Direct and inverse proportion
What are proportions?
Proportion is a statement that defines two specified ratios are equal to each other. Ratio and proportion are explained using fractions. If a fraction is expressed as $$a: b$$, it is called a ratio. On the other hand, when two ratios are equal, it is called a proportion.
Types of proportion:
Direct proportion
Inverse proportion
Here we will discuss direct and inverse proportion in detail.
Direct proportion
Direct proportion is a relation or statement in which the ratio of two variables is constant. For example, two variables, $$ x$$ and $$ y $$, are proportional. It means the value of these two variables will either increase or decrease together. By the above definition, if $$x$$ and $$y$$ are connected by direct proportion, then $$ \frac{x}{y} =\text{ constant}$$.
Inverse proportion/reciprocal proportion
Inverse proportion is a relation or statement in which the product of two variables is a constant. When two variables, $$x$$ and $$y$$, are connected by inverse or reciprocal proportion, the value of one variable increases when the value of the other variable decreases, or vice versa.
If $$(x)$$ and $$(y)$$ are connected by inverse or reciprocal proportion, then by the above definition $$xy=\text{ constant}$$.
E2.8: Express direct and inverse proportion in algebraic terms and use this form of expression to find unknown quantities.
The concepts of positive proportion and inverse proportion are used to solve daily problems. At first, let us discuss the general idea and uses of direct proportion through some examples.
The cost of the food item is directly proportional to its weight; a car's fuel consumption is proportional to the distance covered.
Suppose a train travels $$300\text{ km}$$ in $$6\text{ hours}$$. How much time will it take to cover $$900\text{ km}$$?
Let $$x$$ hours be the time required for the train to travel $$ 900\text{ km}$$.
We know that time taken is directly proportional to the distance covered.
Hence, $$\frac{300}{6} =\frac {900}{x}$$
$$x=\frac{900\times 6}{300}$$
Therefore, $$x=18$$
So, the time taken by the train to cover a distance of $$900\;\text{km}$$ is $$18\text{ hours }$$.
Now let us discuss the general idea and uses of inverse proportion through some examples.
If we increase the speed of a car, the time taken to reach its destination increases. Similarly, if many labours do work, it is done in 15 days. But when a few workers do the same work, it is done in more than 15 days.
Worked examples of proportion
Example 1: If $$12$$ boys can finish their work in $$6\text{ hours}$$, how many boys can do the same work in $$3\text{ hours}$$.
Let $$x$$ boys can complete the same work in $$3$$ hours.
Step 1: Write the equation using inverse proportion.
$$12\times 6=x\times 3$$.
Step 2: Multiply by $$3$$ on both sides in the above equation.
$$x=\frac{12\times 6}{3}$$
Step 3: Answer in preferred notation.
$$x=23$$.
Therefore, $$24$$ boys can do the same work in $$3 \text{ hours}$$.
Example 2: If the cost of $$18\text{ liters}$$ of milk is $$\$200$$, calculate the cost of $$12\text{ liters}$$ of milk.
Let the cost of $$12\text{ liters}$$ of milk be $$x$$.
Step 1: Write the equation using direct proportion.
$$\frac{200}{18}=\frac{x}{12}$$
Step 2: Use the cross-multiplication method.
$$18\times x=200\times 12$$
Step 3: Divide both sides by $$18$$.
$$x=\frac{200\times 18}{12}$$
Step 4: Solve the above equation to get the value of $$x$$.
$$x=300$$
Therefore, the cost of $$12\text{ liters}$$ of milk is $$\$300$$