### Constant Product Rule

First of all consider the situation when or where the constant product rule is applicable.

If the distance between two cities P and Q is 120 km and a car X is travel from P to Q at a constant speed of 30 kmph, then it will take 4 hours to cover the mentioned distance. If the car will increase its speed to 40kmph, then it can cover the distance in 3 hours.

This rule is one of key strategic application in most of the percentage related problems. So the aspirant should be skillful in the effective application of the constant product rule.

Here the rule explained below;

**Method I: Percentage approach.**

If the product of two inversely proportional quantities X and Y is a constant, then;

Variation in X | Resultant effect in Y |
---|---|

r % increase | (100 * r / 100 + r) decrease |

r % decrease | (100 * r / 100 - r) decrease |

**Example 1:**

If the length of a rectangle is increased by 20% then how much percentage decrease should be in its breadth to keep the same area?

**Solution:**

Area = length * breadth.

Area is a constant product of the factor values length and breadth.

20% increment in length → (20 * 100) / (100 + 20) % decrease in breadth.

Hence the breadth should decrease by 16.66% to keep the same area.

**Example 2:**

If the speed of a train is decreased by 25%, then how much percentage more time is required to cover a constant distance?

**Solution:**

25% decrease in speed → (25 * 100) / (100 - 25) % increase in time.

Hence the train will take 33.33% more time to cover the same constant distance.

**Method II: Fractional approach.**

In this fractional approach of the constant product rule, we are expressing the percentage values in its corresponding fractional form. The calculation process with the percentage-equivalent-fractions will support the student to do the mental calculations so easily and quickly.

If the product of two inversely proportional quantities X and Y is a constant, then;

Variation in X | Resultant variation in Y |
---|---|

a/b increase | a/(b+a) decrease |

a/b decrease | a/(b-a) increase |

**Example 1:**

Travelling from city A to B a car keep a constant speed of 30 kmph and while the return journey from B to A, it increased the speed to 40 kmph. So the time taken for the return journey is how much percentage more or less than that for the first half of the trip?

**Solution:**

Speed for A to B = 30 kmph.

Speed for B to A = 40 kmph.

Increase in speed = 10 kmph.

Fraction of increment in speed = 10/30 = 1/3

As per the rule; 1/3 increment in speed → 1/(3+1) decrease in time.

Therefore the time consumption is decreased by 1/4 or 25%.

The above discussed three main pre-requisites are frequently applicable in most of the Profit and Loss questions. Hence, you should prepare well on the above three Percentage applications.

## Basic Terminologies.

**Cost Price (CP) :** This is the real worth of a product. From customer's end **CP **is the amount he spent for

purchasing a product.

**Selling Price (SP):** This is the revenue generated from a product while selling it. From the merchant's end

**SP** is the amount he received from the customer while selling a product to customer.

**Profit (P)/ Gain: **While selling an article if the selling price is greater than cost price then the seller

receives a profit. In this sense **Profit = SP - CP**.

**Loss (L):** if the selling price is lesser than the Cost Price then the seller incurred a loss in the transaction.

Hence **Loss = CP- SP**.

**Profit Percentage:** This is an expression of the profit amount in terms of Cost Price.

Profit % = Profit/Cost Price * 100 or (SP - CP)/Cost Price * 100 .

**Loss Percentage:** Here loss is expressed in terms of Cost Price.

Loss % = Loss % = Loss/Cost Price * 100 or (CP - SP)/Cost Price * 100.

**Marked Price/ Listed Price:** Marked price is the price printed on a product. A customer can see this as the value/price for the product.

**Discount:** If the merchant is selling the product for a price which is lesser than the Marked Price, then he offered a discount to the customer on that particular product.

**Discount = Marked Price - Selling Price.**

**Discount percentage:** Here discount is expressed as the percentage of Marked Price.

Discount Percentage = (Marked Price - Selling Price)/Marked Price * 100 = Discount/Marked Price * 100.

**Illustration:**

If a merchant purchased a product by Rs.100, and he mark up it to Rs.150 finally sold it by Rs.120, then:

CP | SP | MP |
---|---|---|

Rs.100 | Rs.120 | Rs.150 |

- Profit = SP - CP = Rs.120 - Rs.100 = Rs. 20.
- Profit Percentage = Profit/Cost Price * 100 = 20/100 * 100 = 20%
- Mark up by Rs.50 and Mark up to Rs.150.
- Discount = MP - SP = Rs.150 - Rs. 120 = Rs.30.
- Discount Percentage = Discount/Marked Price * 100 = 30/150 * 100 = 20%