# Compound Proportion

“**The proportion involving two or more quantities is called Compound Proportion.**”

__Rules for Solving Compound Proportions__

\[\begin{array}{*{20}{c}} {{\text{Quantity 1}}}&{{\text{Quantity 2}}}&{{\text{Quantity 3}}} \\ {\text{a}}&{\text{b}}&{\text{c}} \\ {\text{d}}&{\text{e}}&x \end{array}\]

**CASE-1**

If quantity 1 and quantity 2 are directly related and quantity 2 and quantity 3 are also directly related, then we use the following rule:

\[\boxed{\frac{{{\text{a x b}}}}{{\text{c}}} = {\text{ }}\frac{{{\text{d x e}}}}{x}}\]

**CASE-2**

If quantity 1 and quantity 2 are directly related and quantity 2 and quantity 3 are inversely related, then we use the following rule:

\[\boxed{\frac{{{\text{b x c}}}}{{\text{a}}}{\text{ = }}\frac{{{\text{e x }}x}}{{\text{d}}}}\]

**CASE-3**

If quantity 1 and quantity 2 are inversely related and quantity 2 and quantity 3 are directly related, then we use the following rule:

\[\boxed{\frac{{{\text{a x b}}}}{{\text{c}}}{\text{ = }}\frac{{{\text{d x e}}}}{x}}\]

**CASE-4**

If quantity 1 and quantity 2 are inversely related and quantity 2 and quantity 3 are also inversely related, then we use the following rule:

\[\boxed{{\text{a x b x c = d x e x }}x}\]

**Example:**

195 men working 10 hours a day can finish a job in 20 days. How many men are employed to finish the job in 15 days if they work 13 hours a day?

**Solution:**

Let $$x$$ be the no. of men required

\[\begin{array}{*{20}{c}} {{\text{Days}}}&{{\text{Hours}}}&{{\text{Men}}} \\ {{\text{20}}}&{{\text{10}}}&{{\text{195}}} \\ {{\text{15}}}&{{\text{13}}}&x \end{array}\]

\[\begin{gathered} 20 \times 10 \times 195 = 15 \times 13{\text{ }} \times x \\ x = \frac{{{\text{20 x 10 x 195}}}}{{{\text{15 x 13}}}} = 200\,men \\ \end{gathered} \]

**Example:**

A soap factory makes 600 units in 9 days with the help of 20 machines. How many units can be made in 12 days with the help of 18 machines?

**Solution:**

\[\begin{array}{*{20}{c}} {{\text{Machines}}}&{{\text{Days}}}&{{\text{Units}}} \\ {{\text{20}}}&{\text{9}}&{{\text{600}}} \\ {{\text{18}}}&{{\text{12}}}&x \end{array}\]

\[\begin{gathered} \frac{{{\text{20 x 9}}}}{{{\text{600}}}}{\text{ = }}\frac{{{\text{18 x 12}}}}{x} \\ 20 \times 9 \times x = {\text{ }}600 \times 18 \times 12 \\ x = \frac{{{\text{600 x 18 x 12}}}}{{{\text{20 x 9}}}} = 720\,units \\ \end{gathered} \]

Qaiser zaman

August 2@ 11:20 pmWhat amout the formula for which the uper side of division is the tails of arrows and the lower side will be the heads of arrows

mohammad siddiq

April 14@ 2:48 pmhow the quantities will be determined as directly related or inversely related.

Asifnawaz

June 19@ 10:50 amIf you increase one quantity and it decrease the 2nd one it means theybare inversely related. For example. Let say men and days. If number of men are less then obvious more days will be required for work and if men are more then kess number of days.

Innocent

October 7@ 8:28 pmHow to use the method of lining to this questions? Just Like inverse proportion.

Abdur Raheem

March 5@ 9:52 amHow we know the 3rd quantity

This is directly or inversely related to 1st value while third quantity is unknown

mba endam

January 17@ 11:13 amYes how do we know if a quantity is inversely or directly related

Nyanamba

January 31@ 1:16 pmPlease assist me with this;

9 tractors of same horsepower can plough 100 hectares of land in 5 days. How many:

a) hectares can be ploughed in 3 days by 8 tractors?

b) days will 3 such tractors take to plough 20 hectares?

Curious

June 22@ 10:26 pmWhat about more than 3 quantities?

Min

February 12@ 3:45 pmYou can learn the arrow head method from YouTube. Easy to learn and we can use it for more than 3 quantities