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HANDBOOK OF PERCENTAGE INCREASE AND PERCENTAGE DECREASE FORMULA APPLICATION APPROACH

2008 EDITION CREATIVE RESEARCHER

CHAUDHARY IMRAN SARWAR, ZAIB SARWAR, CHAUDHARY MUHAMMAD SARWAR, MUHAMMAD ADNAN, CHAUDHARY USMAN SARWAR, ZEESHAN SARWAR, AMEER ANEES GUL & DR MUHAMMAD QAISAR SHAHBAZ

PREFACE

The aim of this handbook is to present formulae to connect two variables such that one variable can be obtained by percentage increase or percentage decrease in the other e.g., to connect cost price with selling price, marked price with discounted price, original salary with new salary and original length with new length etc. Solved examples are also presented to illustrate the different possible scenarios for application of formulae. The above concepts are part of nearly every course on mathematics for secondary school, higher secondary school and undergraduate level throughout the world. So far these formulae are not introduced by any book so the students have to face problem in solving such questions and much of their time is wasted in hit and trial or thinking to solve such questions. The formulae may have extremely high generalizability like that of Pythagorean Theorem. Translation of some formulae in French, German and Italian languages are also given as annexure.

Introduction Nearly all courses on mathematics at middle school, secondary school, higher secondary school and undergraduate level have sections devoted to the concepts of percentage increase or percentage decrease e.g., cost price, selling price, marked price and discounted price etc. Similarly in engineering as well as physics we find concepts of new length obtained after percentage increase or decrease in original length due to temperature changes etc. But no text offers formulae to connect one variable with the other variable obtained after percentage increase or percentage decrease in the first variable. The books (1) and (2) may be consulted for reference. Our formulae will make concepts easier to handle and facilitate in saving time. Various forms of formulae along with the solved examples are presented. The formulae may have extremely high generalizability like that of Pythagorean Theorem.

CHAPTER 1 PERCENTAGE INCREASE AND PERCENTAGE DECREASE; GENERAL APPROACH

General Formulae for Percentage Increase or Percentage Decrease We start with general formula for percentage increase. The formula is to determine the increased amount when original amount and percentage increase are given. The formula is Increased Amount = ((Original Amount)*(100 + Percentage Increase)) / (100) (1)

Example-1 Original amount received by a landlord from his tenants is \$ 10000 per month. What will be the increased amount received after an increase of 15 % in the rent? By applying the above formula (1) we get Increased Amount = ( ( 10000 ) * ( 100 + 15 ) ) / ( 100 ) Increased Amount = \$ 11500

The formula (1) can be re-written for original amount when increased amount and percentage increase are given as Original Amount = ((Increased Amount) * (100)) / (100 + Percentage Increase) (2)

Example-2 Calculate the original amount of rent paid by a tenant if the increased amount is \$ 10000 after an increase of 18 % ? By applying the above formula (2) we get Original Amount = ( ( 10000 ) * ( 100 ) ) / ( 100 + 18 ) Original Amount = \$ 8474.58

When original amount and percentage decrease are given the formula for decreased amount can be given as Decreased Amount = ((Original Amount) * (100)) / (100 – Percentage Decrease) (3)

Example-3 Original amount paid by a traveler for a flight from New York to London is \$ 900. Calculate the decreased amount paid by him after 12 % decrease in the air fare? By applying the above formula (3) we get Decreased Amount = ( ( 900 ) * ( 100 ) ) / ( 100 - 12 ) Decreased Amount = \$ 1022.73

This formula (3) can be re-written for original amount when decreased amount and percentage decrease are given as Original Amount = ((Decreased Amount) * (100 – Percentage Decrease)) / (100) (4)

Example-4 Calculate the original amount of air fare to be paid by a traveler if the decreased amount of air fare is \$ 900 after a decrease of 9 % in air fare? By applying the above formula (4) we get Original Amount = ( ( 900 ) * ( 100 - 9 ) ) / ( 100 ) Original Amount = \$ 819

