WEBVTT 1 00:00:00.000 --> 00:00:03.077 This is Percentages, Changes and Compounding. 2 00:00:03.257 --> 00:00:07.959 This walkthrough uses a generic synthetic narrator, not a recording of Craig Wright. 3 00:00:08.139 --> 00:00:11.006 Pause whenever you need time to calculate. 4 00:00:11.186 --> 00:00:15.668 The original lesson, written walkthrough and exercises accompany this video. 5 00:00:17.348 --> 00:00:19.670 Percentage means parts per hundred. 6 00:00:19.850 --> 00:00:23.553 Percentages are another way to write the fractions from the previous lesson. 7 00:00:23.733 --> 00:00:27.716 Twenty-five per cent and one quarter express the same proportion. 8 00:00:27.896 --> 00:00:34.084 The hundred is a standard comparison scale, not necessarily the actual number of objects involved. 9 00:00:34.264 --> 00:00:40.591 A quarter of twelve items is still twenty-five per cent, even though there are only twelve items. 10 00:00:40.771 --> 00:00:47.377 Keep asking, "A percentage of what?" Twenty-five per cent of a small budget and twenty-five per cent of a 11 00:00:47.557 --> 00:00:51.725 large budget describe the same share but different amounts of money. 12 00:00:51.905 --> 00:00:58.082 In this lesson, all numerical examples are illustrative rather than observations of real prices or returns. 13 00:00:59.762 --> 00:01:01.898 Find a percentage of a quantity. 14 00:01:02.078 --> 00:01:07.639 We can make the arithmetic visible by splitting twelve per cent into ten per cent plus two per cent. 15 00:01:07.819 --> 00:01:12.289 Ten per cent of two hundred and fifty pounds is twenty-five pounds. 16 00:01:12.469 --> 00:01:17.241 One per cent is two pounds fifty, so two per cent is five pounds. 17 00:01:17.421 --> 00:01:21.809 Adding twenty-five and five gives thirty pounds, matching the decimal method. 18 00:01:21.989 --> 00:01:27.725 This is a useful mental check, but the decimal method generalises to awkward percentages as well. 19 00:01:27.905 --> 00:01:30.552 The percentage and its base work together. 20 00:01:30.732 --> 00:01:36.699 Never assume that a percentage reported beside a number must use that number as its base; read the 21 00:01:36.879 --> 00:01:38.470 description carefully. 22 00:01:40.150 --> 00:01:43.168 Multipliers include the amount that remains. 23 00:01:43.348 --> 00:01:47.203 The number one in each formula represents the entire original amount. 24 00:01:47.383 --> 00:01:51.748 With an increase, we keep that original amount and add the extra share. 25 00:01:51.928 --> 00:01:56.003 With a decrease, we keep the original amount minus the removed share. 26 00:01:56.183 --> 00:02:01.686 This explains why a twenty-per-cent discount leaves eighty per cent of the price rather than twenty per 27 00:02:01.866 --> 00:02:03.016 cent. 28 00:02:03.196 --> 00:02:09.639 For a positive starting amount, an increase multiplier is above one and a decrease multiplier lies below 29 00:02:09.819 --> 00:02:10.504 one. 30 00:02:10.684 --> 00:02:13.796 A complete one-hundred-per-cent decrease gives zero. 31 00:02:13.976 --> 00:02:19.804 Larger percentage decreases need careful interpretation and are generally unsuitable for ordinary 32 00:02:19.984 --> 00:02:22.921 non-negative quantities such as stock remaining. 33 00:02:24.601 --> 00:02:28.038 Percentage change uses the starting value. 34 00:02:28.218 --> 00:02:33.175 Calculate the absolute change first: subtract the original value from the new value. 35 00:02:33.355 --> 00:02:36.954 Then compare that change with the original value, not the new one. 36 00:02:37.134 --> 00:02:41.326 A rise of ten may be large or small depending on where we started. 37 00:02:41.506 --> 00:02:46.602 Ten added to forty is a quarter of the starting amount, hence twenty-five per cent. 38 00:02:46.782 --> 00:02:53.110 If the new amount is lower, subtraction gives a negative number and the formula records that direction. 39 00:02:53.290 --> 00:02:59.048 For a starting value of zero, division by zero prevents a finite percentage calculation. 40 00:02:59.228 --> 00:03:01.690 Report the absolute change instead. 41 00:03:01.870 --> 00:03:06.444 Changes involving negative bases also require extra contextual interpretation. 42 00:03:08.124 --> 00:03:11.108 Percentage points compare percentage levels. 43 00:03:11.288 --> 00:03:14.399 The two answers are not competing calculations. 44 00:03:14.579 --> 00:03:16.541 They answer different questions. 45 00:03:16.721 --> 00:03:23.084 Percentage points give the direct distance between two percentage levels, just as degrees give a distance 46 00:03:23.264 --> 00:03:24.657 between temperatures. 