Understanding Percentages - Dr. Frost's Year 7 Math Lesson
Dive into the world of percentages with Dr. Frost in this lesson designed for Year 7 students. Learn how to calculate percentage changes, find percentages of amounts, and solve worded percentage problems. Practice your skills with quickfire questions and exercises to test your understanding. Explore non-calculator methods to find percentage values and changes. Enhance your math skills and master the art of percentages with this comprehensive lesson.
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Year 7 Percentages Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com Objectives: Be able to find the percentage change. Understand use of decimal multipliers to find percentages of an amount and percentage changes. (Extension: Use non-calculator methods to find a value before a percentage change) Last modified: 18th January 2015
RECAP :: Basic Percentages What is: a) 10% of 420? b) 30 as a percentage of 120? c) The cost of a tshirt originally 15 but reduced by 20% ? ? 42 25% ? 12 Learning Outcomes for this topic: 1. Find what % one amount is of another. 2. Find the % change. 3. Find a % of an amount (calculator and non-calculator) 4. Find a value after a % change. 5. Solve more complicated worded percentage puzzles.
Finding the % I had 42 sweets and ate 11 of them. What percentage is this? The fraction is obviously 11 And we know that to convert a fraction to a %, we multiply by 100. 11 42 100 = 26.2% 42 ? The polar bear population fell from 13000 in 2014 to 9300 in 2015. What percentage drop is this? We re interested in the change as a fraction of the original amount. ? ???? = 13000 9300 = 3700 3700 13000 100 = 28.5% ? ? ???? ???? ????? 100 % ? ???? =
Quickfire Questions State what calculation would be needed for the following: (No need to calculate final value) A cat had 57 biscuits and ate 41 of them. What percentage is left? 16 57 100 ? The price of gold per kg increases from 2100 to 2300. 200 2100 100 ? Sales in Tiffin ties declined from 420 in last year to 360. What percentage drop is that? 60 420 100 Matthew Meerkat learnt a third of the 120 pieces in his Piano for Meerkats book followed by another 17 the next week. What percentage did he learn? ? 57 120 100 ?
Percentages of an amount (the non-calculator way) 35% of 64 10%: 10%: 10%: 5%: 6.40 6.40 6.40 3.20_ 22.40
Percentages of an amount (the non-calculator way) 62% of 350 50%: 10%: 2%: 175 35 __ 7__ 217
Check Your Understanding (Without a calculator) Find 28% of 450 126 ? A fridge was originally 480 but sold in a sale for 35% off. What is the sale price? 35% of 480 = 168 480 - 168 = 312 ?
Exercise 1 ? ? What is: [JMC 2001 Q4] 40% of 50% of 60? 12 [JMO 2000 A4] 20% of 30% of 40% of 50? 1.20 5 What percentage is 35 of 80? A cat is reduced in price from 30 to 21. What percentage drop is that? A penguin population increases from 12000 to 47000. What percentage increase is that? 292% 43.75% a b ? 1 ? 30% c [JMO 2004 A3] On Monday the Pied Piper caught 1000 rats in a city. On Tuesday he caught 10% fewer than on Monday. On Wednesday he caught 20% more than on Tuesday. On Thursday he caught 30% fewer than on Wednesday. On Friday he rested. How many rats did he catch in total that week? ? 6 Without use of a calculator, determine: 45% of 120? 54 13% of 350 ? 45.50 64% of 750? 480 360 after a decrease of 85%? 54 450 after an increase of 62%? 729 100, after an increase of 25% followed by a decrease of 25%. 2 ? ??? ? 32% of 150? 48 b d a c e f g ? Solution: 3736 ? [JMC 2013 Q15] The Grand Old Duke of York had 10 000 men. He lost 10% of them on way to the top of the hill, and he lost 15% of the rest as he marched them back down the hill. What percentage of the 10 000 men were still there when they reached the bottom of the hill? A 761 2 % B 75% E 25%Solution: A ? 7 93.75 [IMC 2013 Q4] A standard pack of pumpkin seeds contains 40 seeds. A special pack contains 25% more seeds. Rachel bought a special pack and 70% of the seeds germinated. How many pumpkin plants did Rachel have? Solution: 35 3 C 731 2 % D 662 3 % ? ? [JMC 2009 Q12] Gill is 21 this year. At the famous visit to the clinic in 1988, her weight was calculated to be 5kg, but she now weighs 50kg. What has been the percentage increase in Gill s weight from 1988 to 2009? Solution: 900% 4 [IMC 2008 Q11] What is ? + ? + ? ? Solution: 410 ? 8 ?