CHAPTER 2 Cost Price and Selling Price

Formulae for Cost Price and Selling Price © We can utilize the above concepts for specific applications. We start with the formulae connecting cost price with selling price. First such formula is for determining the selling price when cost price and percentage profit are given. The formula is Selling Price = ((Cost Price) x (100 + Percentage Profit)) / (100) © (5)

Example-5 By purchasing a calculator at \$ 100 and to make a profit of 20 %, at what price must it be sold? By applying the above formula (5) we get Selling Price = ( ( 100 ) * ( 100 + 20 ) ) / ( 100 ) Selling Price = \$ 120

The above formula (5) can be re-written for cost price when selling price and percentage profit are given as Cost Price = ((Selling Price) x (100)) / (100 + Percentage Profit) © (6)

Example-6 To make a profit of 33.3 %, a bicycle is sold for \$ 240. What is the cost price of the bicycle? By applying the above formula (6) we get Cost Price = ( ( 240 ) * ( 100 ) ) / ( 100 + 33.3 ) Cost Price = \$ 180

When cost price and percentage loss are given the formula for selling price can be given as Selling Price = ((Cost Price) x (100 – Percentage Loss)) / (100) © (7)

Example-7 If a car is sold at a loss of 6 % , what is the selling price when the cost price is \$ 18400 for it? By applying the above formula (7) we get Selling Price = ( ( 1840) * ( 100 – 6 ) ) / ( 100) Selling Price = \$ 17296

The above formula (7) can be re-written for cost price when selling price and percentage loss are given as Cost Price = ((Selling Price) x (100)) / (100 – Percentage Loss) © (8)

Example-8 By selling a book for \$ 16.50, a loss of 12 % is encountered. What is the cost price of the book? By applying the above formula (8) we get; Cost Price = ( ( 16.50 ) * ( 100 ) ) / ( 100 – 12 ) Cost Price = \$ 18.75

CHAPTER 3 MARKED PRICE AND DISCOUNTED PRICE

Formulae for Marked Price and Discounted Price© The third set of formulae connects marked price with discounted price. In a pattern similar to that given above in case of discounted price when marked price and percentage discount is given we have Discounted Price = ((Marked Price) x (100 – Percentage Discount)) / (100) © (9)

Example-9 Marked price of a calculator is \$ 200. What will be the discounted price if 16 % discount is offered? By applying the above formula (9) we get; Discounted Price = ( ( 200 ) * ( 100 - 16 ) ) / ( 100 ) Discounted Price = \$ 168

The above formula (9) can be re-written for marked price when discounted price and percentage discount is given as Marked Price = ((Discounted Price) x (100)) / (100 – Percentage Discount) © (10)

Example-10 Calculate the marked price of a book if the discounted price is \$ 125 after a discount of 22 % ? By applying the above formula (10) we get; Marked Price = ( ( 125 ) * ( 100 ) ) / ( 100 - 22 ) Marked Price = \$ 160.26

CHAPTER 4 ORIGINAL SALARY AND NEW SALARY

Formulae for Original Salary and New Salary The fourth set of formulae connects original salary with new salary. When original salary and percentage increase in salary are given the new salary may be calculated as New Salary = ((Original Salary) x (100 + Percentage Increase in Salary)) / (100) (11)

Example-11 Calculate the new salary of an employee if his original salary was \$ 98000 per annum and 14 % increase in salary is announced? By applying the above formula (11) we get; New Salary = ( ( 98000 ) * ( 100 + 14 ) ) / ( 100 ) New Salary = \$ 111720

The above formula (11) can be re-written for original salary when new salary and percentage increase are given as Original Salary = ((New Salary) x (100)) / (100 + Percentage Increase in Salary) (12)

Example-12 If the new salary of an employee is \$ 135000 per annum after an increase of 16 % in the salary then calculate his original salary? By applying the above formula (12) we get; Original Salary = ( ( 135000 ) * ( 100 ) ) / ( 100 + 16 ) Original Salary = \$ 116379.31

In case of percentage decrease in salary when original salary is given the new salary may be determined as New Salary = ((Original Salary) x (100 – Percentage Decrease in Salary)) / (100) (13)

