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Algebra Difficulty 5.9 AIME, harder Find the answer

Four brothers have together forty-eight Kwanzas. If the first brother's money were increased by three Kwanzas, if the second brother's money were decreased by three Kwanzas, if the third brother's money were triplicated and if the last brother's money were reduced by a third, then all brothers would have the same quantity of money. How much money does each brother have?

A number or a short expression. Spacing and $ signs are ignored.

Solution

Let x1,x2,x3, x_1, x_2, x_3, and x4 x_4 be the amounts of money that the first, second, third, and fourth brothers have, respectively. According to the problem, we have the following equation describing their total amount of money:

x1+x2+x3+x4=48 x_1 + x_2 + x_3 + x_4 = 48

We are also given conditions on how these amounts are adjusted:

1. The first brother's money is increased by 3 Kwanzas: x1+3 x_1 + 3 .
2. The second brother's money is decreased by 3 Kwanzas: x23 x_2 - 3 .
3. The third brother's money is tripled: 3x3 3x_3 .
4. The fourth brother's money is reduced by a third: x43 \frac{x_4}{3} .

These adjustments result in all brothers having the same amount of money. Let's denote this common amount by c c . Therefore, we have the following equations:

x1+3=c x_1 + 3 = c
x23=c x_2 - 3 = c
3x3=c 3x_3 = c
x43=c \frac{x_4}{3} = c

Solving these equations for c c in terms of each brother's amount, we get:

x1=c3 x_1 = c - 3
x2=c+3 x_2 = c + 3
x3=c3 x_3 = \frac{c}{3}
x4=3c x_4 = 3c

Substituting these into the total money equation:

(c3)+(c+3)+c3+3c=48 (c - 3) + (c + 3) + \frac{c}{3} + 3c = 48

Simplifying:

2c+c3+3c=48 2c + \frac{c}{3} + 3c = 48

Combine terms:

5c+c3=48 5c + \frac{c}{3} = 48

Multiply the entire equation by 3 to eliminate the fraction:

15c+c=144 15c + c = 144

16c=144 16c = 144

Solving for c c :

c=14416=9 c = \frac{144}{16} = 9

Now, substitute c=9 c = 9 back into the expressions for x1,x2,x3, x_1, x_2, x_3, and x4 x_4 :

x1=93=6 x_1 = 9 - 3 = 6
x2=9+3=12 x_2 = 9 + 3 = 12
x3=93=3 x_3 = \frac{9}{3} = 3
x4=3×9=27 x_4 = 3 \times 9 = 27

Thus, the amounts of money each brother has are:

6,12,3,27 \boxed{6, 12, 3, 27}

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Source: Omni-MATH, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.