+x=16+2x\Rightarrow34-16=2x-x`
`\Rightarrow18=x`

And thus, the time that passes for Rebecca to have twice her Daughter's age is 18 years. Now, we corroborate this by simply seing if when we divide Rebecca's age by her Daughter's age after 18 years, the ratio will be 2, meaning it will be twice her Daughter's age, that is:

`(R+x)/(D+x)=(34+18)/(8+18)=(52)/(26)=2`

Now, this means Rebecca Will be 52 years old while her Daughter will be 26 years old when this happens.