A percentage is a fraction with a denominator (bottom number) of % = 23

Transcription

1 Chapter 5 Percentages We have all seen advertisements on television that use percentages. Shops selling items at a % discount or cereal boxes offering % more than usual for the same price. Percentages are quite often used in school as well. Most of your grades are converted to percentages before giving them back to you. Those percentages are then converted in grades at the end of the year. An A, for example, is a mark between 90% and %. However what does this percentage sign, %, really mean? A percentage is a fraction with a denominator (bottom number) of 23% = Changing Percentages to Fractions A percentage is simply a fraction, However, its denominator will always be. Therefore changing a percentage toafractioncaneasilybedonebyremovingthepercentagesignandsimplyplacingthenumberasthenumerator with a denominator of. Example 1 Change 37% to a fraction 37% = 37 Example 2 Change 18% to a fraction 18% = 18 = 9 In this example, we started by changing the percentage sign into a denominator of. However the fraction was not in its lowest form, so it had to be reduced by dividing the numerator and the denominator by 2. Example 3 Change 1% to a fraction In this example, the percentage is greater than. However the process to change it into a fraction remains the same. However our answer will now be a mixed numeral rather than just a fraction. 1% = 1 = 1 =

2 44 CHAPTER 5. PERCENTAGES Another way of doing this question is the following 1% = 1 = 6 5 It does not matter if you change to a mixed numeral and then reduce, or reduce and then change to a mixed numeral. Either way will give the same answer. = Exercise Change each of the following percentages into a fraction with a denominator of a) 97% b) 72% c) % d) 53% e) 26% f) 19% g) 84% h) 1% 2. Change each of the following percentages into a fraction reduced to its lowest form a) % b) % c) 5% d) 2% e) 13% f) % g) % h) 70% i) 28% j) 72% k) 33% l) 75% m) 48% n) 92% o) 15% p) 47% q) 83% r) 60% s) 35% t) 26% u) 88% v) 45% w) 14% x) 82% y) 12% z) 37% 3. Change each of the following percentages into a whole number or mixed numeral a) % b) 300% c) 700% d) 2% e) 4% f) 1% g) 275% h) 3% i) 560% j) 490% k) 315% l) 285% m) 145% n) 2% o) 176% p) 324% q) 438% r) 113% s) 217% t) 4% u) 241% v) 160% w) 375% x) 191% y) 363% z) 295% 5.03 Changing Fractions to Percentages To change a fraction into a percentage, we first need the denominator to be. Once the denominator is, we can simply remove the denominator and replace it with a percentage sign. Example 4 Change 3 into a percentage 3 = 30 = 30% Example 5 Change 4 5 into a percentage 4 5 = 80 = 80%

3 5.04. EXERCISE Example 6 Change 1 1 into a percentage 1 1 = 1 = 1 = 1% 5.04 Exercise Change each of the following fractions into percentages a) 7 c) 72 e) 90 g) 35 b) 15 d) 56 f) h) Change each of the following fractions into percentages a) 40 h) 7 o) 4 v) 19 b) 15 i) 2 5 p) 19 w) 9 c) 1 2 j) 13 q) 11 x) 8 d) 3 4 k) 2 4 r) 1 4 y) 15 e) 4 5 l) 1 s) 6 z) 8 f) 3 m) 1 5 t) 1 g) 7 n) 9 u) Change each of the following whole numbers or mixed numerals into percentages a) 2 37 h) 4 13 o) 1 6 v) 5 8 b) 1 3 c) 4 d) i) 1 9 j) k) 1 p) q) 2 47 r) w) 1 7 x) y) e) f) l) m) s) 1 9 t) 6 z) 1 16 g) 2 9 n) 5 3 u) Changing Percentages to Decimals To change a percentage to a decimal, start by changing the percentage into a fraction (as we were doing above). Then change the fraction into a decimal. If you are unsure of how to change a fraction into a decimal, have a look through the chapter on decimals. Remember, to change a fraction into a decimal, divide the numerator by the denominator. Because percentages always have a denominator of, when changing a percentage to a decimal, we will always be moving the decimal point of the numerator two places to the left. Example 7 Change 17% into a decimal 17% = 17 = 0.17

