### Compound rule of three

Compound rule of three is a set of two or more simple rule of three. The simple rule of three is a method to solve proportions. If you have three numbers: a, b, c, such that, a / b = c / x, you can calculate the unknown value (x) .

Direct Proportion

If a change in a quantity causes a change in another quantity in the same proportion, the amounts are in direct proportion, or the quantities are directly proportional.
Inverse Proportion

When, the increase of an element causes a decrease in the other one, or by the contrary, a decreasing causing the increase in the other one, this is, an inverse proportion, or the two quantities are inversely proportional.

The expression: a / b = c / x (direct proportion) is now:
(b / a = c / x). For the inverse proportion or (b:a :: c:x), we change the terms of the second equality. The numerator becomes denominator and vice versa.

To solve the compound rule of three is necessary to check whether each number is directly proportional or inversely to the unknown.

Example 1

In a little factory, 8 men assemble 20 personalized motorcycles in 5 days. How many motorcycles must be assembled by 4 men in 16 days?

Problem formulation.

8 workers - 5 days - 20 trucks

4 workers - 16 days - x trucks

Mathematical Formulation:

All of the equalities are direct proportions. For the mathematical formulation, not require any changes.

8 - 5 - 20

4 - 16 - x

x = (4 * 16 *20) / (8 * 5) = 32

x = 32 motorcycles

Example 2:

In 8 hours, 20 trucks unload 160m3 of sand. In 5 hours, how many trucks will be required to download 125m3?
Problem formulation:

160m3 of sand - 8 hours - 20 trucks

125m3 of sand - 5 hours - x trucks

since the second quantity is inversely proportional, we changed the numerator by the denominator and vice versa, so we have:

Mathematical formulation.

160m3 sand - 5 hours - 20 trucks

125m3 sand - 8 hours - x trucks

x = (125 * 8 * 20) /(160 * 5)

x = 25 trucks.

Example 3

If 5 trucks transport 120 tons of goods in 2 days: what goods quantity will 7 trucks transport in 3 days?

Problem formulation.

5 trucks - 2 days - 120 tons

7 trucks - 3 days - x tons

x = (7 * 3 * 120) /( 5 * 2 )

x = 252.

7 trucks will transport 252 tons in 3 days.

Example 4:

It takes 12 days for a team of 12 workers working 10 hours per day to finish an order. How many persons with 8 hours will be needed to realize
the same work in 20 days?

Problem formulation.

12 days - 10 hours - 12 workers

20 days - 8 hours - x workers

Since the first and second quantities is inversely proportional, we changed the numerator by the denominator and vice versa, so we have:

Mathematical formulation.

20 - 8 - 12

12 - 10 - x

x = (12 *10 * 12) /( 20 * 8 ) = 9

x = 9 workers.

How to calculate the Compound Rule of three

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