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I need help to set up this proportion

In Linguistics \$101,\$ the ratio of the number of juniors to the number of seniors is \$3:2\$.  When six more juniors join the class, and one senior drops the class, the ratio of the number of juniors to the number of seniors becomes \$18:7\$.  How many students are in the class after these changes?

Aug 4, 2023

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Call the number of original Juniors = J   Call the number of original Seniors = S

And we know that

J / S  = 3 / 2    so that   J = (3/2)S     (1)

When 6 more Juniors join and one Senior leaves, we  have

[ J + 6 ] / [ S -1 ]  =  18 / 7   (2)

Sub (1) into (2)

[(3/2)S + 6]  / [ S-1] =  18 / 7    cross-multiply

7 [(3/2)S + 6] =  18 [ S - 1]       simplify

(21/2)S + 42 = 18S - 18

42 + 18 =  18S - (21/2)S

60 = (36/2)S - (21/2)S

60 = (15/2)S   multiply through by 2

120 = 15S

S = 120 / 15  =  8 =  number of seniors in original class

J = (3/2)S  (3/2) (8)  =  12 =  number of Juniors in original class

After 6 Juniors join and 1 Senior leaves...we have the ratio  (12 + 6)  / (8 -1)  =  18 / 7

So....there are 25 total students in the  class after the changes

Aug 4, 2023