Let G and H be positive integers. 2G - 10 -H and 3H - 20

Question

Let G and H be positive integers.  2G - 10 > -H  and  3H - 20 < -G . If L is the minimum value of G and M is the maximum value of H, then which of the following pair is (L,M) ?

A) (2,6)
B) (6,2)
C) (5,5)
D) (3,5)
E) (11,1)

chetan2u · Answer

2G-10>-H....(I) 3H-20<-G ......-G>3H-20 or -2G>6H-40....(II) Add I and II 2G-10-2G>-H+6H-40.........-10>5H-40.......5H<30....H<6 So max value of H is 5 Now 2G-10>-H.......6G-30>-3H(I) 3H-20<-G ......-..(II) Add I and II 3H-20-3H<6G-30-G......-20<5G-30........5G>10.......G>2 So minimum value of G is 3 So (L,M) is (3,5) D

garcmillan · Answer

An alternative solution to the one provided by chetan2u is trying out the given points in the two given inequations.

Point (L,M)

A) (2,6)

(i) 2*2-10>-6; -6>-6 (This is not possible, Move on to the next given solution)

B) (6,2)

(i) 2*6-10>-2; -2>--2 (This is not possible, Move on to the next given solution)

C) (5,5)

(i) 2*5-10>-5; 0>-5 (This is OK. Try next inequation)
(ii) 3*5-20<-5; -5<-5 (This is not possible, Move on to the next given solution)

D) (3,5)

(i) 2*3-10>-5; -4>-5 (This is OK. Try next inequation)
(ii) 3*5-20<-3; -5<-3 (This is OK. This is the solution)

OPTION D

Kinshook · Answer

Given: Let G and H be positive integers. 2G - 10 > -H and 3H - 20 < -G . Asked: If L is the minimum value of G and M is the maximum value of H, then which of the following pair is (L,M) ? 2G > 10 - H. G > 5 - H/2. (1) 3H < 20 -G -3H > G -20 (2) Adding (1) & (2) G - 3H > G - H/2 - 15 3H - H/2 < 15 5H/2 < 15 H < 15 * 2/5 = 6 H< 6 (3) M = 5 since M is maximum value of H and is a positive integer -H > -6 10 - H > 10 -6 = 4 2G> 10 - H > 4 G > 2 (4) L = 3 since L is minimum value of G and is a positive integer (L,M) = (3, 5) IMO D

Kinshook · Answer

Given: Let G and H be positive integers. 2G - 10 > -H and 3H - 20 < -G . Asked: If L is the minimum value of G and M is the maximum value of H, then which of the following pair is (L,M) ? 2G + H > 10 (1) G + 3H < 20 (2) 2G + 6H < 40 -2G - 6H > -40 (3) Adding (1) & (3) -5H > - 30 H<6 (4) M = 5 Maximum positive integer value of H Putting H<6 in (1) -H>-6 2G > 10 - H > 10 -6 =4 G>2 (5) L=3 Minimum positive integer value of G (L, M) = (3, 5) IMO D

bumpbot · Answer

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Let G and H be positive integers. 2G - 10 > -H and 3H - 20 < -G. If L

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