Naomi, macy, and sebastian have 234 stamps in all. naomi gives 16 stamps to macy and 24 stamps to sebastian. naomi then has three times as many stamps as macy, and macy has twice as many stamps as sebastian. how many stamps does naomi have at first?

Let the original number of stamps be
x for Naomi,
y for Macy,
z for  Sebastian.

Because the total number of stamps is 234, therefore
x + y + z = 234                (1)

Mary gives 16 stamps to Macy and 24 staps to Sebastian.
The number of stamps owned becomes
x - 40 for Mary,
y + 16  for Macy,
z + 24 for Sebastian.

Mary has 3 times as many stamps as Macy, therefore
x - 40 = 3(y + 16)
x - 3y = 88                  (2)

Macy has twice as many staps as Sebastian, therefore
y + 16 = 2(z + 24)
y - 2z = 32                     (3)

We have the system of equations
x + y + z = 234         (4)
x - 3y      = 88           (5)
      y - 2z = 32          (6)

Subtract (5) from (4).
4y + z = 146              (7)
Substitute (6) into (7).
4(2z + 32) + z = 146
9z =  146 - 128 = 18
z = 2
y = 2z + 32 = 4 + 32 = 36
x = 234 - y - z = 234 - 38 = 196

Naomi originally had 196 stamps.

196 Let's write some equations to express what we know. N = number of stamps Naomi has at the end. M = number of stamps Macy has at the end. S = number of stamps Sebastian has at the end. N+M+S = 234 ; They all have 234 stamps in total N = 3M ; Naomi has 3 times as many stamps as Macy M = 2S ; Macy has 2 times as many stamps as Sebastian Now let's modify the N+M+S = 234 expression. First, copy the expression. N+M+S = 234 Let's replace N with 3M and simplify 3M+M+S = 234 4M+S = 234 Let's replace M with 2S and simplify 4(2S)+S = 234 8S+S = 234 9S = 234 Now solve for S 9S = 234 S = 26 So we now know that Sebastian has 26 stamps at the end. Let's figure out Naomi's and Macy's stamp counts. M = 2S M = 2 * 26 M = 52 So Macy has 52, now for Naomi N = 3M N = 3 * 52 N = 156 And Naomi has 156 at the end. But we want the number of stamps she started with. Since she gave away 16 stamps to Macy and 24 to Sebastian, add those counts to what she has at the end to get her original count. 156 + 16 + 24 = 196 So Naomi started with 196 stamps.