In mathematics, proportions are frequently used for various purposes. It is generally based on fractions & ratios. We know that a term written in the form of p/q is known as a fraction while a term written with a colon sign “p:q” is known as ratios.
Ratios & proportions play a very essential role to understand several concepts of science & mathematics. In this post, we will study all the basic concepts of proportion along with a lot of examples.
What is proportion?
The proportion is an equation equivalent to two ratios. It is referred a part from the whole. It represents the two fractions or ratios that are equivalent. In other words, the comparison of two numbers is said to be the proportion.
In proportion, if two sets of ratios in which one is increasing & other are decreasing, then the ratios are known as inverse ratios. While direct ratios are those ratios in which both sets are increasing or decreasing.
It is denoted by an equal symbol “=” or double ratio symbol “: :”. A proportion is obtained only when both ratios are equal. For example, the interval taken by a car to cover 20 kilometers per hour is equivalent to the interval taken by it to cover the space of 90 kilometers per hour, i.e., 20 kilometers/per hour = 90 kilometers/per hour.
Formula of proportion
To use the formula of proportion, you must be familiar with the formula of ratio. The general formula for ratio is:
p : q or p/q
where p & q are any two quantities or integers. The first term of the numerator of the ratio is said to be antecedent, while the second term or denominator is said to be the consequent.
Similarly, the formula of proportion involves two ratios such as p : q and s : t. The set of two ratios can be written as:
p : q : : s : t
Or
p/q = s/t
The inner terms like q & s are said to be the mean terms while the outer terms like p & t are said to be the extreme terms.
Several ways to write the formula of proportion
The formula of proportion can be written as:
- p/q = s/t ⇒ pt = qs
- p/q = s/t ⇒ q/p = t/s
- p/q = s/t ⇒ p/s = q/t
- p/q = s/t ⇒ (p + q)/q = (s + t)/t
- p/q = s/t ⇒ (p – q)/q = (s – t)/t
Types of proportion
There are two basic types of proportion.
- Direct proportion (DP)
- Inverse or indirect proportion (IP)
Let us discuss the types of proportion briefly.
Direct proportion (DP)
This is the first type of proportion, it states the direct relation among two ratios, integers, or quantities. In simple words, if one ratio or quantity increases the other also increases, or if the one ratio or quantity decreases the other decreases, this kind of relationship is known as DP.
For example, if the speed of the bike increases, it covers more space in a fixed time.
Inverse or indirect proportion (IP)
This is the other type of proportion that states if two ratios are not direct then they must be indirect or inverse. In simple words, the increase in the first quantity causes the decrease in the second quantity, or the decrease in the first quantity causes the increase in the first quantity, this kind of relation is said to be the inverse proportion.
How to solve the problems of ratios & proportion?
Follow the below examples to learn how to solve the problems of ratios & proportion.
Example I
18 vans are required to supply milk in 25 days, how many vans are required to supply the same amount of milk in 15 days.
Solution
Step I: First of all, write the given information about vans & days.
Vans required to supply milk in 25 days = 18
Vans required to supply the same amount of milk in 15 days = x
Step 2: Use the proportion formula & write the vans & days according to the formula.
p : q : : s : t
vans : days : : vans : days
18 : 25 : : x : 15
Step 3: Write the equality sign in the place of the proportion symbol & write both ratios in the form of a fraction.
18/25 = x/15
Step 4: Simplify the above expression by using cross multiplication method to find the proportion “x”.
18/25 = x/15
(18/25) * 15 = x
(18/5) * 3 = x
54/5 = x
X = 10.8
X = 11
Vans required to supply the same amount of milk in 15 days = 18 + 11 = 29
Hence, 11 more Vans are required to supply the same amount of milk in 15 days.
The problem of finding the x term of the proportion can be solved easily by using a proportion calculator to get the result in a few seconds.
Example II
In a cotton bag, there are 15 colors. From these 15 colors, 6 are purple, 4 are yellow, and 5 are black. Calculate the ratio of:
- Purple colors to yellow colors
- yellow colors to total colors
- black colors to purple colors
Solution
Step I: First of all, write the given information of colors.
Total colors = 15
Purple colors = 6
yellow colors = 4
black colors = 5
Step II: Now find the ratio of Purple colors to yellow colors.
Number of Purple colors = 6
Number of yellow colors = 4
The ratio is,
6 : 4 ⇒ 3 : 2
or
6/4 ⇒ 3/2
Step III: Determine the ratio of yellow colors to total colors.
Number of total colors = 15
Number of yellow colors = 4
Subtract the total colors from yellow colors to find the difference = 15 – 4 = 11
Now the ratio is,
4 : 11
Or
4 / 11
Step IV: Now from black colors to purple colors.
Number of black colors = 5
Number of purple colors = 6
Ratio is,
5 : 6
Or
5/ 6
Summary
From the above post, you can grab all the basics of proportion. In this post, we have discussed all about the proportion along with the solved examples. You can easily solve the problems of proportion by learning the above post.