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Business Mathematics

Proportions

Ration and Proportion: Proportions are simple mathematical tools that use ratios to express the relation between multiple quantities. More often, the knowledge of ratio and proportion is applied together to solve day to day problems. For eg – You might have studied 75% of your syllabus for an upcoming test.

The ratio between what you have studied and the total syllabus is 3 : 4. Then in the test of total 40 marks, the ratio of the marks you get, to the total possible will also be the same as 3 : 4. You can easily calculate from this that you’ll get 30 marks out of 40 in that test (assuming that you answered everything you knew correctly!). Thus, many practical scenarios involve the application of ratio and proportion in this way.

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Ratio and Proportion

Representation of Proportions

ratio and proportion

Mathematically, we can thus express proportions as – if four quantities a, b, c, d are in proportion, then the ratio a : b must be equal to c : d, i.e.

\(a : b = c : d\)
$$ \frac{a}{b} = \frac{c}{d} $$

The following conventions are usually employed – the terms a, b, c, d are the terms of the proportion. Out of these the first and the fourth term a and d are extremes (the extreme terms), while the second and the third terms b and c are means (the middle terms).

The product of extremes = The product of the means i.e. ad = bc (The Cross Product Rule)

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Continued Proportions

Three quantities are said to be in continued proportion if a : b = b : c. In fractional form – $$\frac{a}{b} = \frac{a}{c}$$ Here, you can see that b2 = ac. This is also known as the Cross Product Rule. We call b as the mean proportional between a and c in this case. We can extend the concept of continuous proportions for more than three quantities in the following way –

 $$ \frac{a}{b} = \frac{b}{c} = \frac{c}{d} = \frac{d}{e}….. $$

 a : b = b : c = c : d = d : e ….

Properties of Proportions

a : b = c : d ⇔ b : a = d : c

  • Alternendo

a : b = c : d ⇔ a : c = b : d

a : b = c : d ⇔ a + b : b = c + d : d

  • Dividendo

a : b = c : d ⇔ a – b : b = c – d : d

  • Componendo and Dividendo

a : b = c : d ⇔ a + b : a – b = c + d : c – d

  • Equality of Addendoes

a : b = c : d = e : f ….. = k (say)

Then any term of the form a + c + e ….. : b + d + f…, is known as an Addendo. And, all of such ratios (Addendoes) are equal to the original ratio i.e.

a + c : b + d = a + e : b + f = a + c + e : b + d + f = …… = k

All of these properties can simply be derived if you express the ratios in a fractional form. Still, it is important to recognize one when you see them in the ratio form as well. Now go through the solved examples below to clear all of your doubts, if any.

Solved Examples for You

Question 1: The numbers 14, 16, 35, 42 are not in proportion. The fourth term for which they will be in proportion is?

Solution: Let the fourth term be x. By the definition of the ratios being in proportion – $$\frac{14}{16} = \frac{35}{x}$$ $$x = \frac{35 \times 16}{14}$$ $$ x = 40 $$ Clearly, the fourth term has to be equal to 40, not 42, for the numbers given to be in proportion.

Question 2: If x/4 = y/3 = z/2, then what would be the value of (5x + y – 2z)/3y?

Solution: Let us assume that the given ratios in the continued proportion are all equal to an integer k. Then, $$ \frac{x}{4} = \frac{y}{3} = \frac{z}{2} = k$$ Considering the ratios now one at a time, we get – $$x = 4k$$ $$y = 3k$$ $$z = 2k$$
Use these values to evaluate the required expression – $$\frac{5x + y – 2z}{3y}$$ $$ = \frac{5\times 4k + 3k – 2\times 2k}{3\times 3k}$$ $$ = \frac{19k}{9k}$$ $$ = \frac{19}{9}$$ Hence we saw the applications of ratio and proportion with the help of these examples.

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