Jack and Jill have certain number of apples with them. The total number of apples with both of them is less than 80. If

[M7MBA]

Jack and Jill have a certain number of apples with them. The total number of apples with both of them is less than 80. If Jill gives a certain number of apples to Jack, then Jack will have four times the number of apples Jill has. If Jack gives the same number of apples to Jill, then Jack will have thrice the number of apples that Jill has. What can be the number of apples that Jack initially has?

A. 31
B. 36
C. 45
D. 62
E. 63

Answer: A

Source: e-GMAT

[Scott@TargetTestPrep]

Jack and Jill have a certain number of apples with them. The total number of apples with both of them is less than 80. If Jill gives a certain number of apples to Jack, then Jack will have four times the number of apples Jill has. If Jack gives the same number of apples to Jill, then Jack will have thrice the number of apples that Jill has. What can be the number of apples that Jack initially has?

A. 31
B. 36
C. 45
D. 62
E. 63

Answer: A

Solution:

Let K and L be the number of apples Jack and Jill have, respectively, and let a be the number of apples they give to each other. We can create the following equations and inequalities:

K + L < 80

4(L - a) = K + a

4L - 4a = K + a (Eq. 1)

3(L + a) = K - a

3L + 3a = K - a (Eq. 2)

If we subtract Eq. 2 from Eq. 1, we have:

L - 7a = 2a

L = 9a

If we add the two equations, we have:

7L - a = 2K

Substituting 9a for L, we have:

7(9a) - a = 2K

62a = 2K

31a = K

Substituting 31a for K and 9a for L in the inequality, we have:

31a + 9a < 80

40a < 80

a < 2

Since a must be a positive integer, a = 1, and therefore, K = 31a = 31 x 1 = 31.

Answer: A

[swerve]

Jack and Jill have a certain number of apples with them. The total number of apples with both of them is less than 80. If Jill gives a certain number of apples to Jack, then Jack will have four times the number of apples Jill has. If Jack gives the same number of apples to Jill, then Jack will have thrice the number of apples that Jill has. What can be the number of apples that Jack initially has?

A. 31
B. 36
C. 45
D. 62
E. 63

Answer: A

Source: e-GMAT

Let \(K\) be the apples with Jack. Total apples \(80-x\). Hence apples with Jill \((80-x-K)\)

Jill gives A apples to Jack. The new arrangement
Jack \(= K+A\)
Jill \(= (80-x-K)-A\)
Given that, \((K+A) = 4(80-x-K)- A \qquad (1)\)

Similarly, the other arrangement
Jack \(= K-A\)
Jill \(= (80-x-K)+A\)
Given that, \((K-A) = 3(80-x-K)+ A \qquad (2)\)

Solving the \(2\) equations, we get
\(A = \dfrac{k}{31}\)
Plugging values, we get that \(K = 31\) (gives \(A =1\)) satisfies the equations \((1) \& (2)\)

Therefore, A