47 00:03:24.837 --> 00:03:28.854 A percentage change measures that distance relative to the original level. 48 00:03:29.034 --> 00:03:35.663 Our increase of six points is one fifth of the original thirty-per-cent level, so its relative size is 49 00:03:35.843 --> 00:03:37.724 twenty per cent. 50 00:03:37.904 --> 00:03:43.477 Always include the words "percentage points" when you mean a subtraction between percentage levels. 51 00:03:43.657 --> 00:03:49.508 Saying only that a rate "rose six per cent" would usually suggest a different relative calculation and could 52 00:03:49.688 --> 00:03:51.081 mislead your reader. 53 00:03:52.761 --> 00:03:55.583 Opposite percentages do not usually cancel. 54 00:03:55.763 --> 00:03:57.876 Slow down at the change of base. 55 00:03:58.056 --> 00:04:02.050 The first ten per cent is ten per cent of one hundred, which is ten. 56 00:04:02.230 --> 00:04:06.374 The second is ten per cent of one hundred and ten, which is eleven. 57 00:04:06.554 --> 00:04:10.978 Removing eleven after adding ten leaves us one pound below the start. 58 00:04:11.158 --> 00:04:18.182 This is why adding the signed percentages, plus ten and minus ten, gives the wrong answer for successive 59 00:04:18.362 --> 00:04:19.209 changes. 60 00:04:19.389 --> 00:04:22.605 Multipliers retain the changing base automatically. 61 00:04:22.785 --> 00:04:26.907 Equal increases and decreases are not reverse operations. 62 00:04:27.087 --> 00:04:32.729 To undo a known multiplier, divide by that multiplier rather than applying the same percentage with the 63 00:04:32.909 --> 00:04:34.012 opposite sign. 64 00:04:35.692 --> 00:04:38.281 Compounding builds on earlier changes. 65 00:04:38.461 --> 00:04:44.789 An exponent is a compact way to record repeated multiplication: a multiplier squared means the multiplier 66 00:04:44.969 --> 00:04:46.339 times itself. 67 00:04:46.519 --> 00:04:48.968 Three periods require three copies. 68 00:04:49.148 --> 00:04:55.360 The percentage changes may describe output, costs or another quantity; compounding is a mathematical 69 00:04:55.540 --> 00:04:59.127 structure, not a promise about real-world performance. 70 00:04:59.307 --> 00:05:02.883 It differs from repeatedly adding the same original-base amount. 71 00:05:03.063 --> 00:05:08.705 Before using the formula, identify the length of each period and make sure the rate refers to that same 72 00:05:08.885 --> 00:05:09.814 period. 73 00:05:09.994 --> 00:05:14.615 An annual percentage cannot simply be relabelled as a monthly percentage. 74 00:05:14.795 --> 00:05:20.855 Fees, withdrawals, capacity limits or changing rates would need to be included separately in the model. 75 00:05:22.535 --> 00:05:26.216 Worked example: discount an illustrative price. 76 00:05:26.396 --> 00:05:31.992 This question specifies the order and the base of the charge, so we do not need to guess how the seller 77 00:05:32.172 --> 00:05:33.449 calculates it. 78 00:05:33.629 --> 00:05:35.336 Begin with the discount. 79 00:05:35.516 --> 00:05:42.644 Fifteen divided by one hundred is zero point one five, and multiplying one hundred and twenty by that 80 00:05:42.824 --> 00:05:44.786 decimal gives eighteen pounds. 81 00:05:44.966 --> 00:05:49.668 Subtract eighteen from one hundred and twenty to get one hundred and two pounds. 82 00:05:49.848 --> 00:05:54.736 The multiplier route reaches the same intermediate value because eighty-five per cent remains. 83 00:05:54.916 --> 00:05:57.540 Write that intermediate value down clearly. 84 00:05:57.720 --> 00:06:03.420 The five-per-cent charge is attached to this discounted amount, rather than to the original list price. 85 00:06:05.100 --> 00:06:08.676 Worked example: apply the charge and compare. 86 00:06:08.856 --> 00:06:15.509 Five per cent of one hundred and two pounds is five pounds ten, so adding the charge produces one hundred 87 00:06:15.689 --> 00:06:17.175 and seven pounds ten. 88 00:06:17.355 --> 00:06:21.279 We then compare that final total with the original list price. 89 00:06:21.459 --> 00:06:27.554 The reduction is twelve pounds ninety, which is ten point seven five per cent of one hundred and twenty 90 00:06:27.734 --> 00:06:28.721 pounds. 91 00:06:28.901 --> 00:06:35.507 The combined multiplier provides a second check: zero point eight nine two five means the final amount is 92 00:06:35.687 --> 00:06:38.520 eighty-nine point two five per cent of the original. 93 00:06:38.700 --> 00:06:45.260 Its distance below one hundred per cent is ten point seven five percentage points on that original-price 94 00:06:45.440 --> 00:06:46.287 index. 95 00:06:47.967 --> 00:06:51.357 Worked example: set up repeated growth. 