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## Proportional Division

### Directly Proportional Division

The directly proportional division consists of dividing a number, in a directly proportional parts. The higher of the proportion, the greater is the part.
Example 01:
Division of household expenses of \$ 1,800.00; in proportion to the incomes of participants, as follow: A = \$ 5,000.00; B = \$ 3,000.00; C = \$ 1,000.00;
Proportions:
a = 5000;
b = 3000;
c = 1000;
Solution:
N = 1800;
X = a + b + c = 900;
p1, p2 and p3 are the participants;
Divide the value of N, for X to find constant K.
Multiply the proportion of each participant by the constant K.
K = N / X = 18000 / (5000 + 3000 + 1000) = 0.20;
So, we have;
Participants:
P1 = a * K = 5000 * 0.20 = \$1,000.00;
P2 = b * K = 3000 * 0.20 = \$  600.00;
P3 = c * K = 1000 * 0.20 = \$  200.00.
p1 + p2 + p3 = 1800;

Example 02:
Division of 6000 directly proportional to 3, 5 and 8;
Solution:
N = 6000;
Proportions:
a = 3;
b = 5;
b = 8;
X = a + b + c = 16;
K = N / X = 6000/16 = 375;
p1 = 3 * 375 = 1125;
P2 = 5 * 375 = 1875;
P3 = 8 * 375 = 3000;
p1 + p2 + p3 = 6000;
How to calculate the Directly Proportional Division.

### Inversely Proportional Division

A inversely proportional division consists of dividing one number in parts, inversely proportional. The higher of the proportion, smaller is the part.
Example 1:
A father decided to give, to her three daughters in inverse proportion of their salaries, the monthly value of \$ 3,000.00;
Salaries:
Emily  =   \$4,000.00;
Diana  =   \$3,000.00;
Gracie =  \$2,000.00;
N = 3000;
Proportions:
a = 4000;
b = 3,000;
c = 2000;
Solution:
Calculating the constant K;
K = N / ((1 / a) + (1 / b) + (1 / c));
K = 2769230, 77;
Result:
P1 = (1 / a) * K = 692;
P2 = (1 / b) * K = 923;
P3 = (1 / c) * K; = 1385;
p1 + p2 + p3 = 3000;
So, we have;
Emily =  \$  692.00;
Diana =  \$  923.00;
Gracie =\$1,385.00;

Example 2:
Division of 500 in parts inversely proportional to 3, 5 and 7.
Solution:
N =500;
a, b, c - proportions;
a = 3; b = 5; c = 7;
p1, p2, p3 - Parts;
K - Constant;
K = N / ((1 / a) + (1 / b) + (1 / c));
K = 500 /((1/3) + (1/5) + (1/7)) = 739.44;
P1 = (1 / a) * k = (1/3) * 739.44 = 246;
P2 = (1 / b) * k = (1/5) * 739.44 = 148;
p3 = (1 / c ) * k = (1/7) * 739.44= 106;
p1 + p2 + p3 = 500;
How to calculate the Inversely Proportional Division

### Division, simultaneously, direct and inversely proportional

Example:
Division of 800 in parts directly proportional to 2; 4; 6 and inversely to 3; 5 7.;
Solution:

N = 800;
n1, n2, n3 - dirtect proportions;
n1 = 2; n2 = 4; n3 = 6;
d1, d2, d3, - inverse proportions;
d1 = 3; d2 = 5; d3 = 7;
K = Constant;
p1, p2 e p3 - Parts;
K = N / (n1 /d1) + (n2 / d2) + (n3 / d3);
K = 800 / ((2 /3) + (4 / 5) + (6 / 7)) = 344,26;
P1 = (n1 /d1) * k = (2/3) * 344,26 = 230;
P2 = (n2 /d2) * k = (4/5) * 344,26 = 275;
p3 = (n3 /d3) * k = (6/7) * 344,26 = 295;
p1 + p2 + p3 = 800;