4 46 CHAPTER 5. PERCENTAGES Example 8 Change 3% into a decimal 3% = 3 = 0.03 It is very important in this example to move the decimal point two places. Moving the decimal point once will give.3 or 0.3. Moving the decimal place for a second time to the left gives.03 or written correctly, Example 9 Change 135% into a decimal 135% = 135 = 1.35 After changing the percentage into a fraction, you may be tempted to reduce it into its lowest forms. However we want to leave the fraction as it is without reducing because dividing by is easy, we just have to move the decimal point. If the fraction is first reduced, we will no longer be dividing by a power of and the division then becomes a lot more difficult Exercise Change each of the following percentages into decimals a) 40% b) 67% c) 98% d) 78% e) 4% f) 51% g) 35% h) 84% i) 31% j) 70% k) 2% l) 82% m) 13% n) 41% o) 19% p) 30% q) 95% r) 15% s) 72% t) 96% u) 48% v) 97% w) 55% x) 92% y) 83% z) 24% 2. Change each of the following percentages into decimals a) 286% b) 8% c) 444% d) 335% e) 443% f) 130% g) 331% h) 345% i) 282% j) 189% k) 152% l) 261% m) 305% n) 463% o) 341% p) 291% q) 214% r) 496% s) 3% t) 265% u) 277% v) 495% w) 170% x) 317% y) 323% z) 470% 5.07 Changing Decimals to Percentages To change a decimal into a percentage, first change the decimal into a fraction. Then change the fraction (by multiplying the numerator and denominator by the same number) so that the denominator is. However a percentage is simply a fraction with a denominator of, so all that is left is to remove the denominator and replace it with a percentage sign. If you are unsure on how to change a decimal into a fraction, have a look through the section on decimals. Example Change 0.18 into a percentage 0.18 = 18 = 18%

5 5.08. EXERCISE Example 11 Change 0.4 into a percentage Example 12 Change 2.63 into a percentage 0.4 = 4 = 40 = 40% 2.63 = 2 63 = 263 = 263% 5.08 Exercise Change each of the following decimals into a percentage a) 0.58 b) 0.95 c) 0.04 d) 0.76 e) 0.1 f) 0.52 g) 0.79 h) 0.09 i) 0.31 j) 0.83 k) 0.40 l) 0.05 m) 0.75 n) 0. o) 0.37 p) 0.44 q) 0.8 r) 0.94 s) 0.27 t) 0.32 u) 0.06 v) 0.13 w) 0.96 x) 0.87 y) 0.62 z) Change each of the following decimals into a percentage a) 1.99 b) 1.80 c) 3.33 d) 2 e) 4.59 f) 1.94 g) 2.65 h) 4.7 i) 3.02 j) 5 k) 1.16 l) 1.58 m) 3.8 n) 2.89 o) 4.94 p) 1.77 q) 1.32 r) 2.3 s) 1.52 t) 1.69 u) 1.38 v) 2.53 w) 2.26 x) 3.81 y) 3.45 z) Finding Percentages of Quantities Often we use percentages to describe a portion of a quantity. For example, 15% of primary school students are in year 6. If we know the total amount of students in the primary school, we can work out how many students are actually in year 6. Alternatively, if you received 80% on your last Maths test and there were 40 questions on the test, how many questions did you actually correctly answer? When finding a percentage of a quantity, start by changing the percentage to a decimal. Then multiply the decimal and the quantity together to get the final answer. In the following examples, the working out has not been shown for the final multiplication. However please make sure you show your working for your questions. If you are unsure on how to multiply when it involves decimals, have a look at the decimals chapter. Example 13 Find % of 30 % of 30 = = 15

6 48 CHAPTER 5. PERCENTAGES Example 14 Find % of 80 % of 80 = = 16 Example 15 Find % of 2.8m % of 2.8m = m = 0.7m Always remember to include units in your final answer where needed. 5. Exercise Evalulate the following a) 6% of 16 b) 8% of 22 c) 5% of 88 d) 1% of 57 e) 7% of 90 f) 9% of 33 g) 4% of h) 2% of 84 i) 3% of Evaluate the following a) 5% of 62 b) 8% of c) 2% of 75 d) 60% of 86 e) 83% of f) % of 87 g) % of 76 h) 82% of 64 i) 5% of 77 j) 8% of 87 k) % of 62 l) 1% of 49 m) 9% of 21 n) 7% of 83 o) 3% of 47 p) 4% of q) % of 51 r) 9% of 458 j) 79% of 129 k) 55% of 235 l) 92% of 448 m) % of 361 n) 4% of 147 o) 51% of 423 p) 85% of 156 q) 33% of 352 r) 49% of 285 s) 1% of 295 t) 6% of 654 u) 2% of 831 v) 5% of 418 w) 7% of 371 x) 4% of 293 y) 3% of 7 z) 8% of 568 s) 96% of 440 t) 6% of 347 u) 57% of 132 v) 11% of 227 w) 71% of 342 x) 30% of 160 y) 26% of 498 z) 1% of Exercise Out of the students in the class, 15 of them are girls and the rest are boys. What percentage of the students in the class are girls? 2. On a very long and difficult exam, you achieved a mark of 156 out of a possible 0. What percentage did you receive for the exam? 3. On tuckshop day, 14 students in the class ordered tuckshop. What percentage of students ordered tuckshop if there are students in the class? 4. A soft drink claims to have 5% real fruit juice contained in it. If you buy a 380mL can of soft drink, how much real fruit juice is contained in it? 5. Over the Christmas period, a store received back 3 of all the Christmas lights it sold because they were faulty. What percentage of lights were not returned? 6. A survey taken at a shopping center revealed that 35% of all people entering the shopping center will leave without buying anything. What fraction of customers buy something while in the shopping center?

