A Ratio is used to represent the relation between two or more values. For example: if there are 10 pencils and 18 pens then we can write the ratio as:
RatioBack to Top
A: B or A / B,
Here we can easily see that ‘A/B’ is in rational form. Now we take a phrase and according to that we have to write the ratio between two quantities. There are 15 hens and 9 ducks in a park. Now express the ratio between hens and ducks. If we represent hens with ‘A’ and ducks with ‘B’ than according to ratio property
A / B = 15 / 9 = 5 / 3,
So the ratio is 5: 3. To understand that ratio topic more deeply we take an example where we have to find the ratio. If we have the value of B = 6 and the ratio A: B = 5: 2 then find the value of ‘A’.
Here we can easily see that the ratio A: B =5: 2 and we can also express that
A / B = 5 / 2,
Now we can put the value of B in the above equation
A / 6 = 5 / 2,
After cross multiplication
A * 2 = 5 * 6,
A = (5 * 6) / 2 = 30 / 2 = 15,
Let’s take an example where we have to find ratios of two numbers. We have the value of A = 4 and the ratio A: B = 2: 5. Now we have to find the value of ‘B’. Here we can easily see that ratio of ‘A’ and ‘B’ is
A: B = 2: 5,
A / B = 2 / 5,
4 / B = 2 / 5,
B = (5 * 4) / 2 = 10.
ProportionsBack to Top
In order to solve the terms which are given in the proportion, we Mean that:
a) p, q, r and s are called first, second, third and the fourth terms of the proportion.
b) Here in proportion we say ‘p’ and ‘s’ are the extremes or we say them as extreme terms and ‘q’ and ‘r’ are called the mean terms or we also call them the means.
c) We must always remember that in case of proportions, we must have product of means = product of extremes.
d) We also call‘s’ as the fourth proportion to the terms p, q, and r terms.
e) In case we are given p: q:: r : s, then p, q, and r are said to be in continued proportions.
f) Here we call ‘r’ as the third proportion to ‘p’ and ‘q’ also we say that ‘r’ is the fourth proportion to p, q, q.
g) Also the Point to be remembered in this continued proportion is that p/q = r / s.
Thus we say that q2 = p * r.
Or we say that b = root (p * r).
Properties of ProportionBack to Top
Proportion is the relation between two ratios when they are equal. In other words, we can say that any two ratios are said to be in proportion if two ratios are equivalent.
Now we will explain how four terms of the two ratios involved in a proportion are related. Let the two ratios that are in proportion be a: b and c: d. Since they are in proportion, we write it as, a : b :: c : d. This is read as ‘a’ ratio ‘b’ is proportional to ‘c’ ratio ‘d’. Also a: b = c: d. The following points are associated with such a proportion and are called as the properties of Proportions.
# a , b , c , d are called first , second , third & fourth term respectively.
# a and d are called the extremes , while b and c are defined as the means.
# The product of means & the product of extremes are equal, i.e, a * d = b * c
# d is called the fourth proportion to a , b , c.
The following properties of proportion should be remembered:
If a : b :: b : c , then we say that,
# a , b , c are in continued proportion.
# c is called as the third proportional to a ,b and fourth proportional to a , b , d.
# b is called the Mean proportional or geometric mean between a and c and is given by b^2 = ac.
Means and ExtremesBack to Top
a : b :: c : d,
We read the above expression as a ratio b is proportional to c ratio d. Here the term ‘a’ is called as the first proportion, ‘b’ is called as the second proportion, ‘c’ is called third proportion and ‘d’ is called the fourth proportion. We call the first and the fourth term as the extreme terms and the second and the third term is called the mean terms. So here we say that the given terms are in proportion if we write the product of first and fourth term is equal to the product of the second and third term. Now we will look, at the following example where we have the two ratios.
2: 5 and 4 : 10. We need to check that the given ratios are in the proportion or not. So we will write it as follows:
2 : 5 : : 4 : 10, here we will first find the product of extremes and then find the product of means and compare if the two values are equal or not.
Here we write product of extremes = 2 * 10 = 20 and the product of means = 5 * 4 = 20
Thus in both the cases we get the same answer. So the two ratios are in proportion.