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Percentages

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CAT 2025 Lesson : Percentages - Product Constancy

5. Product Constancy

Product constancy arises when

2

variables are inversely proportional to each other. For example,
- Expenditure = Price

\times

Quantity
- Area = Length

\times

Breadth
- Distance = Speed

\times

Time

The two variables in the RHS (Right-Hand Side) of the equations above are inversely proportional. In other words in the first equation, if expenditure remains constant, then an increase in price will result in a decrease in quantity.

To compute the change in values, we recommend the following method.

We first convert the percentage to a fraction. We do this by replacing % symbol with its value

\dfrac{1}{100}

. This is detailed in Section 2.4. For instance, 25% = 25

\times \dfrac{1}{100} = \dfrac{1}{14}

The product constancy rule for increase/decrease expressed as a fraction, with 1 in the numerator is very simple and is the recommended method.

Where the product of two variables -

\bm{x}

and

\bm{y}

, is constant,

- if

\bm{x}

increases by

\dfrac{a}{b}

of itself, then

\bm{y}

decreases by

\dfrac{a}{b + a}

of itself

- if

\bm{x}

decreases by

\dfrac{a}{b}

of itself, then

\bm{y}

increases by

\dfrac{a}{b - a}

of itself

Example 18

If the length of a rectangle is to be increased by

20\%

, by what percentage should the breadth be reduced to ensure that the area remains the same?

Solution

Area of Rectangle = Length

\times

Breadth

Length has increased by 20% or

\dfrac{1}{5}

.

∴ Breadth has decreased by

\dfrac{1}{5 + 1} = \dfrac{1}{6} = \dfrac{1}{6} \times

100% = 16.67%

Answer:

16.67\%

Example 19

A family spends a fixed amount on rice every month. When the price of rice increased to Rs.

6

/kg, it reduced the quantity purchased by

8

kg. When the price reduced to Rs.

4

/kg, the quantity purchased by the family increased by

12

kg. If the total amount spent on rice remained the same throughout, what is it?

Solution

Expense = Price

\times

Quantity

Let

x

kg be the quantity of rice purchased at Rs. 6/kg. Quantity purchased at Rs. 4/kg is 8 + 12 = 20 kg more than that purchased at Rs. 6/kg.

Price (Rs./kg)	Quantity (kg)
6	$x$
4	$x + 20$

∴ When price per kg reduced from Rs. 6 to Rs. 4, quantity of rice increased by 20 kg.

When price decreases by

\dfrac{1}{3}

from Rs. 6, the quantity increases by

\dfrac{1}{3 - 1} = \dfrac{1}{2}

from

x

kg.

∴

\dfrac{1}{2} \times x = 20

⇒

x

= 40 kg

Amount spent on rice =

6 \times 40

= Rs. 240

Answer: Rs. 240

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