## How Old Are They?

Mary’s and Bob’s ages combined are twice Jane’s age. Mary is $8$ years older than Bob. Jane’s age plus Bob’s age is $20$ years. How old are they?
Source: NCTM Mathematics Teacher, September 2006

Solution
Let $b,j$, and $m$ represent the ages of Bob, Jane, and Mary respectively.
$b+m=2j\qquad (1)$
$m-b=8\qquad\; (2)$
$b+j=20\qquad\; (3)$
Add Eq. $(1)$ and Eq. $(2)$
$b+m=2j$
$m-b=8$
——————
$2m=2j+8$
$m=j+4\qquad (4)$
Add Eq. $(2)$ and Eq. $(3)$
$m-b=8$
$b+j=20$
——————
$m+j=28\quad\;\; (5)$
Substitute the value of $m$ in Eq. $(4)$  into Eq. $(5)$
$(j+4)+j=28$
$2j=24$
$j=12$
Jane’s age equals $12$.
Substitute the value of Jane’s age into Eq. $(4)$
$m=12+4=16$
Mary’s age equals $16$.
Substitute the value of Jane’s age into Eq. $(3)$
$b+12=20$
$b=8$
Bob’s age equals $8$.

Answer: Bob is $8$, Jane $12$, Mary $16$