In this article, we will discuss fourth, mean, and third proportional in detail along with the examples. But before proceeding to discuss these terms, first, let us see what is a proportion.

What is a Proportion?

We know that in a proportion, we use different ratios to calculate the unknown quantities. A proportion is in fact the equality of different ratios. For instance, if we have the quantities that are represented by w, x, y, and z, then we can write these quantities in proportional form like this:

\frac{w} {x} = \frac {y}{z}

Remember that we take a ratio of two similar quantities that have the same units. Hence, we also say that a ratio has no unit. Now, that you know what is a ratio and a proportion, let us proceed to discuss fourth, mean, and third proportional.

 

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What is a Fourth Proportional?

Suppose w:x :: y:z is a proportion of two ratios. We can write it as w:x = y:z. The quantity z is known as fourth proportional to the quantities w, x, and y.

For example, if we write the quantities 7,8,9, and 10 in the proportional form 7:8 :: 9:10, then 10 is the fourth proportional to 7, 8, and 9.

 

What is a Third Proportional?

Suppose a:b :: c:d is a proportion of two ratios. We can write it as a:b = c:d. The quantity c is known as third proportional to the quantities a, b, and d.

For example, if we write the quantities 7,8,9, and 10 in the proportional form 7:8 :: 9:10, then 9 is the third proportional to 7, 8, and 10.

 

What is a Mean Proportional?

Consider a proportion a : b :: c : d. The mean proportional between two terms of the ratio is computed by taking the square root of the product of the two quantities in the ratio. For example, in the proportion a : b :: c : d, we can compute the mean proportional for the ratio a : b by taking the square root of the product of the quantities a and b. Mathematically, we can write it as:

Mean proportional = \sqrt{ab}

 

In the next section, we will solve some examples in which we will calculate third, fourth and mean proportional.

 

Example 1

Find the fourth proportional to 3, 5 and 6.

Solution

Suppose the fourth proportional is x. We can write the above numbers in proportional form like this:

3 : 5 :: 6 : x

\frac {3}{5} = \frac{6}{x}

Take x on the left hand side of the equation and the fraction \frac{3}{5} on the right hand side.

x = \frac {5}{3} \cdot 6

x = 5 \cdot 2

x = 10

Hence, the fourth proportional is 10.

 

Example 2

Find the fourth proportional to 11, 15 and 22.

Solution

Suppose the fourth proportional is x. We can write the above numbers in proportional form like this:

11 : 15 :: 22 : x

\frac {11}{15} = \frac{22}{x}

Take x on the left hand side of the equation and the fraction \frac{11}{15} on the right hand side.

x = \frac {15}{11} \cdot 22

x = 15 \cdot 2

x = 30

Hence, the fourth proportional is 30.

 

Example 3

Find the fourth proportional to x + 3x, x + 2 and 5x, if x = 3.

Solution

In this example, first, we will calculate the quantities by substituting x = 3.

First quantity = 3 + 3(3) = 12

Second quantity = x + 2 = 5

Third quantity = 5x = 5 (3) = 15

Suppose the fourth proportional is x. We can write the above numbers in proportional form like this:

12 : 5 :: 15 : x

\frac {12}{5} = \frac{15}{x}

Take x on the left hand side of the equation and the fraction \frac{12}{5} on the right hand side.

x = \frac {5}{12} \cdot 15

x = \frac {75}{12}

x = \frac {25}{4}

Hence, the fourth proportional is \frac {25}{4}.

 

Example 4

Find the third proportional to 2, 9 and 15.

Solution

Suppose the third proportional is x. We can write the above numbers in proportional form like this:

2 : 9 :: x : 15

\frac {2}{9} = \frac{x}{15}

Multiply both sides by 15 to isolate x on the either side of the equation:

x = \frac {2}{9} \cdot 15

x = \frac {10}{3}

Hence, the third proportional is \frac {10}{3}.

 

Example 5

Find the third proportional to 11, 13 and 26.

Solution

Suppose the third proportional is x. We can write the above numbers in proportional form like this:

11 : 13 :: x : 26

\frac {11}{13} = \frac{x}{26}

Multiply both sides by 26 to isolate x on the either side of the equation:

x = \frac {11}{13} \cdot 26

x = 11 \cdot 2

x = 22

Hence, the third proportional is 22.

 

Example 6

Find the third proportional to 3x, x + 5 and 5x - 1, if x = 2.

Solution

First, we will compute the values of these quantities like this:

First quantity = 3x = 3(2) = 6

Second quantity = x + 5 = 2 + 5 = 7

Fourth quantity = 5x - 1 = 5 (2) - 1 = 9

Suppose the third proportional is y. We can write the above numbers in proportional form like this:

6 : 7 :: y : 9

\frac {6}{7} = \frac{y}{9}

Multiply both sides by 9 to isolate y on the either side of the equation:

y = \frac {6}{7} \cdot 9

y = \frac {54}{7}

Hence, the third proportional is \frac {54}{7}.

 

Example 7

Find the mean proportional between the quantities 6\frac {4}{2} and 9.

Solution

To find the mean proportional, we need to multiply the quantities and take the square root of them.

6 \frac {4}{2} is equal to \frac {16}{2} = 8. The product of 8 and 9 is 72. The square root of 72 is the mean proportional of 6\frac {4}{2} and 9.

Mean Proportional = \sqrt{72}

 

Example 8

Find the mean proportional between the quantities 5\frac {14} {7} and 225.

Solution

5 \frac {14}{7} is equal to \frac {49}{7} = 7. First, we will take the product of 7 and 225 and then we will take the square root of the resulting term to get the mean proportional.

Mean proportional = \sqrt {225 \cdot 7}

= \sqrt {1575}

 

Example 9

Find the mean proportional between the quantities 4\frac {17} {2} and 50.

Solution

4 \frac {17}{2} is equal to \frac {25}{2}. First, we will take the product of 50 and \frac {25}{2} and then we will take the square root of the resulting term to get the mean proportional.

Mean proportional = \sqrt {\frac {25}{2} \cdot 50}

= \sqrt {625}

= 25

Hence, the mean proportional is 25.

 

Example 10

Find the third proportional to 7x, x + 4 and 2x - 1, if x = 4.

Solution

First, we will compute the values of these quantities like this:

First quantity = 7x = 7(4) = 28

Second quantity = x + 4 = 4 + 4 = 8

Fourth quantity = 2x - 1 = 2 (4) - 1 = 7

Suppose the third proportional is y. We can write the above numbers in proportional form like this:

28 : 8 :: y : 7

\frac {28}{8} = \frac{y}{7}

Multiply both sides by 7 to isolate y on the either side of the equation:

y = \frac {8}{28} \cdot 7

y = 2

Hence, the third proportional is 2.

 

Example 11

Find the fourth proportional to 5x + 3, 4x + 2 and 6x, if x = 1.

Solution

In this example, first, we will calculate the quantities by substituting x = 1.

First quantity = 5x + 3 = 5 (1) + 3 = 8

Second quantity = 4x + 2 = 4 (1) + 2 = 4 + 2 = 6

Third quantity = 6x = 6 (1) = 6

Suppose the fourth proportional is y. We can write the above numbers in proportional form like this:

8 : 6 :: 6 : y

\frac {8}{6} = \frac{6}{y}

Take y on the left hand side of the equation and the fraction \frac{8}{6} on the right hand side.

x = \frac {6}{8} \cdot 6

x = \frac {36}{8}

x = \frac {9}{2}

Hence, the fourth proportional is \frac {9}{2}.

 

 

 

 

 

 

 

 

 

 

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Rafia Shabbir