Question

Lionel and Ronaldo had a discussion on the ages of Jose’s sons. Ronaldo made following statements about Jose’s sons:
i. Jose has three sons.
ii. The sum of the ages of Jose’s sons is 13.
iii. The product of the ages of the sons is the same as the age of Lionel.
iv. Jose’s eldest son, Zizou weighs 32 kilos.
v. The sum of the ages of the younger sons of Jose is 4.
vi. Jose has fathered a twin.
vii. Jose is not the father of a triplet.
viii. The LCM of the ages of Jose’s sons is more than the sum of their ages.
Which of the following combination gives information sufficient to determine the ages of Jose’s sons?

• A. i, ii, iii and iv
• B. i, ii, iv and vi
• C. i, ii, iii and v
• D. i, ii, v and vii
• E. i, ii, v and vi

Hint:

When given the sum of two of three numbers, compute the third immediately.
“Has fathered a twin” implies exactly two children share the same age among the siblings.
Test sufficiency by checking whether multiple solutions remain possible without a particular statement.

Answer 1

Answer: (E)

Step 1: Let the sons’ ages in nondecreasing order be \(a\le b\le c\). From (i) and (ii), there are three sons with \(a+b+c=13\).
Step 2: From (v), the sum of the two younger sons is \(a+b=4\). Hence the eldest son’s age is \[ c=13-(a+b)=13-4=9. \] Step 3: From (vi) “Jose has fathered a twin” \(\Rightarrow\) two of the sons are of equal age. As the eldest is \(9\), the twins must be the two younger sons, so \(a=b\). With \(a+b=4\), we get \(a=b=2\).
Therefore the ages are \((2,2,9)\), uniquely determined using (i), (ii), (v), (vi).
Why others are insufficient:
(A) and (C) without (vi) allow \((1,3,9)\) or \((2,2,9)\) etc.
(B) lacks (v), so twins with sum \(13\) could be \((4,4,5)\), \((3,3,7)\), etc.
(D) “not a triplet” still allows multiple possibilities like \((1,3,9)\) and \((2,2,9)\). Hence only (E) suffices.