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CAT 2025 Lesson : Percentages - Product Constancy

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5. Product Constancy

Product constancy arises when 22 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 1100\dfrac{1}{100}. This is detailed in Section 2.4. For instance, 25% = 25 ×1100=114\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 - x\bm{x} and y\bm{y}, is constant,

- if x\bm{x} increases by ab\dfrac{a}{b} of itself, then y\bm{y} decreases by ab+a\dfrac{a}{b + a} of itself

- if x\bm{x} decreases by ab\dfrac{a}{b} of itself, then y\bm{y} increases by ab−a\dfrac{a}{b - a} of itself

Example 18

If the length of a rectangle is to be increased by 20%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 15\dfrac{1}{5}.

∴ Breadth has decreased by 15+1=16=16×\dfrac{1}{5 + 1} = \dfrac{1}{6} = \dfrac{1}{6} \times 100% = 16.67%

Answer: 16.67%16.67\%

Example 19

A family spends a fixed amount on rice every month. When the price of rice increased to Rs. 66/kg, it reduced the quantity purchased by 88 kg. When the price reduced to Rs. 44/kg, the quantity purchased by the family increased by 1212 kg. If the total amount spent on rice remained the same throughout, what is it?

Solution

Expense = Price ×\times Quantity

Let xx 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 xx
4 x+20x + 20

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

When price decreases by 13\dfrac{1}{3} from Rs. 6, the quantity increases by 13−1=12\dfrac{1}{3 - 1} = \dfrac{1}{2} from xx kg.

∴ 12×x=20\dfrac{1}{2} \times x = 20

⇒ xx = 40 kg

Amount spent on rice = 6×406 \times 40 = Rs. 240

Answer: Rs. 240

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