Ratios and Proportions

Proportionals (Third, Fourth and Mean)

In the proportion, we saw how we can use the ratio of various quantities to analyze and compute the quantities themselves. Another very important concept that arises from the concept of proportion is the definition of proportional. In the following section, we shall define what we call a proportional and also solve many interesting examples of this concept. We will also define what we mean by the fourth proportional, the third proportional and the mean proportionals. Let us begin!

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Proportionals (Third, Fourth and Mean)

Let us start by defining what we mean by the proportion. The equality of any two ratios is called a proportion. For example, if we have any four numbers or quantities that we represent as ‘a’, ‘b’, ‘c’, and ‘d’ respectively, then we may write the proportion of these four quantities as:


a:b = c:d or a:b :: c:d. From this, we will now define the proportionals. Let us begin by defining the fourth proportional.

Fourth Proportional

If a:b :: c:d or in other words a:b = c: d, then the quantity ‘d’ is what we call the fourth proportional to a, b and c.

For example, if we have 2, 3 and 4, 5 are in the proportion such that 2 and 5 are the extremes, then 5 is the fourth proportional to 2, 3, and 4.

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Third Proportional

Similar to the f=definition of the fourth proportional, we define the term known as the third proportional. The third proportional of a proportion is the second term of the mean terms. For example, if we have a:b = c:d, then the term ‘c’ is the third proportional to ‘a’ and ‘b’.

Mean Proportional

The mean proportional between the two terms of a ratio in a proportional is the square root of the product of these two. For example, in the proportion a:b :: c:d, we can define the mean proportional for the ratio a:b as the square root of the product of the two terms of the ratio or √ab.

Solved Examples For You

Part I

Example 1: Find the fourth proportional to 4, 9 and 12?

A) 18               B) 21                   C) 24                 D) 27

Answer: Let the fourth proportional to the numbers 4, 9, and 12 be x. Then from the definition of the fourth proportional, we must have:

4: 9 :: 12 : x or in terms of the fractions we can write;

4x = 9 × 12. In other words, we have x = {9 × 12}/4 = 27. Thus the correct option is D) 27.

Example 2: The third proportional to 16 and 36 will be:

A) 9                 B) 27                  C) 45                      D) 81

Answer: Let the third proportional to the two numbers 16 and 36 be x. Therefore from the definition of the third proportional, we have:

16: 36 :: 36 : x or in other words we can write:

16x = {36 × 36}/16 = 81. Therefore the correct option is D) 81.

Example 3: Find the mean proportional between 0.08 and 0.18?

A) 0.12                    B) 0.09                    C) 0.12                   D) 0.16

Answer: We have the two terms as 0.08 and 0.18. Let us find their square root first. We have sqrt{0.08 + 0.18} = sqrt{8/100 + 18/100} = 12/10 or  0.12. Thus the correct option here is C) 0.12.

Example 4: If we divide 76 into four parts proportional to 7, 5, 3, 4, then the smallest part is equal to:

A) 11                      B) 12                      C) 13                         D) 14

Answer: The ratio that we have here 7: 5: 3: 4. Then the smallest part will correspond to the factor of ‘3’. The sum of the terms of the ratio = 7 + 5 + 3 + 4 = 19.

Hence the smallest part will be = [76×{3/19}] = 12. Hence the correct option is B) 12.

Part II

Example 5: A and B are two alloys of gold and copper prepared by mixing metals in the ratio of 7: 2 and 7: 11 respectively. If we melt equal quantities of the alloys to form a third alloy C, then the ratio of the quantity of gold or the proportion of gold to the quantity of copper in C will be:

A) 6: 5                     B) 5: 7                        C) 7: 5                        D) 9: 8

Answer: Let us find the amount of gold in C first. The amount of gold in C = [7/9 + 7/18] units. The units could be grams or kgs or anything else that determines the concentration. Therefore we can write: [7/9 + 7/18] = 7/6 units.

Similarly, we can find the amount of copper in C as equal to [2/9 + 11/18] units. We can simplify it with the help of cross multiplication and write this as equal to 5/6 units.

Thus, the ratio of the concentration of Gold: Copper = 7/6: 5/6 = 7: 5. Thus the correct option here is C) 7: 5.

Example 6: Two numbers are respectively 20% and 50% more than a third number. the ratio of the two numbers is:

A) 3: 4                 B) 4: 5                     C) 5: 6                       D) 6: 7

Answer: Let the third number be equal to x. Now the first number is 20% more than x. In other words we can say that this number is 120% of x = 120x/100 = 6x/5.

Similarl,y the second number is 50% more than x. therefore we can write this number as 150%(x) = 150x/100 = 3x/2.

Thus the ratio of the two numbers can be written as: 6x/5 : 3x/2 = 12x: 15x or this can be written as 4: 5. Therefore the correct option is B) 4: 5.

Practice Questions

Q 1: A sum of Rs. 53 is divided among A, B, C in such a way that A gets Rs. 7 more than what we give to B. Also, B gets eight rupees more than what we give to C. Then the ratio of their shares is:

A) 25: 18: 10           B) 32: 65: 98                     C) 24: 76: 65                  D) 98: 53: 87

Ans: A) 25: 18: 10.

Q 2: What is the ratio whose terms differ by 40 and the measure of which is 2/7?

A) 12: 87                  B) 96: 23                    C) 16: 56                   D) 51: 12

Ans: C) 16: 56.

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