Question

Mrs. Smith is currently 27 years older than her daughter Kate. In 8 years Mrs. Smith will be twice as old as Kate. How old are Mrs. Smith and Kate and how old will they be in 8 years?

Answer 1

Kate is 19 years old and Mrs. Smith is 46 years old now.

Explanation:

Suppose the age of Kate at present = `x`
Then age of Mrs. Smith now = `x+27`

Now after 8 years the age of Mrs. Smith will be twice as that of Kate.

After 8 years, the age of Kate = `x+8`
and, age of Mrs. Smith = `x+27+8`
As Mrs. Smiths's age will be twice as that of Kate after 8 years so, twice the age Kate will make it equal to Mrs. Smiths's age at that time.

Writing that in equation form,
2(`x+8`) = `x+27+8`
or, `2x + 16 = x + 35`

Bringing the x terms together and the number terms together

`2x-x = 35-16`
or, `x = 19`
Therefore, Mrs. Smiths's age now = `x+27` = `19+27` = 46 years

You can check your answer by adding 8 years to both ages and see that Mrs. Smith's age is twice that of Kate after 8 years.

Answer 2

See a solution process below"

Explanation:

First, let's call Mrs. Smith's age: `s` and we can call Kate's age: `k`

Now we know from above Mrs. Smith is 27 years older than Kate so we can write:

`s = k + 27`

We also know in 8 years Mrs. Smith will be twice as old as Kate so we can write:

`s + 8 = 2(k + 8)`

We can now substitute `(k + 27)` from the first equation for `s` in the second equation and solve for `k`:

`s + 8 = 2(k + 8)` becomes:

`(k + 27) + 8 = 2(k + 8)`

`k + 27 + 8 = (2 xx k) + (2 xx 8)`

`k + 35 = 2k + 16`

`-color(red)(k) + k + 35 - color(blue)(16) = -color(red)(k) + 2k + 16 - color(blue)(16)`

`0 + 19 = -color(red)(1k) + 2k + 0`

`19 = (-color(red)(1) + 2)k`

`19 = 1k`

`19 = k`

`k = 19`

We can now substitute `19` for `k` in the first equation and calculate `s`:

`s = k + 27` becomes:

`s = 19 + 27`

`s = 46`

Mrs. Smith is 46 and Kate is 19

In 8 years Mrs Smith will be 54 and Kate will be 27