Question #b3354

2 Answers
Jul 5, 2017

Answer:

Today Jane is 3 years old and Kenny is 7 years old

Explanation:

Their situation after 7 years:

#(x+7) + (4+x+7) = 24#

The first term is for Jane and the second term is for Kenny.

Therefore when you solve the above written equation, you will get Jane is 3 years old (x=3). Kenny is 7 years old (right now).

3 years later (starting now)

Jane will be 6 and Kenny will be 10 years old. The equation for this situation is:

#(x+3) + (x+4+3) = 16#

Jul 5, 2017

Answer:

Solution to a#-># formula for age after 3 years #2x+10=T_3#

We are not asked to find the value of #T_3#

Solution to b#->#Jane's age now #=3#

General case for n years: #2x+2n+4=T_n#

Explanation:

#color(red)("Note that TODAY Jane is x years old")#

#color(blue)("Initial condition")#

Jane =#x#
Kenny #=x+4#

#color(blue)("Ages after seven years:")#

It is given that the combined age after 7 years is 24.

Jane #=x+7#
Kenny #=(x+4)+7=x+11#

#[x+7]+[x+11]=24" "->" "2x+18=24#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Answering part b first as it determines " x)#

Jane's age today#->x#

So we need to determine #x#

Using: #2x+18=24#

Subtract 18 from both sides (get rid of it from the left)

#2x=24-18#

#2x=6#

Divide both sides by 2

#x=3 larr" age at initial condition which is now"#

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Answer part a")#

Only need the formula !!!!

Let the combined age at initial condition be #T_0# (T for total)
Let the combined age at the time interval #n# be #T_n#
So the combined age at year 3 will be #T_3#

Initial condition: # [x]+[x+4] = T_0#

Just added the #T# for completeness. We do not really need to know its value.

After 3 years #color(red)(ul("each"))# of them would have gained 3 years, which when added give a total increase of 6 as the sum of their years.

#[xcolor(red)(+3)]+[x+4color(red)(+3)]=T_0color(red)(+3+3)=T_3#

#2x+10=T_3larr" We are not asked to find the value of "T_3#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("General case "->" Not asked for in the question")#

#[x+n]+[x+4+n]=T_n#

#x+x+4+n+n=T_n#

#2x+4+2n=T_n#

#2x+2n+4=T_n#