# 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 x | Decrease in y | Change 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 x | Increase in y | Change 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%

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%

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%