Example-13 Calculate the new salary of an employee if her original salary was \$ 112000 per annum and she has to face a 4 % decrease in salary due to car loan? By applying the above formula (13) we get; New Salary = ( ( 112000 ) * ( 100 - 4 ) ) / ( 100 ) New Salary = \$ 107520

The above formula (13) can be re-written for original salary when new salary and percentage decrease in salary are given as Original Salary = ((New Salary) x (100)) / (100 – Percentage Decrease in Salary) (14)

Example-14 Calculate the original salary of an employee if her new salary is \$ 164000 per annum and she has to face a 3 % decrease in salary due to car loan? By applying the above formula (14) we get; Original Salary = ( ( 164000 ) * ( 100 ) ) / ( 100 - 3 ) Original Salary = \$ 169072.16

CHAPTER 5 Original Length and New Length

Formulae for Original Length and New Length The fifth set of formulae connects original length with new length. When original length and percentage increase in length are given the new length may be determined by New Length = ((Original Length) x (100 + Percentage Increase in Length)) / (100) (15)

Example-15 Due to increase in temperature the length of an alloy rod increases by 6 % . If the original length was 94 meters find the new length?

```By applying the above formula (15) we get;
```

New Length = ( ( 94 ) * ( 100 + 6 ) ) / ( 100 ) New Length = 99.64 meters The above formula (15) can be re-written for original length when new length and percentage increase in length are given as Original Length = ((New Length) x (100)) / (100 + Percentage Increase in Length) (16)

Example-16 Due to increase in temperature the length of an alloy rod increases by 4 % . Calculate the original length if the new length is 115 meters?

```By applying the above formula (16) we get;
```

Original Length = ( ( 115 ) * ( 100 ) ) / ( 100 + 3 ) Original Length = 111.65 meters

In case of percentage decrease in length when original length is given the new length may be determined by New Length = ((Original Length) x (100 – Percentage Decrease in Length)) / (100) (17)

Example-17 Due to decrease in temperature the length of an alloy rod decreases by 8 % . If the original length was 164 meters find the new length?

```By applying the above formula (17) we get;
```

New Length = ( ( 164 ) * ( 100 - 8 ) ) / ( 100 ) New Length = 150.88 meters

The above formula (17) can be re-written for original length when new length and percentage decrease in length are given as Original Length = ((New Length) x (100)) / (100 – Percentage Decrease in Length) (18)

Example-18 Due to decrease in temperature the length of an alloy rod decreases by 7 % . Calculate the original length if the new length is 209 meters?

```By applying the above formula (18) we get;
```

Original Length = ( ( 164 ) * ( 100 ) ) / ( 100 - 7 ) Original Length = 176.34 meters

References 1. Seng, T. K. and Keong, L. C. (1996). New Syllabus D Mathematics 2. Yee, L. P. (Ed). Karachi: Oxford University Press. 2. Lin, L. S., and Hoong, H. C. K. (1999). New Mathematics for Secondary Schools: Syllabus D1. Meng, S. K. (Ed). Singapore: Paramount Publishing Enterprise.

Annexure-I FORMULAE IN ENGLISH

FORMULA - 1. Increased Amount = ((Original Amount)*(100 + Percentage Increase)) / (100)

FORMULA - 2. Original Amount = ((Increased Amount) * (100)) / (100 + Percentage Increase)

FORMULA - 3. Decreased Amount = ((Original Amount) * (100)) / (100 – Percentage Decrease)

FORMULA - 4. Original Amount = ((Decreased Amount) * (100 – Percentage Decrease)) / (100)

FORMULA - 5. Selling Price = ((Cost Price) x (100 + Percentage Profit)) / (100)

FORMULA - 6. Cost Price = ((Selling Price) x (100)) / (100 + Percentage Profit)

FORMULA - 7. Selling Price = ((Cost Price) x (100 – Percentage Loss)) / (100)

FORMULA - 8. Cost Price = ((Selling Price) x (100)) / (100 – Percentage Loss)

FORMULA - 9. Discounted Price = ((Marked Price) x (100 – Percentage Discount)) / (100)

FORMULA - 10. Marked Price = ((Discounted Price) x (100)) / (100 – Percentage Discount)

FORMULA - 11. New Salary = ((Original Salary) x (100 + Percentage Increase in Salary)) / (100)

