# Question #b3354

Jul 5, 2017

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

#### Explanation:

Their situation after 7 years:

$\left(x + 7\right) + \left(4 + x + 7\right) = 24$

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:

$\left(x + 3\right) + \left(x + 4 + 3\right) = 16$

Jul 5, 2017

Solution to a$\to$ formula for age after 3 years $2 x + 10 = {T}_{3}$

We are not asked to find the value of ${T}_{3}$

Solution to b$\to$Jane's age now $= 3$

General case for n years: $2 x + 2 n + 4 = {T}_{n}$

#### Explanation:

$\textcolor{red}{\text{Note that TODAY Jane is x years old}}$

$\textcolor{b l u e}{\text{Initial condition}}$

Jane =$x$
Kenny $= x + 4$

$\textcolor{b l u e}{\text{Ages after seven years:}}$

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

Jane $= x + 7$
Kenny $= \left(x + 4\right) + 7 = x + 11$

$\left[x + 7\right] + \left[x + 11\right] = 24 \text{ "->" } 2 x + 18 = 24$
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$\textcolor{b l u e}{\text{Answering part b first as it determines } x}$

Jane's age today$\to x$

So we need to determine $x$

Using: $2 x + 18 = 24$

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

$2 x = 24 - 18$

$2 x = 6$

Divide both sides by 2

$x = 3 \leftarrow \text{ age at initial condition which is now}$

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$\textcolor{b l u e}{\text{Answer part a}}$

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: $\left[x\right] + \left[x + 4\right] = {T}_{0}$

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

After 3 years $\textcolor{red}{\underline{\text{each}}}$ of them would have gained 3 years, which when added give a total increase of 6 as the sum of their years.

$\left[x \textcolor{red}{+ 3}\right] + \left[x + 4 \textcolor{red}{+ 3}\right] = {T}_{0} \textcolor{red}{+ 3 + 3} = {T}_{3}$

$2 x + 10 = {T}_{3} \leftarrow \text{ We are not asked to find the value of } {T}_{3}$
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$\textcolor{b l u e}{\text{General case "->" Not asked for in the question}}$

$\left[x + n\right] + \left[x + 4 + n\right] = {T}_{n}$

$x + x + 4 + n + n = {T}_{n}$

$2 x + 4 + 2 n = {T}_{n}$

$2 x + 2 n + 4 = {T}_{n}$