Proportion

We will discuss about the proportion of four quantities.

Four quantities w, x, y, and z are in proportion if w : x :: y : z or, \(\frac{w}{x}\) = \(\frac{y}{z}\).


Definition: If four quantities w, x, y and z are such that the ratio w : x is equal to the ratio y : z then we say w, x, y and z are in proportion or, w, x, y and z are proportional. We express it by writing w : x :: y : z.

w : x :: y : z if and only if \(\frac{w}{x}\) = \(\frac{y}{z}\) i.e., wz = xy.

In w : x : : y : z, w, x, y and z are the first, second, third and fourth terms respectively.

Also, w and z are called the extreme terms while x and y are called the middle terms or mean terms.

That is in a proportion the first and fourth terms are called extremes, while the second and third terms are called means.

Product of extremes = product of means

or, wz = xy.



If w, x, y and z are in proportion then the last term z is also called the fourth proportional.

For example, 2 : 7 :: 8 : 28 because \(\frac{2}{7}\) = \(\frac{8}{28}\).


Solved examples on Proportion:

Check whether the following numbers form a proportion or not.

(i) 2.5, 4.5, 5.5, 9.9

(ii) 1\(\frac{1}{4}\), 1\(\frac{3}{4}\), 1.5, 1.4

Solution:

(i) 2.5 : 4.5 = \(\frac{2.5}{4.5}\) = \(\frac{25}{45}\) = \(\frac{5}{9}\)

5.5 : 9.9 = \(\frac{5.5}{9.9}\) = \(\frac{55}{99}\) = \(\frac{5}{9}\)

Therefore, \(\frac{2.5}{4.5}\) = \(\frac{5.5}{9.9}\)

Hence, 2.5, 4.5, 5.5 and 9.9 are in proportion.


(ii) 1\(\frac{1}{4}\) : 1\(\frac{3}{4}\) = \(\frac{5}{4}\) : \(\frac{7}{4}\) = \(\frac{5}{4}\) × 4: \(\frac{7}{4}\) × 4 = 5 : 7 = \(\frac{5}{7}\)

1.5 : 1.4 = \(\frac{1.5}{1.4}\) = \(\frac{15}{14}\)

Since, \(\frac{5}{7}\)  ≠ \(\frac{15}{14}\)

Hence, 1\(\frac{1}{4}\), 1\(\frac{3}{4}\), 1.5 and 1.4 are not in proportion.

 










10th Grade Math

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