Question

Question #b3354

Answer 1

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`

Answer 2

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`
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`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"`

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`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`
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`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`