2 friends meet after a long time. One friend says the product of his 2

Question

2 friends meet after a long time. One friend says the product of his 2 children's age is 96. If the sum of their ages is 2 more than twice the difference, which of the following could be true.

I. One child is 4 years older than the other
II. The difference between their ages is 10
III. Sum of their ages is 22

A. Only I
B. Only I and II
C. Only III
D. Only II and III
E. I,II, III

ankushsambare · Answer

Let the children’s ages be x and y.
Given:
 
 Product of ages = 96
xy = 96 
 Sum is 2 more than twice the difference.  
Assume x > y.
Then:
x + y = 2(x − y) + 2
Simplify:
x + y = 2x − 2y + 2
x − 3y + 2 = 0
x = 3y − 2
Substitute into xy = 96:
(3y − 2)y = 96
3y2 − 2y − 96 = 0
Factor:
(3y + 16)(y − 6) = 0
So:
y = 6
x = 16
Children’s ages are 16 and 6.
Check statements:
I. One child is 4 years older than the other
16 − 6 = 10 → False
II. Difference between ages is 10
True
III. Sum of ages is 22
16 + 6 = 22 → True
 Correct Answer: D) Only II and III

Kinshook · Answer

2 friends meet after a long time. One friend says the product of his 2 children's age is 96. If the sum of their ages is 2 more than twice the difference, which of the following could be true.

Let the ages of children be x & y. Let us also assume that x>y.
x*y = 96
x+y = 2 + 2(x-y)
x+y = 2 + 2x - 2y
x=3y-2

y(3y-2) = 96
3y^2-2y-96=0
y=6
x=16

I. One child is 4 years older than the other; x-y=16-6=10; False
II. The difference between their ages is 10; x-y=16-6=10; True
III. Sum of their ages is 22; x+y=16+6=22; True

A. Only I
B. Only I and II
C. Only III
D. Only II and III
E. I,II, III

IMO D

Nav_124 · Answer

Let the children’s ages be x and y.

We are given:

Product of ages =96
xy=96
“The sum of their ages is 2 more than twice the difference.”

Assume older child is x, younger is y.

Then:

x+y=2(x−y)+2

Simplify:

x+y=2x−2y+2
x−3y+2=0
x=3y−2

Now substitute into the product condition:

(3y−2)y=96
3y^2−2y−96=0

Factor:

3y^2−18y+16y−96=0
3y(y−6)+16(y−6)=0
(3y+16)(y−6)=0

So,

y=6

Then:

x=3(6)−2=16

So the children are 16 and 6 years old.

Now check the statements:

I. One child is 4 years older than the other

Difference: 16−6=10
False.

II. The difference between their ages is 10

True.

III. Sum of their ages is 22
16+6=22
True.

Therefore, the correct answer is:

D. Only II and III

KarishmaB · Answer

Look for some pairs of factors of 96:
8 and 12 (closest factors. Since 96 is close to 100, we start with 10, 10 and we arrive at 8, 12)
6 and 16
...
1 and 96

I. One child is 4 years older than the other

8 and 12 is possible but it doesn't satisfy the given criteria - Sum is 20; difference is 4; Twice the difference is 8. Sum is not 2 more than 8. 
In no other case can the diff between the ages be 4. It will be more than 4 as the factors become extreme.

II. The difference between their ages is 10

With 6 and 16, difference is 10. Possible. 
  
III. Sum of their ages is 22
  
With 6 and 16, sum is 22. Possible.

Answer (D)

2 friends meet after a long time. One friend says the product of his 2

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2 friends meet after a long time. One friend says the product of his 2

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