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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)
Example1 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 rewritten for original amount when increased amount and percentage increase are given as Original Amount = ((Increased Amount) * (100)) / (100 + Percentage Increase) (2)
Example2 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)
Example3 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 rewritten for original amount when decreased amount and percentage decrease are given as Original Amount = ((Decreased Amount) * (100 – Percentage Decrease)) / (100) (4)
Example4 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)
Example5 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 rewritten for cost price when selling price and percentage profit are given as Cost Price = ((Selling Price) x (100)) / (100 + Percentage Profit) © (6)
Example6 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)
Example7 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 rewritten for cost price when selling price and percentage loss are given as Cost Price = ((Selling Price) x (100)) / (100 – Percentage Loss) © (8)
Example8 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)
Example9 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 rewritten for marked price when discounted price and percentage discount is given as Marked Price = ((Discounted Price) x (100)) / (100 – Percentage Discount) © (10)
Example10 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)
Example11 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 rewritten for original salary when new salary and percentage increase are given as Original Salary = ((New Salary) x (100)) / (100 + Percentage Increase in Salary) (12)
Example12 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)
Example13 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 rewritten 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)
Example14 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)
Example15 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 rewritten 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)
Example16 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)
Example17 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 rewritten 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)
Example18 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.
AnnexureI 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 + ProzentsatzZunahme)) /(100)
FORMEL  2. Ursprüngliche Menge = ((erhöhte Menge) * (100)) /(100 + ProzentsatzZunahme)
FORMEL  3. Verringerte Menge = ((ursprüngliche Menge) * (100)) /(100  ProzentsatzVerlust)
FORMEL  4. Ursprüngliche Menge = ((verringerte Menge) * (100  ProzentsatzAbnahme)) /(100)
FORMEL  5. Verkaufspreis = ((Kostenpreis) x (100 + ProzentsatzProfit)) /(100)
FORMEL  6. Kostenpreis = ((Verkaufspreis) x (100)) /(100 + ProzentsatzProfit)
FORMEL  7. Verkaufspreis = ((Kostenpreis) x (100  ProzentsatzVerlust)) /(100)
FORMEL  8. Kostenpreis = ((Verkaufspreis) x (100)) /(100  ProzentsatzVerlust)
FORMEL  9. Gediskontierter Preis = ((markierter Preis) x (100 
ProzentsatzDiskont)) /(100)
FORMEL  10. Markierter Preis = ((gediskontierter Preis) x (100)) /(100  ProzentsatzDiskont)
FORMEL  11. Neues Gehalt = ((ursprüngliches Gehalt) x (100 + ProzentsatzZunahme des Gehaltes)) /(100)
FORMEL  12. Ursprüngliches Gehalt = ((neues Gehalt) x (100)) /(100 + ProzentsatzZunahme des Gehaltes)
FORMEL  13. Neues Gehalt = ((ursprüngliches Gehalt) x (100  ProzentsatzAbnahme am Gehalt)) /(100)
FORMEL  14. Ursprüngliches Gehalt = ((neues Gehalt) x (100)) /(100  ProzentsatzAbnahme am Gehalt)
FORMEL  15. Neue Länge = ((ursprüngliche Länge) x (100 + ProzentsatzZunahme der Länge)) /(100)
FORMEL  16. Ursprüngliche Länge = ((neue Länge) x (100)) /(100 + ProzentsatzZunahme der Länge)
FORMEL  17. Neue Länge = ((ursprüngliche Länge) x (100  ProzentsatzAbnahme an der Länge)) /(100)
FORMEL  18. Ursprüngliche Länge = ((neue Länge) x (100)) /(100  ProzentsatzAbnahme 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)
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