Concept of “Percentage” [“प्रतिशत”]–07

Question:  Find a single equivalent increase, if a number is successively increased by 10%, 15% & 20%. [एकल समतुल्य वृद्धि निकालें; यदि एक संख्या को क्रमिक रूप से 10%, 15% तथा 20% से बढ़ा दिया जाता है.] [Ans. (a) 51.8%]

(a) 51.8% (b) 58.1% (c) 50.8% (d) 51.2% (e) None of these  

Solution:

Detailed Explanation: The successive increase in a number are 10%, 15% & 20%.We have to find a single equivalent increase.

First Method:

Concept: The order of percentage change can be changed and by doing this, the equivalent change is not affected. Easier value like 10% is used in the last, so that calculation becomes easy.

Let the number is 100 units. When it increased by 20%, it becomes 100 + 20% of `00 = 100 + 20 = 120

It is again increased by 15%, then it becomes 120 + 15% of 120 = 120 + 18 = 138

Finally, it is increased by 10%, then it becomes 138 + 10% of 138 = 138 + 13.8 = 151.8

So, the single equivalent increase = (151.8-100)% = 51.8% Ans.

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Second Method: When any quantity is changed by x%, y% and z% successively, then the quantity is changed by

x + y + z + (xy + yz + zx)/100 + (xyz)/(100²)%

= 10 + 15 + 20 + (10×15 + 15×20 +20×10)/100 + (10×15×20)/(100²)%

= (45 + 6.5 + 0.3)% = 51.8% Ans.

Third Method: When any quantity is changed by x% and y% successively, then the quantity is changed by

[x + y + (xy)/100]%

We have three successive change, so we take two of them first and equivalent change to these two is determined, then again apply the formula for this single equivalent change and the third change, so that the final single equivalent change is determined.

The single equivalent change of 10% and 15% is given by

[10 + 15 + (10×15)/100]% = [25 + 1.5] % = 26.5%

Again using the formula to determine the single equivalent change of 26.5% and 20%, we get final single equivalent change as

[26.5 + 20 + (26.5×20)/100]% = [46.5 + 5.3] % = 51.8% Ans.