Concept:Set up equations for combined ages of parents and children at different times, considering the number of children.
Explanation:Let
X = combined age of man and wife,
Y = combined age of all children, and
N = number of children.
From condition 1:
X=6Y …(1)
Two years ago: parents' combined age =
X−4, children's combined age =
Y−2N.
Condition 2:
X−4=10(Y−2N)Simplify:
X−4=10Y−20N ⇒
X−10Y+20N=4 …(2)
Six years later: parents' combined age =
X+12, children's combined age =
Y+6N.
Condition 3:
X+12=3(Y+6N)Simplify:
X+12=3Y+18N ⇒
X−3Y−18N=−12 …(3)
Substitute
X=6Y from (1) into (2) and (3):
From (2):
6Y−10Y+20N=4 ⇒
−4Y+20N=4 ⇒
5N−Y=1 …(4)
From (3):
6Y−3Y−18N=−12 ⇒
3Y−18N=−12 ⇒
Y−6N=−4 …(5)
Add equations (4) and (5):
(5N−Y)+(Y−6N)=1+(−4) ⇒
−N=−3 ⇒
N=3.
Answer:The number of children is
3.