Decimal Multipliers Suppose we wanted to find 100% of 14. What would we multiply it by? 14 ? ? This feels rights, because multiplying by a number just less than 1 makes the number just slightly smaller, which we expect. What if we wanted to find 80% of it? 14 ?.? ? What if we wanted to increase it by 10%? (think that this will just slightly increase the number) 14 ?.? ? Reduce it by 15%? 14 ?.?? ? So it seems if we turn the percentage into a decimal, we can just multiply by it
More examples Bro Tip: For % increases and decreases, just think that everything starts at 100% of its value. So if its value increases by 5%, it s now at 105% of its value, which is 1.05 as a decimal. Effect Multiplier ? ? 1 0.15 0.03 ? ? ? ? ? ? No change Find 15% of it. Find 3% of it. 1.1 0.7 1.03 0.96 1.024 Increase by 10% Decrease by 30% Increase by 3% Decrease by 4% Increase by 2.4%
Check Your Understanding Show the calculation needed (using only a single multiplication) ? ?.?? ?.?? ?? ?.? ?? ?.?? ?? ?.?? ?.?? ?.?? ?.?? ?.?? ?? ?.? ?.? Find 38% of 4.20 Find 70% of 35 60 after an increase of 12% 85 after an increase of 7% 3.20 after a decrease of 12% 8.24 after a decrease of 1% 10 after an increase of 10% followed by a decrease of 10% ? ? ? ? ? ?
Exercise 2 (Calculator use encouraged!) Calculate, giving your answer to the nearest penny: (and showing the multiplication you use) 13% of 56 74% of 3.20 130% of 77.10 31% of 8.41 1.5% of 75 80 after an increase of 20% 35 after a decrease of 10% 46 after an increase of 2% 103 after a decrease of 30% 70 after an increase of 5.5% $840 after an increase of 125% 1 ? ? ? ? ? ? ? ? ? ? ? 7.28 2.37 100.23 2.61 1.13 96 31.50 46.92 67.90 73.85 $1890 a b c d e f g h i j k [Kangaroo Grey 2004 Q7] Maija has a rectangular patio in her garden. She decides to make the patio larger by increasing both its length and width by 10%. What is the percentage increase in the area of the patio? 21% ? 3 Pippin the Cat initially weighs 10kg. She gets 10% heavier one year, followed by 20% heavier the next year. Felix the Cat again initially weighs 10kg. She gets 15% heavier one year, followed by 15% the next. Who is the heaviest after the two years? Pippin is 13.2kg. Felix is 13.225kg, so is heavier. ? 2
Tips for Percentage Puzzle Solving A 25% increase is not the opposite of a 25% decrease. #1 You have 100. It increases by 25%. What percentage will it have to decrease by to get you back to 100? 100 after 25% is added is 125. We need to reduce it by 25. But 25 is a fifth (i.e. 20%) of 125. So the reduction is 20%. ? Another way to see this is by fractions. To increase by 25%, we multiply by 5 Note that 5 5 of the new amount (i.e. reducing by 20%) will cancel out the effect. 4. 4 4 5= 1. So finding 4 [IMC 2003 Q16] After a year s training, Minnie Midriffe increased her average speed in the London Marathon by 25%. By what percentage did her time decrease? 20%?
Tips for Percentage Puzzle Solving Be able to deal with reverse percentages. #2 By reverse percentages, I mean find a value before a percentage change or before we took a percentage of an amount. You will cover this properly in Year 9, but we will look at a simple non-calculator technique for the moment. [JMC 2010 Q16] One of the examination papers for Amy s Advanced Arithmetic Award was worth 18% of the final total. The maximum possible mark on this paper was 108 marks. How many marks were available overall? Hint: Use the same approach we used for reverse fractions of amounts . 18% is 108 1% is 6 100% is 600 ?
Tips for Percentage Puzzle Solving Be able to deal with reverse percentages. #2 The price of a hamster is reduced by 20% to 640. What was the original cost? Bro Side Question: Why can t we just add 20% on to 640? Because the 20% was of the original amount, not of 640. ? 80% is 640 So 10% is 80 So 100% is 800. ? Quickfire Questions: ? ? ?? If 60% of a number is 420. What is the number? A lobster is reduced by 30% to 3500. What was the original cost? If 22% of a number is 66, what is the number? A hamster is reduced by 10% to 3600. What was the original cost? 4000 700 7000 300
Tips for Percentage Puzzle Solving Try a particular value (e.g. 100) when the original value doesn t matter. #3 [IMC 2011 Q17] A shop advertised Everything half price in our sale , but also now advertises that there is An additional 15% off sale prices . Overall, this is equivalent to what reduction on the original prices? ? If item was 100. After half of, it s 50. 15% of 50 is 7.50, so new price is 42.50. That represents a 57.5% reduction. [IMC 2005 Q22] Inspector Remorse had a difficult year in 2004. A crime wave in Camford meant that he had 20% more cases to solve than in 2003, but his success rate dropped. In 2003, he solved 80% of his cases, but in 2004 he solved only 60% of them. What was the percentage change in the number of cases he solved in 2004 compared with 2003? A Down by 10% B Down by 8% C No change D Up by 8% E Up by 10% Suppose he had 100 cases in 2003. He solved 80% = 80 of these. Therefore in 2004 he had 120 cases. He solved 60% = 72 of these. The drop from 80 to 72 is 10%, so the answer is A. ?
