Things to Learn (Key words, Notation & Formulae)

Transcription

1 Things to Learn (Key words, Notation & Formulae) Key words: Percentage This means per 100 or out of 100 Equivalent Equivalent fractions, decimals and percentages have the same value. Example words Rise, inflate, added, growth, more, extra, up, etc. for Increase Example words for Decrease Fall, drop, decline, reduction, discount, cut, deflate, shrink, less, down, decay, etc Key skills: Percentage of amounts E.g. Find 10% of 340 You could spot that 10% is 1 10 so finding 10% is equivalent to finding 1 10 of something. To find 1 we divide by 10 so the answer is E.g. Find 15% of 420 You could find 10% or 1 as seen above 10 then use this to find 5% by halving the amount and add them to make 15%. 10% = 42 so 5% = 21 15% = = 63 You could spot that 15% is 3 and then 20 calculate = 63 1 You could spot that 15% is 3 20 so you could divide by 20 then multiply by = = 63 There are always many ways of getting to a particular percentage. You simply have to be confident finding key values like 1%, 2%, 5%, 10%, 20%, 25%, 50% and 75% and you can build up percentages from those. Strong fraction and FDP skills also help a lot with quicker methods! E.g. Find 17.5% of 225 A useful calculator skill for more complex numbers is to type in either of the following: 17.5% = as normal decimal so or alternatively, newer calculators have a % button = so % = (or to nearest penny)

2 Expressing one number as a percentage of another E.g. A score of 3 marks out of 10 on a test as a % You could use equivalent fractions to find 3 10 = = 30% For nice denominators which convert into 100 easily this is a good method. E.g. A score of 5 out of 9 on a test as a % You could calculate the division 3 10 = 0.3 Then multiply by = 30% You could spot or know that 1 = = 0. 5 (0.555 ) = % As a calculator method you can type in = % or % This method work is equivalent to calculating = 30% 1 10 This example has no easy equivalent fraction which converts to 100 so we must use either fraction knowledge or a calculator method. Increasing or decreasing an amount by a percentage E.g. Tom s savings One method is to find the 2% amount of 500 increase (using the above methods) and then add by 2% in one it on. year, how much 2% of 500 does he have = 10 after one year? = 510 A calculator method to find a 2% increase is to add that 2% onto the whole starting amount, which is 100%. 102% = 1.02 as a decimal so or % will calculate an answer of 510 as well E.g. A shop is having a sale of 20% off. A T-shirt originally costs 22, what is the sale price? One method is to find the 20% amount (using the above methods) and then subtract it off. 20% of = Is the same as = = A calculator method to find a 20% decrease is to subtract 20% from the whole starting amount, which is 100%. 80% = 0.8 as a decimal so or 22 80% will calculate an answer of as well

3 Section 1. Finding a percentage of an amount: 1) Easy questions (non-calculator): a) Find 10% of 20 b) Find 20% of 45 c) Find 15% of 60 d) Find 35% of 80 e) Find 75% of 120 f) Find 150% of 40 g) Find 200% of 27 2) Harder questions (non-calculator): a) Find 7.5% of 40 b) Find 17.5% of 60 c) Find 12.5% of 88 d) Find 87.5% of 56 e) Find 48% of 200 f) Find 89% of 300 3) Calculator questions: a) Find 1.2% of 155

4 b) Find 3.4% of 120,000 c) Find 0.01% of 30,500 Section 2. Expressing one number as a percentage of another: 1) Express the shaded part of each diagram as a percentage of the whole. a) b) c) d) 2) Answer the following percentage questions: a) What is 7 out of 10 as a percentage? b) What is 11 out of 20 as a percentage? c) What is 24 out of 75 as a percentage? d) What is 3 out of 8 as a percentage? e) What is 124 out of 200 as a percentage? f) What is 4 out of 11 as a percentage? (hint: there is a nice pattern to elevenths) 3) Answer the following worded percentage problems: a) A crisp packet weighing 25g contains 7g of fat. What percentage is this?

5 b) Joey scores 37 out of 40 in a French test. What percentage is this? c) On Monday, 3 of my class of 29 students were late for school. What percentage were on time? (Calculator question) Section 3. Increasing or decreasing an amount by a percentage: 1) Increase by a percentage questions (non-calculator): a) Increase 40 by 10% b) Increase 50 by 20% c) Increase 4 by 50% d) Increase 16 by 75% e) Increase 95 by 100% f) Increase 15 by 200% g) Increase 200 by 2% h) Increase 124 by 17.5% (calculator question) 2) Decrease by a percentage questions: a) Decrease 50 by 30% b) Decrease 60 by 15% c) Decrease 200 by 80% d) Decrease 24 by 75% e) Decrease 1,234,567,890 by 100% f) Decrease 405 by 1.3% (calculator question) 3) Worded percentage change questions: a) Sarah s salary is a year. She receives a pay rise of 1%, what is her new salary?