7 5.11. EXERCISE Your parents tell you that for your next exam you need to achieve a mark of at least 60%. If the exam has 45 questions on it, how many questions must you answer correctly to achieve 60%? 8. On a school excursion, 78% of the students chose to attend. If there are 3 students enrolled at the school, how many went on the excursion? 9. The government asks that you pay tax on all money that you earn. If you earn $78 and are required to pay % in tax, how much do you have left to spend?. 85% of people are right-handed. Express this as a fraction. 11. A household estimated that % of all their rubbish could actually be recycled instead. If they used to throw away 32kg of rubbish each week, how much will they be throwing away now that they are recycling? 12. Of the numbers -29 (inclusive), what percentage of them are divisible by 3? kg of sand was purchased to build a sand pit for the prep students. However, only 38kg was delivered. What percentage of sand was received out of the total amount ordered? 14. A ticket for a concert is $75. 65% of the money received from each ticket is used to pay expenses and the rest is profit. How much profit is made from each ticket? 15. Tina saves % of the money she makes each week. How much will she have saved after 4 weeks if she earns $ a week?

8 CHAPTER 5. PERCENTAGES Answers Exercise a) 97 b) 72 h) 7 i) 7 w) 7 x) 41 l) 2 17 m) 1 9 c) d) 53 j) 18 k) 33 y) 3 z) 37 n) 2 13 o) 1 19 e) 26 f) 19 g) 84 h) 1 2a) 1 b) 1 5 l) 3 4 m) 12 n) 23 o) 3 p) 47 q) 83 3a) 1 b) 3 c) 7 d) e) f) p) 3 6 q) 4 19 r) 1 13 s) 2 17 t) u) 2 41 c) 1 r) 3 5 g) v) d) 1 s) 7 h) 3 1 w) e) 13 f) 1 4 g) 1 2 t) 13 u) 22 v) 9 i) j) 4 9 k) 3 3 x) 1 91 y) 3 63 z) 2 19 Exercise 5.04 a) 7% b) 15% c) 72% d) 56% e) 90% f) % g) 35% h) 84% 2a) 80% b) 30% c) % d) 75% e) 80% f) 30% g) 35% h) 70% i) 40% j) 26% k) % l) % m) % n) 45% o) 40% p) 76% q) 55% r) % s) 60% t) 4% u) 80% v) 95% w) 90% x) 32% y) 75% z) 80% 3a) 237% b) 130% c) 400% d) 2% e) 1% f) 3% g) 290% h) 465% i) 118% j) 3% k) % l) 5% m) 280% n) 515% o) 160% p) 375% q) 294% r) 440% s) 118% t) 600% u) 395% v) 532% w) 170% x) 260% y) 4% z) 164% Exercise a) 0.4 b) 0.67 c) 0.98 d) 0.78 e) 0.04 f) 0.51 g) 0.35 h) 0.84 i) 0.31 j) 0.7 k) 0.02 l) 0.82 m) 0.13 n) 0.41 o) 0.19 p) 0.3 q) 0.95 r) 0.15 s) 0.72 t) 0.96 u) 0.48 v) 0.97 w) 0.55 x) 0.92 y) 0.83 z) a) 2.86 b) 2.58 c) 4.44 d) 3.35 e) 4.43 f) 1.3 g) 3.31 h) 3.45 i) 2.82 j) 1.89 k) 1.52 l) 2.61 m) 3.05 n) 4.63 o) 3.41 p) 2.91 q) 2.14 r) 4.96 s) 3.1 t) 2.65 u) 2.77 v) 4.95 w) 1.7 x) 3.17 y) 323 z) 470

9 5.11. EXERCISE Exercise a) 58% b) 95% c) 4% d) 76% e) % f) 52% g) 79% h) 9% i) 31% j) 83% k) 40% l) 5% m) 75% n) % o) 37% p) 44% q) 80% r) 94% s) 27% t) 32% u) 6% v) 13% w) 96% x) 87% y) 62% z) 3% 2a) 199% b) 180% c) 333% d) 0% e) 459% f) 194% g) 265% h) 470% i) 302% j) 0% k) 116% l) 158% m) 380% n) 289% o) 494% p) 177% q) 132% r) 230% s) 152% t) 169% u) 138% v) 3% w) 226% x) 381% y) 345% z) 4% Exercise 5. 1a) 0.96 b) 1.76 c) 4.4 d) 0.57 e) 6.3 f) 2.97 g) 2 h) 1.68 i) 0.72 j) 6.96 k) 6.2 l) 0.49 m) 1.89 n) 5.81 o) 1.41 p) 0.4 q) 5.1 r) s) 2.95 t) u) v).9 w).97 x) y) 3.21 z) a) 3.1 b) 4 c) 1.5 d) 51.6 e).75 f) 87 g) 19 h) i) 3.85 j) 1.91 k) 129. l) m) 36.1 n) 5.88 o) p) q) r) s) t).82 u) v) w) x) 48 y) z) 2.66 Exercise ) 60% 2) 78% 3) 70% 4) 19mL 5) 85% 6) 13 7) 27 8) 273 9) $70. ) 17 11) 24kg 12) 30% 13) 95% 14) $26. 15) $40