Check Your Understanding [IMC 1999 Q11] 20% off everything , screamed the sale posters. I paid 60. What should I have paid before the sale? Solution: 75 ? 1 [JMO 2007 B3] a) Yesterday evening, my journey home took 25% longer than usual. By what percentage was my average speed reduced compared to normal? 20% ? 2 [STMC Regional 2012/13 Q8] In a sale, the price of a computer is reduced by 20%. At this reduced price the shopkeeper still makes a profit of 20%. What would have been his percentage profit if the computer had been sold at full price? (Hint: 100 makes the numbers awkward. Try 300) Suppose computer was originally 300. Sale price is 240. If this was 120% of its cost price, cost price is 200. Thus from 200 to 300 a profit of 50% is made. ?
Exercise 3 (Questions on provided sheet) 6 Two racoons are initially the same price. One racoon is reduced in price by 60%. What percentage more expensive is the dearer racoon? Solution: 150% If 40% of a number is 24, what is the original number? Solution: 60 1 ? ? If a price is reduced by 30% to 49, what was the original price? 2 Solution: 70 ? [SMC 2006 Q8] Matt black paint absorbs 97% of light, the remainder being reflected. Scientists have developed a new superblack coating, 10 times blacker than matt black paint, meaning it reflects 1 reflected by matt black paint. What percentage of light does the new coating absorb? A 9.7 B 90.3 C 99.7 D 99.9 E 970 7 A burger is originally sold for double the price of a hot dog. If the burger is reduced in price by 10%, what percentage more expensive is the burger now? Solution: 80% 3 ? 10 of the light [JMO 2012 A2] Mike drank 60% of his glass of milk. Afterwards, 80 ml of milk remained in the glass. What volume of milk was initially in the glass? Solution: 200ml 4 ? Solution: C ? 5 [Kangaroo Pink 2007 Q3] An international organisation has 32 members. Every year the number of members increases by 50%. How many members will it have three years from now? Solution: 108 ?
Exercise 3 (Questions on provided sheet) [JMC 2009 Q16] The kettle in Keith s kitchen is 80% full. After 20% of the water in it has been poured out, there are 1152ml of water left. What volume of water does Keith s kitchen kettle hold when it is full? A 1400ml B 1600ml C 1700ml D 1800ml E 2000ml [TMC Regional 2013 Q7] A shopkeeper paid 30 (cost price) for a coat. She wishes to place a price tag on it so that she can offer a 10% discount on the price marked on the tag and still make a profit of 20% on the cost price. What price should she mark on the tag? Solution: 40 8 11 Solution: D ? ? [JMO 1996 A7] When a barrel is 30% empty it contains 30 litres more than when it is 30% full. How many litres does the barrel hold when full? Solution: 75 9 [SMC 2003 Q9] Mary received a 10% pay rise, and Margaret received a 5% pay rise. This gave them both salaries of 23 100 per year. How much more, per year, did Margaret earn than Mary before they received these pay rises? 12 ? [TMC Regional 2011 Q8] There are 100 students in a school year group. On one day 99% of the students are present. Only 98% of the students with brown hair are present. How many of the 100 students have brown hair? Solution: 50 students (the one absent person represents 2% of brown-haired people) ? 10 Solution: 1000 ?
Exercise 3 (Questions on provided sheet) [JMO 2001 A6] One third of the population now has access to the internet; this is 50% more than one year ago. What fraction of the population had access to the internet one year ago? Solution: ? ? ? 13 16 [Kangaroo Pink 2010 Q2] Ivan, Tibor and Alex sat a test and achieved 85%, 90% and 100% respectively. Tibor scored just one more mark than Ivan. How many marks did Alex get? Solution: 20 ? 14 [IMC 2003 Q16] After a year s training, Minnie Midriffe increased her average speed in the London Marathon by 25%. By what percentage did her time decrease? Solution: 20% ? [IMC 2000 Q20] The populations of five cities A, B, C, D, E in 1988 and 1998 are shown on these scales. 17 [JMO 2003 A5] Part of a bridge spans a river 35 metres wide. One third of the length of the bridge is on one side of the river and 20% of the length of the bridge is on the other side. How long is the bridge? Solution: 75m (? the other, representing ? bridge spans the 35m) 15 Which of the five cities showed the largest percentage increase in population from 1988 to 1998? A B C D E Solution: C ? ? was on one side and ? ??, thus ? ? ? on ?? of the
Exercise 3 (Questions on provided sheet) [SMC 2007 Q9] In a sale, a shopkeeper reduced the advertised selling price of a dress by 20%. This resulted in a profit of 4% over the cost price of the dress. What percentage profit would the shopkeeper have made if the dress had then been sold at the original selling price? 18 Solution: 30%. Sadly the try a specific value strategy doesn t work well here as the numbers are too awkward. If selling price is S and cost price is C, then ? ?.? ? = ?.?? ? ? =?.?? ?.? ? = ?.?? This corresponds to a profit of 30%.