FORMULA - 12. Original Salary = ((New Salary) x (100)) / (100 + Percentage Increase in Salary)

FORMULA - 13. New Salary = ((Original Salary) x (100 – Percentage Decrease in Salary)) / (100)

FORMULA - 14. Original Salary = ((New Salary) x (100)) / (100 – Percentage Decrease in Salary)

FORMULA - 15. New Length = ((Original Length) x (100 + Percentage Increase in Length)) / (100)

FORMULA - 16. Original Length = ((New Length) x (100)) / (100 + Percentage Increase in Length)

FORMULA - 17. New Length = ((Original Length) x (100 – Percentage Decrease in Length)) / (100)

FORMULA - 18. Original Length = ((New Length) x (100)) / (100 – Percentage Decrease in Length)

FORMULAE IN FRENCH

FORMULE - 1. Quantité accrue = ((quantité originale) * (100 + augmentation de pourcentage)) /(100)

FORMULE - 2. Quantité originale = ((quantité accrue) * (100)) /(100 + augmentation de pourcentage)

FORMULE - 3. Quantité diminuée = ((quantité originale) * (100)) /(100 - perte de pourcentage)

FORMULE - 4. Quantité originale = ((quantité diminuée) * (100 - diminution de pourcentage)) /(100)

FORMULE - 5. Prix de vente = ((prix d'achat) x (100 + bénéfice de pourcentage)) /(100)

FORMULE - 6. Prix d'achat = ((prix de vente) x (100)) /(100 + bénéfice de pourcentage)

FORMULE - 7. Prix de vente = ((prix d'achat) x (100 - perte de pourcentage)) /(100)

FORMULE - 8. Prix d'achat = ((prix de vente) x (100)) /(100 - perte de pourcentage)

FORMULE - 9. Prix escompté = ((prix marqué) x (100 - escompte de pourcentage)) /(100)

FORMULE - 10. Prix marqué = ((prix escompté) x (100)) /(100 - escompte de pourcentage)

FORMULE - 11. Nouveau salaire = ((salaire original) x (100 + augmentation de pourcentage de salaire)) /(100)

FORMULE - 12. Salaire original = ((nouveau salaire) x (100)) /(100 + augmentation de pourcentage de salaire)

FORMULE - 13. Nouveau salaire = ((salaire original) x (100 - diminution de pourcentage de salaire)) /(100)

FORMULE - 14. Salaire original = ((nouveau salaire) x (100)) /(100 - diminution de pourcentage de salaire)

FORMULE - 15. Nouvelle longueur = ((longueur originale) x (100 + augmentation de pourcentage de longueur)) /(100)

FORMULE - 16. Longueur originale = ((nouvelle longueur) x (100)) /(100 + augmentation de pourcentage de longueur)

FORMULE - 17. Nouvelle longueur = ((longueur originale) x (100 - diminution de pourcentage de longueur)) /(100) FORMULE - 18. Longueur originale = ((nouvelle longueur) x (100)) /(100 - diminution de pourcentage de longueur)

FORMULAE IN GERMAN

FORMEL - 1. Erhöhte Menge = ((ursprüngliche Menge) * (100 + Prozentsatz-Zunahme)) /(100)

FORMEL - 2. Ursprüngliche Menge = ((erhöhte Menge) * (100)) /(100 + Prozentsatz-Zunahme)

FORMEL - 3. Verringerte Menge = ((ursprüngliche Menge) * (100)) /(100 - Prozentsatz-Verlust)

FORMEL - 4. Ursprüngliche Menge = ((verringerte Menge) * (100 - Prozentsatz-Abnahme)) /(100)

FORMEL - 5. Verkaufspreis = ((Kostenpreis) x (100 + Prozentsatz-Profit)) /(100)

FORMEL - 6. Kostenpreis = ((Verkaufspreis) x (100)) /(100 + Prozentsatz-Profit)

FORMEL - 7. Verkaufspreis = ((Kostenpreis) x (100 - Prozentsatz-Verlust)) /(100)

FORMEL - 8. Kostenpreis = ((Verkaufspreis) x (100)) /(100 - Prozentsatz-Verlust)