6 b) A car is worth and depreciates in value by 12% in a year. What is the value of the car next year?

7 Section 1. Finding a percentage of an amount: 1) Easy questions (non-calculator): a) Find 10% of 20 You should be able to recognise that 10% means divide the amount by ten, since 10% And multiplying by 1/10 is the same as dividing by 10. So: 10% of 20 = = = 2 b) Find 20% of 45 Like the above, you should be able to see that 20% means divide the amount by five. So: 20% of 45 = = 45 5 = 9 c) Find 15% of 60 A bit trickier than the first two. For cases like this, it s always easier to convert to a fraction. 15% % of 60 = = = = 9 You might be able to see you can cross-cancel, which makes it easier to multiply numbers together, so in the last example: = = = 9 d) Find 35% of 80 We ll use the same method as before, but using cross-cancelling to make the multiplication easier. e) Find 75% of % % of 80 = = = = 28 75% % of 120 = = = = 90

8 f) Find 150% of 40 If the percentage is greater than 100, then our answer will increase. We can still use the same method as before! 150% = = % of 40 = = = = 60 g) Find 200% of 27 You should be able to see that 200% of an amount means twice that amount (follow through the similar steps as before if you re unsure). So: 200% of 27 = 2 27 = 54 2) Harder questions (non-calculator): a) Find 7.5% of 40 These questions are harder, but the method is exactly the same as before: 7.5% % of 40 = = = = 3 Another way is to find easier percentages and add them together, so 10% of 40 = 4 You should be able to see that 5% of 40 is half of that amount, so 5% of 40 = 2, and 2.5% of 40 is half of that, so 2.5% of 40 = 1. Now we can write 7.5% of 40 = (5% of 40) + (2.5% of 40) = = 3 b) Find 17.5% of 60 You can use either method for this calculation, so either: 17.5% = = % of 60 = = = = 21 2 (or 10.5) Or: 10% of 60 = 6, so 5% of 60 = 3, and 2.5% of 60 = % of 60 = (10% of 60) + (5% of 60) + (2.5% of 60) = = 10.5

9 c) Find 12.5% of 88 Again, you can use either method. So: 12.5% (It is useful to remember that 12.5% 1/8) 12.5% of 88 = = = = 11 Or: 10% of 88 = 8.8, so 5% of 88 = 4.4, and 2.5% of 88 = % of 88 = (10% of 88) + (2.5% of 88) = = 11 d) Find 87.5% of 56 Again, you can use either method. So: 87.5% (It is useful to remember that 87.5% 7/8) 87.5% of 56 = = = = 49 Or: 50% of 56 = 28, so 25% of 56 = 14, and 12.5% of 56 = % of 56 = (50% of 56) + (25% of 56) + (12.5% of 56) = = 49 e) Find 48% of 200 Again, you can use either method, so either: 48% % of 200 = = = = 96 Sometimes, it can be helpful to not convert the percentage into a fraction in its simplest form. In that last example, if we write 48% 48/100, then the calculation becomes = = = 96

10 For the other method, it s slightly different. Rather than adding percentages of amounts together, we can also subtract percentages of amounts. So: 50% of 200 = 100, and 1% of 200 = 2, so 2% of 200 = 4 48% of 200 = (50% of 200) (2% of 200) = = 96 f) Find 89% of 300 As before, the first method always works, so: 89% % of 300 = = = = 267 For the second method, we can use something similar to what we did with the previous question, but take it one step further. If you want to calculate a large percentage of something, you can calculate the percentage of the whole that you don t want, and subtract it from the whole. We can write that as: 10% of 300 = 30, and 1% of 300 = 3, so 89% of 300 = 100% of % of % of 300 = (10% of 300) + (1% of 300) = = 33 89% of 300 = (100% of 300) (11% of 300) = = 267 3) Calculator questions: a) Find 1.2% of 155 When using a calculator, you can either convert the percentage into a fraction (as we ve done before): 1.2% % of 155 = = Or you can in your calculator (if it s modern enough!): % = To work out the answer, which is 1.86

11 b) Find 3.4% of 120,000 Using either method: c) Find 0.01% of 30,500 Using either method: 3.4% of 120,000 = ,000 = % of 30,500 = 1 30,500 = Section 2. Expressing one number as a percentage of another: 1) Express the shaded part of each diagram as a percentage of the whole. a) b) c) d) a) There are 5 5 squares in total, which is 25 squares. 6 of the squares are shaded, so as a fraction, the shaded amount is 6/25. To express this as a percentage, you want to make the denominator = 4, so the shaded region as a percentage is: = = 24% b) There are 8 squares in total, and 3 of them are shaded, so the fraction of shaded squares is 3/ = 12.5, so the shaded region as a percentage is: = = 37.5% c) There are 9 squares in total, and 4 of them are shaded, so the fraction of shaded squares is 4/ = 11. 1, so the shaded region as a percentage is: = = % d) There are 25 squares in total, and 1 of them is shaded, so the fraction of shaded squares is 1/ = 4, so the shaded region as a percentage is: = = 4%

