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Ratios and Proportions Geometry

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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:

10:18; or we can say that for every 10 pencils there are 18 pens.

In mathematics if an equation written in the form of

$\frac{PQ}{RS}$

where both the ratios PQ and RS are equal, are said to be proportion.

For example:

$\frac{70}{42}$ = $\frac{y}{52}$

⇒(70) (52) = (42) (Y);

Now solve these values for ‘Y’.

⇒3640 = 42 Y;

Here we find the value of ‘Y’.

⇒42 Y = 3640;

⇒Y = $\frac{3640}{42}$

⇒Y = 86.66;

After solving we get the value of Y = 86.66;

There is 86.66 liter water for 52 students.

Ratio

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Ratio is generally used to measure the relation between two Numbers. It is basically comparison of two numbers or quantities. If we have two numbers ‘A’ and ‘B’ then the ratio between ‘A’ and ‘B’ is

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.

Proportions

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When we find the fraction of the related terms, we call it ratio. Word Ratio means to represent the data in form of fraction such that the units of the two terms are equal. We can no find ratio or proportion for any two terms which are unlike. In this session we will study how to solve proportions. Let’s first see what all proportions are. Any four Numbers are said to be I proportion, if p: q = r : s and we write the relation of proportion as follows: p: q:: r : s. Now if we are given that the four terms p, q, r, s are in proportion, then it means p: q:: r : s.
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 Proportion

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Ratio is a fraction of two quantities that are of same kind and they can also be expressed in the same units. Example the Ratio of 30 minutes to 12 hours is written as 30 min: 12hours = 30min: 720min = 1 : 24
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 Extremes

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Means and extremes are the terms that are commonly used in Ratio and proportion. We say that, two ratios are in proportion, when the product of the means is equal to the product of the extremes. Let us first look at the Mean and extreme terms. If we have two ratios as a : b and c : d and we write that a: b is proportional to c : d, then it is expressed mathematically as :
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.