WOODBROOK SECONDARY SCHOOL MATHEMATICS PERCENTAGES FORM 4 % 1 100

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1 A percentage is a fraction whose denominator is. It is represented using the symbol %, where: % 1 Ex. 5% = 5 1 = 5 Ex. 115% = = 115 Ex % = = = 7 2 Ex % = = = 1 8

2 A percentage is directly related to fractions and decimals by performing simple calculations to convert from one to the next. To convert a percentage into fraction, the value is written over (denominator) and it is reduced into it lowest terms. Ex. 45% = = To convert a percentage into a decimal, divide the value by or move the decimal point two places to the left. NB. Any whole number can be written as a decimal where all the digits after the decimal point is zero. Ex. 27% = 27.0 = 0.27 Ex % = 42.5 = 0.425

3 The following table shows some basic conversions: Percent Decimal Fraction 1% / 5% / 20 10% / 10 12½% / 8 20% / 5 25% / / 3 % / 3 50% / 2 75% / 4 80% / 5 90% / 10 99% / % % / 4 150% / 2 200% NOTE: 1. % is equivalent to Any percent that is less than % is a proper fraction and is less than Any percent that is more than % is an improper fraction and is greater than 1.

4 Percentage of A Quantity To calculate the percentage of a quantity, the percentage is written as a fraction and it is multiplied by the quantity. The following examples illustrates how to find the percentage of a quantity: Example: Calculate 25% of 80 Therefore 25% of 80 is % = = 20 4 Example: 15% of 200 apples were bad. How many apples were bad? 15% = = 30 Therefore 30 apples were bad.

5 Ex. In a class of 120 pupils, 60% were boys. How many boys are there in the class and hence find the number of girls in the class Number of boys in class: = Number of girls in class: 120 = Ex. In a survey done on smoking, 150 questionnaires were distributed. Of the 150 participants, 80% were male and 20% were female. 30% said they smoked as a hobby, 45% said they smoked to relieve stress and 25% said they smoked due to peer pressure. Calculate: a) The number of males and female who took part in the survey b) The number of people who smoked as a hobby c) The number of people who smoked to relieve stress d) The number of people who smoked due to peer pressure

6 Percentage Increase and Percentage Decrease A percentage increase in a quantity represents an increase in the quantity by a certain amount based on the percentage given. Let n be the quantity being increased by x%. There are two ways to determine the new value of n after the increase: 1. Find x% of n then add it to n 2. Find ( + x)% of n The alternative way can be proven to be true by following this procedure: ( + x)% n = (% + x%) n = ( + x ) n = (1 + x ) n This is just the first way of calculating the percentage increase = n + ( x n) Ex. Samantha s allowance for a week was $60. She got an increase of 25%. Determine her new allowance. First determine the amount her allowance increased by: = Add this value to her current allowance: 60 + = Her new allowance is therefore: $

7 Alternative: Samantha s percentage increase is ( + 25)% = 125% Her new allowance is therefore: = = 5 15 = $75 Now consider the following converse problem: Ex. After a 25% increase in Samantha s allowance, she received $75. What was her allowance before the increase? It is known that % represents Samantha s allowance before the increase. Since her allowance increased by 25% then 125% is equivalent to $75. Therefore: 125 n = 75 n = n = 3 5 n = 3 20 n = $60 By multiplying the increased amount by the reciprocal of the percentage increase, the original amount, n can be found.

8 1. In 2002, the price of a doubles was $2.00. In 2014, the price of a double went up by %. Find the price of a doubles in A house was tiled using 10 x 10 tiles. A contractor was hired to tile over the house by using a tile that is 80% larger. What is the dimensions of the tile that has to be bought? 3. After a 10% increase in a boy s height, he was 160cm tall. How tall was the boy originally? 4. After 5 years the population on an island increased by 85% resulting in persons living on the island. How many people were on the island before the increase? 5. In previous years, the pipeline used to deliver water to communities measured Due to recent upgrades and increased pressure in the water supply, the pipeline was increase by % to facilitate the increase in pressure. What is the size if the new pipeline being used? 6. In an experiment to determine the rate at which a seedling grows in bright light and dark conditions, it was found that the seedling in the dark grew 40% the length of a seedling in bright light. If the length of a seedling in dark measured 210cm, find the length of the seedling in bright conditions assuming they both started growing at the same time.

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10 A percentage decrease in a quantity represents a decrease in the quantity by a certain amount based on the percentage given. Let n be the quantity being decreased by x%. There are two ways to determine the new value of n after the decrease: 1. Find x% of n then subtract it from n 2. Find ( x)% of n The alternative way can be proven to be true by following this procedure: ( x)% n = (% x%) n = ( x ) n = (1 x ) n This is just the first way of calculating the percentage decrease = n ( x n) Ex. Decrease 80 by 40%. = = 4 8 = 32 Therefore the new value is: = 48

11 Alternative: WOODBROOK SECONDARY SCHOOL Since % represents the total value (80), after a 40% decrease the value would be equivalent to % - 40% = 60%. Therefore = 6 8 = 48 Now consider the converse problem. Ex. After a 40% decrease in a number, the new value is 48. What was the original value? Since 40% was deducted from the number the remaining is equivalent to 60%. Therefore 60 n = 48 n = n = 8 10 n = 80 By multiplying the decreased amount by the reciprocal of the percentage decrease, the original amount, n can be found. Hence the original number is 80.

12 1. A roll-on roll-off car dealer imports 120 cars per year. Due to slow sales, he was forced to reduce the number of cars he imported by %. How many cars does he import now? 2. During the rainy season, the measured rainfall was 30mm however in the dry season, this decreased by 60%. What is the new measured rainfall in the dry season? 3. Due to high inflation rates, a company had to let go 30% of their employees. If 300 employees were let go, how many employees worked at the company? 4. A show on network television gets picked up or cancelled due to viewership by an audience. The show premiered with 12 million viewers on the pilot episode and its viewership was dropped by 55%. How many people watch the show now? 5. A large tank that supplies water to a small remote village decreased by 62% after 4 days. The water remaining in the tank is 1140 Litres. What is the capacity of the tank? 6. A marine biologist observed that the population of fish in a particular river has decreased by 12% its original value. It is assumed that the population of fish that remained is How much fish was there in the river before the decrease?