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?

2 Answers
Dec 19, 2017

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.

Dec 19, 2017

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