# Product Stability Ratio

In a number of topics/concepts, we encounter the relationship where the product of two quantities equals the third quantity.For example,

Speed Ã— Time = Distance

Price Ã— Consumption = Expenditure

Number of persons Ã— Days = Work done

Length Ã— Breadth = Area of rectangle

If we generalize product stability ratio, it can be written like

Speed Ã— Time = Distance

Price Ã— Consumption = Expenditure

Number of persons Ã— Days = Work done

Length Ã— Breadth = Area of rectangle

Apart form these cases, a number of times we see cases where one quantity is increased to get another quantity, e.g., if we increase cost price to obtain a certain profit, we obtain selling price or if we increase principal, we obtain amount.

If we generalize product stability ratio, it can be written like

A Ã— B = P

Now, if A is increased by a certain percentage, then B is required to be decreased by certain percentage so that the product (P) remains stable.

For example, if we increase A by 25% and P has to be constant, then B is required to be decreased by 20%.

This procedure can be summed up in the following way:

So, if A is increased by 25%, then we need to decrease B by 20% to maintain the product stable.

**This one mathematical information can be used in so many forms:**

**Percentage**If A is 25% more than B, then by how much percentage B is less than A?**Profit and Loss**An article is sold for Rs 125 at a profit of 25%. What is the cost price of the article?**Time, Speed****and Distance (TSD)**When speed of a car is increased by 25%, time taken reduces by 20 minutes in covering a certain distance. What is the actual time taken to cover the same distance by actual speed?**TSD**Mayank goes to his office from his home at a speed of 20 kmph and gets late by 10 minutes. However, when he increases his speed to 25 kmph, he is 20 minutes early. What is the distance from his office to his home?**Time and Work**Efficiency of Amit is 25% more than Vinit. Vinit takes 20 days to complete a work. How many days will Amit take to do the same work?**Time and Work**20 men can do some job in 50 days. In how many days will 25 men do the same job?**Simple Interest (SI)**Rate of interest is 12.5% per annum SI. What is the principal if amount obtained after two years is Rs 1250?**Percentage**Due to a price hike of 25%, 5 kg less sugar is available for Rs 100. What is the original price per kg?**Mensuration**Length of a rectangle is increased by 25%. By what percentage should the breadth be decreased so that area remains constant?

In all the above written situations, just one mathematical information has been used, i.e., if A is increased by 25%, then B decreases by 20%. Let us see the solution of all the questions given above.

*Solution 1*__Normal Method__

Let us assume B = 100, then A = 125

Now, B is 25 less than A.

Percentage of B is less than A = 25/125 Ã— 100 = 20%

**Product Stability Ratio Method**

Using product stability rule, since A is 25% more than B, so B is 20% less than A.

*Solution 2*__Normal Method__

CP Ã— 1.25 = SP

So, CP = SP/1.25 = 125/1.25 = Rs 100

**Product Stability Ratio Method**

If we increase CP by 25%, we will get SP.

So, if we decrease SP by 20%, we will get CP.

Hence CP = Rs 100

Solution 3Solution 3

__Normal Method__

Since we know S = V Ã— T (Distance = Speed Ã— Time)

New speed = 1.25 V, so new time = T/1.25

So, reduction in time = Tâ€“T/1.25 = 0.25 T/1.25 = T/5

T/5 = 20 min â‡’ T = 100 min

Product Stability Ratio Method

Product Stability Ratio Method

**Since speed has been increased by 25%, so time will reduce by 20%.**

Now, 20% T(Time) = 20 min

So, Total time = 100 min

**Solution 4**

__Normal Method__

Let us assume that distance = D

So, D/20 â€“ D/25 = 30/60 hr. = Â½

So, D = 50 km

**Product Stability Ratio Method**

**S = V Ã— T**

25% â†‘ 20%â†“

So, 20% T = 30 min

â‡’ T = 150 mins = 2Â½

So, total distance = 20 Ã— 2Â½ = 50 km

*Solution 5*__Normal Method__
Vinit is taking 20 days to complete the work i.e., Vinit is doing 100% work in 20 days. So, Vinit is doing 5% work in a day. Since efficiency of Amit is 25% more than Vinit, so Amit is doing 6.25% work per day.

So, number of days taken by Amit = 100/6.25 days = 16 days

**Product Stability Ratio Method**

**Efficiency of Amit is 25% more than Vinit. So, Amit will take 20% less days than Vinit.**

So, number of days taken by Amit = 16 days

**Solution 6**__Normal Method__

Using Work = Number of persons Ã— Number of days

Work = 20Ã—50 = 1000

Now, 1000 = 25Ã—D

So, D = 40

**Product Stability Ratio Method**

Number of persons increases by 25%, so number of days will decrease by 20%.

So, number of days = 40 days

**Solution 7**__Normal Method__

Using the formula for SI = PRT/100

P = (SIÃ—100)/RT

Putting the values gives us P = Rs 1,000

**Product Stability Ratio Method**

**Interest for two years = 25%**

So, if we decrease the amount by 20%, then we will get the principal.

Hence, Principal = Rs 1,000

*Solution 8*__Normal Method__

Let us assume that original price per kg = Rs P per kg

So, final price per kg = Rs 1.25 P

Hence, (120/P) â€“ (120/1.25P) = 5

Solving this equation gives P = Rs 4 per kg

**Product Stability Ratio Method**

Since, the price hike is 25%, 20% less quantity of sugar will be available for Rs 100.

20% = 5 kg â‡’ 100% = 25 kg

So, 25 kgs were available for Rs 100 initially.

So, Price = Rs 4/kg

*Solution 9*__Normal Method__

Length |
Breadth |
Area |
||||

Initiallyâ€¦ | â†’ | 10 | Ã— | 10 | = | 100 |

Finallyâ€¦ | â†’ | 12.5 | Ã— | B | = | 100 |

So, B = 8

Percentage decrease = 20%

**Product Stability Ratio Method**

Till now, it must have become very obvious that the breadth will decrease by 20% to keep the area constant.

# Extension of Product Stability Ratio

This table is a two-way table, i.e., if we decrease A by 50%, then B is needed to be decreased by 100%.If we express the percentage figures given in the product stability table in ratios, then it comes like the following:

So, corresponds to . It means that if we increase A by 2%, then B is needed to be decreased by 1.96% (approx.) so that P remains constant.

Similarly, it can be done with all the reciprocals.
But the problem which lies with this table is that it has values which are reciprocals only.
So, what are we required to do if we increase A by 15%?
Take this as

Similarly, it can be done with all the reciprocals.