FORMEL - 9. Gediskontierter Preis = ((markierter Preis) x (100 - Prozentsatz-Diskont)) /(100)

FORMEL - 10. Markierter Preis = ((gediskontierter Preis) x (100)) /(100 - Prozentsatz-Diskont)

FORMEL - 11. Neues Gehalt = ((ursprüngliches Gehalt) x (100 + Prozentsatz-Zunahme des Gehaltes)) /(100)

FORMEL - 12. Ursprüngliches Gehalt = ((neues Gehalt) x (100)) /(100 + Prozentsatz-Zunahme des Gehaltes)

FORMEL - 13. Neues Gehalt = ((ursprüngliches Gehalt) x (100 - Prozentsatz-Abnahme am Gehalt)) /(100)

FORMEL - 14. Ursprüngliches Gehalt = ((neues Gehalt) x (100)) /(100 - Prozentsatz-Abnahme am Gehalt)

FORMEL - 15. Neue Länge = ((ursprüngliche Länge) x (100 + Prozentsatz-Zunahme der Länge)) /(100)

FORMEL - 16. Ursprüngliche Länge = ((neue Länge) x (100)) /(100 + Prozentsatz-Zunahme der Länge)

FORMEL - 17. Neue Länge = ((ursprüngliche Länge) x (100 - Prozentsatz-Abnahme an der Länge)) /(100)

FORMEL - 18. Ursprüngliche Länge = ((neue Länge) x (100)) /(100 - Prozentsatz-Abnahme an der Länge)

FORMULAE IN ITALIAN

FORMULA - 1. Importo aumentato = ((importo originale) * (100 + aumento di percentuale)) /(100)

FORMULA - 2. Importo originale = ((importo aumentato) * (100)) /(100 + aumento di percentuale)

FORMULA - 3. Importo diminuito = ((importo originale) * (100)) /(100 - perdita di percentuale)

FORMULA - 4. Importo originale = ((importo diminuito) * (100 - diminuzione di percentuale)) /(100)

FORMULA - 5. Prezzo di vendita di vendita = ((prezzo di costo) x (100 + profitto di percentuale)) /(100)

FORMULA - 6. Prezzo di costo = ((prezzo di vendita di vendita) x (100)) /(100 + profitto di percentuale)

FORMULA - 7. Prezzo di vendita di vendita = ((prezzo di costo) x (100 - perdita di percentuale)) /(100)

FORMULA - 8. Prezzo di costo = ((prezzo di vendita di vendita) x (100)) /(100 - perdita di percentuale)

FORMULA - 9. Prezzo scontato = ((profondo prezzo) x (100 - sconto di percentuale)) /(100)

FORMULA - 10. Profondo prezzo = ((prezzo scontato) x (100)) /(100 - sconto di percentuale)

FORMULA - 11. Nuovo stipendio = ((stipendio originale) x (100 + aumento di percentuale nello stipendio)) /(100)

FORMULA - 12. Stipendio originale = ((nuovo stipendio) x (100)) /(100 + aumento di percentuale nello stipendio)

FORMULA - 13. Nuovo stipendio = ((stipendio originale) x (100 - diminuzione di percentuale nello stipendio)) /(100)

FORMULA - 14. Stipendio originale = ((nuovo stipendio) x (100)) /(100 - diminuzione di percentuale nello stipendio)

FORMULA - 15. Nuova lunghezza = ((lunghezza originale) x (100 + aumento di percentuale di lunghezza)) /(100)

FORMULA - 16. Lunghezza originale = ((nuova lunghezza) x (100)) /(100 + aumento di percentuale di lunghezza)

FORMULA - 17. Nuova lunghezza = ((lunghezza originale) x (100 - diminuzione di percentuale di lunghezza)) /(100) FORMULA - 18. Lunghezza originale = ((nuova lunghezza) x (100)) /(100 - diminuzione di percentuale di lunghezza)

CONTACT DETAILS 20, A-I, TOWNSHIP, LAHORE, PAKISTAN 54000 TELEPHONE # +0923454634211, +092425007777 E-MAIL: creativeresearcher111@yahoo.com

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