# Concept of Base Change (xy=constant)

Concept of base change is also applicable when product of two variables is constant.

Let there be two quantities x and y that multiply to form a quantity z. We can say: z = x × y

E.g. Total Expenditure = Price × Quantity
Total Sales = Sales in volume × Price per unit
Total Work = Efficiency × Time
Distance = Speed × Time
Area of rectangle = Length × Breadth

• Say x or y is increased by a%, then:
z = x (1 + 𝑎/100) × y = x y (1 + 𝑎/100) = z (1 + 𝑎/100)
So increasing either x or y means that z is increased by the same percentage.

Hence, we can compensate the increase in x by decreasing x in a proportionate manner.
Or, we can compensate the increase in x by decreasing y in a proportionate manner.
Or, we can compensate the increase in z by decreasing z in a proportionate manner.
Answer will be the same in all the three cases.

• If the x (say price of an item) goes up by p%, then y (the quantity consumed) should be decreased by ($\frac{p}{100+p}$) × 𝟏𝟎𝟎% so that the total expenditure remains the same.
Increase in xDecrease in yChange in z
20%16.66%0
25%20%0
33.33%25%0
50%33.33%0
100%50%0
• If x (say the price of an item) goes down by p%, then y (the quantity consumed) should be increased by ($\frac{p}{100-p}$) × 𝟏𝟎𝟎% so that the total expenditure remains the same.
Decrease in xIncrease in yChange in z
16.66%20%0
20%25%0
25%33.33%0
33.33%50%0
50%100%0
###### Q. Price of a commodity has increased by 60%. By what percent must a consumer reduce the consumption of the commodity so as not to increase the expenditure ?
(a) 37%         (b) 37.5%          (c) 40.5%          (d) 60%

Explanations :

Explanation 1: Using Percentage Method
Expenditure = Amount consumed × Price
So, 60% increase in price is equivalent to 60% increase in expenditure.

Let initial value of Expenditure be 100
60% increase - 160

So, to revert back 160 to 100, the required percentage reduction needed in Expenditure = (60/160) × 100 = (3/8) × 100 = 37.5%
Explanation 2: Using Fraction Method
60% = 3/5

5 corresponds to original number; 3 is the change
As it’s an increase, the final number = 5 + 3 = 8

So, to revert back 8 to 5, the required percentage reduction needed in Expenditure = (3/8) × 100 = 37.5%
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