12 2) Answer the following percentage questions: a) What is 7 out of 10 as a percentage? We can use the same method as before: convert to a fraction out of 100, and that is the percentage = 10, so: 7 10 = = = 70% b) What is 11 out of 20 as a percentage? = 5, so: = = = 55% c) What is 24 out of 75 as a percentage? Some questions are a lot easier if you put the fraction in its simplest form first! = = 8 25 Now we do as before = 4, so: 8 25 = = = 32% d) What is 3 out of 8 as a percentage? = 12.5, so: 3 8 = = = 37.5% e) What is 124 out of 200 as a percentage? Again, try to simplify the fraction a bit first, and the question becomes easier! = = = 62% f) What is 4 out of 11 as a percentage? (hint: there is a nice pattern to elevenths) If we work out 1/11, we see there is a pattern of 0. 09, i.e. the pattern repeats forever as: = This means we can write 4/11 as: 4 11 = = = So as a percentage, we can write 4/11 as: 4 11 = = 100 = %

13 3) Answer the following worded percentage problems: a) A crisp packet weighing 25g contains 7g of fat. What percentage is this? First, write what the fraction of fat is: 7g fat out of 25g = = 4, so as a percentage: 7 25 = = = 28% b) Joey scores 37 out of 40 in a French test. What percentage is this? Again, we write what the fraction of marks Joey scored: 37 out of 40 = = 2.5 (you can show this by using long division!), so as a percentage: = (37 2) + (37 0.5) = = = = 92.5% c) On Monday, 3 of my class of 29 students were late for school. What percentage were on time? (Calculator question) If we put 3/29 into a calculator, we get We can write this as a percentage by multiplying the amount by 100, which means the percentage of late students is: = % (to 1 decimal place) Section 3. Increasing or decreasing an amount by a percentage: 1) Increase by a percentage questions (non-calculator): a) Increase 40 by 10% Start by finding 10% of 40, which is 4. An increase by 10% means we add this number to the whole, so: Increase 40 by 10% = 40 + (10% of 40) = = 44 Note that increasing something by 10% is the same as saying 110% of something, so in the previous example: 110% of 40 = = 40 = 40 = = = 44

14 b) Increase 50 by 20% 20% of 50 = 10, so an increase of 20% is: Increase 50 by 20% = 50 + (20% of 50) = = 60 c) Increase 4 by 50% 50% of 4 = 2, so an increase of 50% is: Increase 4 by 50% = 4 + (50% of 4) = = 6 d) Increase 16 by 75% So to find the new amount: 75% of = = = 12 Increase 16 by 75% = 16 + (75% of 16) = = 28 e) Increase 95 by 100% 100% of 95 = 95, so an increase of 100% is: Increase 95 by 100% = 95 + (100% of 95) = = 190 f) Increase 15 by 200% 200% of = 30, so: Increase 15 by 200% = 15 + (200% of 15) = = 45 g) Increase 200 by 2% 2% of 200 = = = = 4 Therefore: Increase 200 by 2% = (2% of 200) = = 204 h) Increase 124 by 17.5% (calculator question) The easy way to type this into a calculator is to realise that an increase by 17.5% is the same as 117.5% of the whole. So you would type in a calculator: which gives an answer of % =

15 2) Decrease by a percentage questions: a) Decrease 50 by 30% The method for these questions is the same as the previous ones, but this time a decrease by a percentage means you subtract that amount from the whole. So for this question: 30% of 50 = = = = 15 Therefore: Decrease 50 by 30% = 50 (30% of 50) = = 35 b) Decrease 60 by 15% 15% of 60 (10% of 60) + (5% of 60) = = 9 Therefore: Decrease 60 by 15% = 60 (15% of 60) = 60 9 = 51 c) Decrease 200 by 80% 80% of 200 = = = = 160 Therefore: Decrease 200 by 80% = 200 (80% of 200) = = 40 d) Decrease 24 by 75% 75% of 24 = = = = 18 Therefore: Decrease 24 by 75% = 24 (75% of 24) = = 6 e) Decrease 1,234,567,890 by 100% Decreasing the whole of something by 100% means you are subtracting the whole from the whole, so the answer is 0 f) Decrease 405 by 1.3% (calculator question) A decrease of something by 1.3% is the same as saying 98.7% of something. We can put this in a calculator as: % = which gives an answer of

16 3) Worded percentage change questions: a) Sarah s salary is a year. She receives a pay rise of 1%, what is her new salary? A pay rise of 1% means her salary has increased by 1%. 1% of = = = = Therefore: A pay rise of 1% on = (1% of 24000) = = b) A car is worth and depreciates in value by 12% in a year. What is the value of the car next year? Depreciates by 12% means the value has decreased by % of = = = = = 1680 Therefore: A depreciation of 12% = (12% of 14000) = = 12320