Percentage Change,Ratios, Indices (Excel)DHICTadapted by Dr Colin FuICT 1 Assignment 1• Have you started your assignment 1?• Go watch the videos • More videos in Week 8 Revision• It is good that you go watch them now rather than waitingtill Week 8?Assignment 1 3 Percentage• A percentage is a means of expressing a fraction in partsof a hundred• The words “per cent” simply mean “per hundred”, so“percentage“ mean “out of a hundred”.• The expression “20 per cent” means 20 out of 100.Mathematically, it is written as 20100• The per cent sign is %, so 20 per cent is written as 20%. Examples• Percentage Changes Percentage Changes• It is often useful to know by how much an item’s valuesor properties change over time.• The percentage change measures an item’s change invalue relative to its original value.• It has numerous applications in real-life scenarios andmost fields of study. Example 1• At KFC, the $2 Toonie Tuesday special had a priceincrease to $2.39. Calculate the % increase in its price. Example 1 Answer• At KFC, the $2 Toonie Tuesday special had a priceincrease to $2.39. Calculate the % increase in its price. Example 2• An air turbine decreases its rotational speed from 5 rpm(rotations per minute) to 3 rpm. Calculate the % decreasein its speed. Example 2 Answer• An air turbine decreases its rotational speed from 5 rpm(rotations per minute) to 3 rpm. Calculate the % decrease in itsspeed.• Negative sign here means the speed slows down• Notice that a positive percentage change denotes apercentage increase, whereas a negative percentagechange denotes a percentage decrease. For example, adecrease is the same as a change. Example 3• Becky dropped a ball off of a balcony and calculated thatit would take 2.0 seconds to hit the ground. In actuality,the ball took 2.12 seconds to hit the ground. Find the %error in her time. Example 3 Answer• Becky dropped a ball off of a balcony and calculated thatit would take 2.0 seconds to hit the ground. In actuality,the ball took 2.12 seconds to hit the ground. Find the %error in her time.• Note % error is actually the same as % change! % Error?• % error is actually the same as % change! Example 4• Andy shot a flare, calculating that it would reach amaximum height of 400m when it actually reached aheight of 450m. Find the % error in his calculation. Example 4 Answer• Andy shot a flare, calculating that it would reach amaximum height of 400m when it actually reached aheight of 450m. Find the % error in his calculation. Example 5• Dylan estimated that it would take him 5 hours to drive toOttawa from Toronto, but when he made the journey, ittook him 4 hours. Find the % error in his estimation. Example 5 Answer• Dylan estimated that it would take him 5 hours to drive toOttawa from Toronto, but when he made the journey, ittook him 4 hours. Find the % error in his estimation. Excel• There is no percentage-change formula in Excel. Youhave to sub the values into the formula to find what youneed to find. Ratios What is ratio?Ratio is a comparison between twoquantities of the same kindRatio of two quantities, a and b, is written asa : b abRead as ‘a is to b’ Ratios• A ratio is a way of expressing the relationship betweentwo quantities.• It is essential that the two quantities are expressed in thesame units of measurement – pence, number of people,etc. – or the comparison will not be valid• Note that the ratio itself is not in any particular unit – itjust shows the relationship between quantities of thesame unit. Ratios• We use a special symbol (the colon symbol : ) in expressing ratios• 200: 87 (pronounced “200 to 87”)• Ratio of two quantities, a and b, is writtenasa : b abRead as ‘a is to b’ What is the ratio of women to men?3:2 What is the ratio of men to women?2:3 Ratios – rules• They should always be reduced to their lowest possibleterms• Consider the ratio of girls to boys in a group of 85 girlsand 17 boys.– The ratio would be 85 : 17. This can be reduced asfollows:– The ratio of 85 to 17 is, therefore, 5 : 1 Ratios – rules• Ratios should always be expressed in whole numbers.No fractions and decimals in them• Consider the ratio of average miles per gallon for carswith petrol engines to that for cars with diesel engines,where the respective figures are 33½ mpg and 47 mpg.– We would not express this as 33½ : 47, but convertthe figures to whole numbers by, here, multiplying by2 to give:– 67 : 94 Example 1• What is the ratio of blue balls to pink? Example 1 – Answer9 : 3 or 3 : 1• What is the ratio of blue balls to pink? • What is the ratio of pink balls toblue?Example 2 • What is the ratio of pink balls toblue?9 : 6 or 3 : 2Example 2 – Answer Example 3 – Calculating ratio• Consider the case of three partners – Ansell, Boddingtonand Devenish – who share the profits of their partnershipin the ratio of 3 : 1 : 5. If the profit for a year is £18,000,how much does each partner receive? Example 3 – Answer• Example 3 – Answer cont’ed• Equivalent ratios• An equivalent ratio can be obtained whenwe multiply or divide both quantities in aratio by the same number2 16 3×3 =x3812÷ 4 ÷ 4=2 3 Example 5 – Equivalent?2 : 36 : 9 Ratio in business• Ratios are used extensively in business to provideinformation about the way in which one element relatesto another.