96 00:06:51.537 --> 00:06:57.110 This is a deliberately simple model with a constant growth rate and no rounding to whole units. 97 00:06:57.290 --> 00:07:02.607 The units could describe a divisible measure of output, so fractional values are allowed. 98 00:07:02.787 --> 00:07:09.173 First turn four per cent into the decimal zero point zero four, then add one to include the original output. 99 00:07:09.353 --> 00:07:14.368 Multiplying five hundred by one point zero four gives five hundred and twenty. 100 00:07:14.548 --> 00:07:20.075 You can check that by finding four per cent of five hundred, which is twenty, and adding it to the start. 101 00:07:20.255 --> 00:07:24.794 The crucial next step is to use five hundred and twenty as the new base. 102 00:07:26.474 --> 00:07:30.468 Worked example: finish and measure total growth. 103 00:07:30.648 --> 00:07:36.291 The second increase is twenty point eight units, slightly larger than the first increase of twenty. 104 00:07:36.471 --> 00:07:41.904 That additional zero point eight is four per cent of the first year's extra twenty units. 105 00:07:42.084 --> 00:07:47.506 Adding the second increase to five hundred and twenty gives five hundred and forty point eight. 106 00:07:47.686 --> 00:07:52.005 To find the overall rate, return to the original five hundred-unit base. 107 00:07:52.185 --> 00:07:58.744 The gain of forty point eight divided by five hundred is zero point zero eight one six, or eight point one 108 00:07:58.924 --> 00:08:00.225 six per cent. 109 00:08:00.405 --> 00:08:06.546 Adding two lots of twenty would produce five hundred and forty, which corresponds to a different, 110 00:08:06.726 --> 00:08:08.131 non-compounded rule. 111 00:08:09.811 --> 00:08:12.691 Reverse a percentage change by dividing. 112 00:08:12.871 --> 00:08:17.863 Working backwards means undoing the operation that produced the observed value. 113 00:08:18.043 --> 00:08:25.229 A fifteen-per-cent increase multiplies the original by one point one five, so division by one point one five 114 00:08:25.409 --> 00:08:26.454 reverses it. 115 00:08:26.634 --> 00:08:33.194 Sixty pounds is the result, and multiplying sixty by one point one five checks the answer. 116 00:08:33.374 --> 00:08:39.121 Do not remove fifteen per cent of sixty-nine: that would reverse a different operation based on sixty-nine 117 00:08:39.301 --> 00:08:40.241 itself. 118 00:08:40.421 --> 00:08:44.415 For a discount, the same logic applies with a multiplier below one. 119 00:08:44.595 --> 00:08:50.470 If a full one-hundred-per-cent discount leaves zero, however, the original cannot be recovered from that 120 00:08:50.650 --> 00:08:52.287 final zero alone. 121 00:08:53.967 --> 00:08:57.113 Report the base, period and rounding. 122 00:08:57.293 --> 00:09:02.645 Suppose a tiny activity doubles while a much larger activity grows only slightly. 123 00:09:02.825 --> 00:09:08.410 The tiny activity has the larger percentage increase, but it may still add fewer units. 124 00:09:08.590 --> 00:09:12.851 This is why absolute changes and relative changes often belong together. 125 00:09:13.031 --> 00:09:18.882 Time periods matter too: a growth figure for one month cannot be compared fairly with a whole-year figure 126 00:09:19.062 --> 00:09:21.454 without additional analysis. 127 00:09:21.634 --> 00:09:27.787 When rounding a repeated calculation, keep the full intermediate values so that early rounding does not 128 00:09:27.967 --> 00:09:28.896 accumulate. 129 00:09:29.076 --> 00:09:33.267 Finally, a percentage describes a relationship between numbers. 130 00:09:33.447 --> 00:09:38.335 It does not show whether a policy, a price change or some other event caused the movement. 131 00:09:40.015 --> 00:09:42.244 A reliable percentage decision tree. 132 00:09:42.424 --> 00:09:46.870 Before attempting the exercises, practise choosing the operation in words. 133 00:09:47.050 --> 00:09:53.622 Are you finding part of a whole, moving a quantity forwards, comparing endpoints, or working backwards? 134 00:09:53.802 --> 00:09:57.900 Those are different tasks even when the same percentage appears in the question. 135 00:09:58.080 --> 00:10:03.467 Write the base next to each calculation and label percentage-point answers explicitly. 136 00:10:03.647 --> 00:10:09.882 For successive changes, a small timeline can help: starting amount, first new amount, final amount. 137 00:10:10.062 --> 00:10:13.092 Check the final answer against a combined multiplier. 138 00:10:13.272 --> 00:10:20.087 Enter a percentage such as seven point five as 7.5 when the field already displays the percent unit, rather 139 00:10:20.267 --> 00:10:22.775 than entering its decimal multiplier.