The Smith family has 13 children, all born exactly 2 years apart. How old will the second-youngest child be when the oldest child is five ti

The Smith family has 13 children, all born exactly 2 years apart. How old will the second-youngest child be when the oldest child is five times the age of the youngest child?

$$\small{\text{$
\begin{array}{rcl}
$arithmetical sequence $
~
\underbrace{c_1}_{child ~1} ~~
\underbrace{(c_1+1\cdot 2)}_{child ~2} ~~
\underbrace{(c_1+2\cdot 2)}_{child ~3} ~~
\underbrace{(c_1+3\cdot 2)}_{child ~4} ~~
\cdots\\\\
c_n=c_1+(n-1)\cdot2\\\\
c_{13}=c_1+12\cdot2\\
c_{13}=c_1+24\\
\end{array}
$}}\\\\
\small{\text{$
\begin{array}{rcl}
c_{13}=c_1+24 = 5\cdot c_1\\
c_1+24 = 5\cdot c_1\\
4\cdot c_1 = 24 \\
\mathbf{c_1 = 6} \\\\
c_2 = c_1+2\\
c_2 = 6+2\\
\mathbf{c_2 = 8}
\end{array}
$}}$$

The second-youngest child is 8 years old

The oldest child is 24 years older than the youngest

Let y be the time elapsed from the birth of the youngest to the time when the oldest is 5 times that age......so we have....

5y    = 24 + y     subtract y from each side

4y = 24     then y = 6

So, six years from the birth of the youngest, the oldest will be 30 and the youngest will be 6....thus.....the oldest is five times as old as the youngest at that time......

Then, the second to the youngest will be 8  at that time....two years older than the youngest....!!!

The Smith family has 13 children, all born exactly 2 years apart. How old will the second-youngest child be when the oldest child is five times the age of the youngest child?

When the youngest child turns 6, the oldest will be 30, which makes him/her 5X older than the youngest one.

The second-youngest child will be 8 years old.