• For example, a common way of analysing businessperformance is to compare profits with sales.• If we look at the ratio of profits to sales across two years,we may be able to see if this aspect of businessperformance is improving, staying the same ordeteriorating.• (This will be examined elsewhere in your studies.) Rates Rates• A rate is a ratio that compares two quantities measured indifferent units.• A unit rate is a rate whose denominator is 1 when it iswritten as a fraction.• To change a rate to a unit rate, first write the rate as afraction and then divide both the numerator anddenominator by the denominator. Example 1• Sue walks 6 yards and passes 24 security lights set alongthe sidewalk. How many security lights does she pass in1 yard? Example 1 – Answer Example 2• A dog walks 696 steps in 12 minutes. How many stepsdoes the dog take in 1 minute? Example 2 – Answer Unit Price• A unit price is the price of one unit of an item. The unitused depends on how the item is sold. The table showssome examples. Example 3• A 12-ounce sports drink costs $0.99, and a 16- ouncesports drink costs $1.19. Which size is the best buy? Example 3 – Answer Proportions In simple proportions, all you need to do is examine the fractions.If the fractions both reduce to the same value, the proportionis true.This is a true proportion, since both fractions reduce to1/3.Are theseproportional?andA proportion is an equation that states that two ratios are equal,such as: 5152 65 5 115 5 3=2 2 16 2 3= • Find the value of a quantity, given theratio and value of another quantity.What is x?Example 1 To determine theunknown value,you must crossmultiply.In simple proportions, you can use this same approach whensolving for a missing part of a proportion.Remember thatboth fractionsmust reduce tothe same value.Check yourproportion(3)(x) = (2)(9)(3)(6) = (2)(9)18 = 18(3)(x) = (2)(9)3x = 18x = 6Find the value of a quantity, given the ratio and valueof another quantity. What is x?Example 1 – Answer • 1) Are the following true proportions?Example 2 2 103 5=46 12x= 1) Are the following true proportions?Nox=8Example 2 – Answer 2 103 5=46 12x= 3) Solve for x:a) If 4 tickets to ashow cost $9.00,find the cost of 14tickets.b) A house which isappraised for $10,000pays $300 in taxes.What should the tax beon a house appraisedat $15,000.x=10=$31.50 =$450 25 5x 2= A house painter mixes yellow paint with blue paint to getgreen paint. The ratio of the volume of the yellow paint to thevolume of the blue paint is 5:3. if the difference in volumes ofthe two paints is 500ml, find the volume of the green paint.Example 3 5 : 3Difference involumesbetweenyellow andblue = 500ml+ =Volume of green paintDifference in volumesbetween yellow and blue=Volume of green paint500 ml=Volume of green paintExample 3 – Answer 5 35 3+ –824 5002000mlml= = Indices and Roots Indices• Indices are found when we multiply a number by itselfone or more times. Indices• The power of a number is the result of multiplying thatnumber a specified number of times. Thus, 8 is the thirdpower of 2 (23) and 81 is the fourth power of 3 (34)? Indices• A negative exponent indicates a reciprocal. Indices and Roots• Indices also do not have to be whole numbers – they canbe fractional=• •which is• Area=3 x 3= 3²=9Area=4 x 4= 4²=163 94 16√ Calculate the area of the squarexxArea=x x x = x²Area=5 x 5 = 5² = 255 25If x² = 36 what is x ?x = √36 = √(6 x 6) = 6 Volume = 3 x 3 x 3= 3³ = 27Volume = 4 x 4 x 4= 4³= 643 274 64 y yyWhat is the volume ofthis cube?Volume =y x y x y= y³If y³ = 125 what is y?y = 125=y = 5(5 x 5 x 5) Standard Form (aka ScientificNotation) Expressing Numbers in StandardForm• 93,825,000,000,000 × 843,605,000,000,000 =?• These are very big numbers and you would probablyhave great difficulty stating them, let alone multiplyingthem• The method of expressing numbers in standard formreduces the number to a value between 1 and 10,followed by the number of 10s you need to multiply it byin order to make it up to the correct size. • • •Expressing Numbers in StandardForm Expressing Numbers in StandardForm• Just as with indices we can also have numbersexpressed in standard form using a negative index Scientific Notation• Scientific Notation is also referred to as scientific form orstandard index form, or standard form in the UK) is a way ofexpressing numbers that are too big or too small to beconveniently written in decimal form. ScientificNotationStandardNotation1.23 ×1021231.23 × 1031,2301.23 ×10412,3001.23 × 105123,0001.23 ×1061,230,000 ScientificNotationStandardNotation1.23 ×10-20.01231.23 × 10-30.001231.23 ×10-40.0001231.23 × 10-50.00001231.23 ×10-60.00000123 Power of 10 Example 1• Change to standard form Example 1 – Answer• Change to standard form Example 2• Change to standard form Example 2 – Answer• Change to standard form Example 3• Change to standard form Example 3 – Answer• Change to standard form Fractions, Decimals and Indices? Fractions, Decimals and Indices?• Fractions, Decimals and Indices?• Factorial Factorials• Factorial is expressed in !• 4! = = 24• 4 factorial is written as 4! (and is sometimes read as “fourbang”) and

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