Ratios and Proportions

Invertendo, and Alternendo


To solve the questions of ratio and proportion you need to remember the different set of rules. Particularly while solving proportion, you are required to remember five set of laws. They are, Invertendo, Alternendo, Componendo, Dividendo and Componendo and Dividendo. And today we are going to discuss Invertendo and Alternendo laws of proportion.

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Alternendo and Invertendo Laws of Proportion

Before understanding the laws let’s look into what proportion is? A ratio becomes a proportion when they are equated with each other. For example, a, b, c, and d are in proportion when a/b = c/d. Also, if a : b = b : c = c : d, then a/b = b/c = c/d is said to be in continued proportion.


Invertendo Law

In Invertendo, if a : b = c : d then b : a = d : c. In this law, you need to remember that if the ratio is in proportion then the inverse of that ratio is also proportion to each other.

For, example 8 : 10 = 16 : 25.

Here using the Invertendo law, 10 : 8 = 25 : 16 => 5 : 4 = 5 : 4.

Alternendo Law

In this law, if a : b : : c : d, then a : c : : b : d. In this law, the property of continued proportion is used. Suppose you are given two ratios and the values in both the ratios are proportional to each other. In this law, when you replace the denominator of the first ratio with the numerator of the second ratio, the two ratio remains proportional to each other. Here, the important word is proportional because unlike Invertendo you are not required to find the equal ratio but the proportional ratio.

For example, 3 : 4 = 6 : 8, then using Alternendo law, the ratios will be 3 : 6 : : 4 : 8 => 1 : 2 = 1 : 2.

Solved Example

Q. A cat pursues a dog and takes 5 leaps for every 6 leaps of the dog, but it is given that 4 leaps of cat are equal to 5 leaps of the dog. What will be the speed of the cat and the dog?

Ans: This type of question is a common example of Invertendo law. In an exam, you won’t find a straightforward question from the law. Because that would be very simple! Instead, you need to apply the law to find the speed of the two moving objects. Now, in the question, 4 leaps of cat are equal to 5 leaps of dog. It is also given that 5 leaps of cat are also equal to 6 leaps of dog.

Thus, from the given conditions, we can find that 5 leaps of cat = 25/4 leaps of dog. So, the required ratio of the speed of cat and speed of dog will be, 25/4: 6 = 25: 24.

Further, you can also solve this type question by inverting the second ratio given to you and then multiplying the numerator of both the ratio and simultaneously multiplying the denominator of both the ratios.

Practice Questions

1. The proportion of sugar and milk in 3 of the given samples is 2:1, 3: 2, and 5:3. A mixture is prepared which comprises the three samples. What will be the proportion of sugar and milk will be in that mixture?

A. 227: 133                  B. 224: 132
C. 225: 135                  D. None of the above

The correct answer is A.

2. Divide 465 into three parts such that they are in the ratio of 1/2: 1/3: 1/5.

A. 220, 160, 80               B. 225, 150, 90
C. 230, 150, 80               D. 225, 160, 90

The correct answer is B.

3. If A : B = 3 : 4 and B : C = 5 : 6 find A : B : C.

A. 2 : 3: 4                        B. 3: 4: 5
C. 15: 20: 24                  D. None of the above

The correct answer is C.

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