The combined age of X, Y, and Z is $190$ years. The difference between the ages of X and Y is twice the difference between the ages of Y and Z. If the ratio of Y's age to Z's age is $7 : 9$, and it is given that $X

Question

The combined age of X, Y, and Z is $190$ years. The difference between the ages of X and Y is twice the difference between the ages of Y and Z. If the ratio of Y's age to Z's age is $7 : 9$, and it is given that $X < Y$, with all ages being integers, what is the minimum possible age of X (in years)?

Accepted Answer

Let the ages of Y and Z be 7x and 9x
 respectively (ratio 7:9).

Sum of ages: X+Y+Z=190 ⇒
 X+7x+9x=190 ⇒
 X+16x=190
 ⇒ X=190-16x.
 Given Y−X=2(Z−Y): 7x−X=2(9x−7x) 
⇒ 7x−X=22x=4x
 ⇒ 7x−4x=X 
⇒ X=3x
. Equate: 3x=190−16x 
⇒ 19x=190 ⇒ x=10.

Compute X: X=3x=310=30. Check: Y=710=70, Z=910=90, so 30<70<90 holds. 
Answer: X=30 years.

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