1. Percentages

2. Percentages are just fractions

3. Percentages are just fractions. They are designed to describe these numbers differently,

4. Percentages are just fractions. They are designed to describe these numbers differently, so that percentages usually fall in the range from 1 – 100.

5. We seem to find it easier to do this than dealing with the range from 0.01 – 1.00 (or 1/100 – 1)

6. Percentages are just fractions. They are designed to describe these numbers differently, so that percentages usually fall in the range from 1 – 100. We seem to find it easier to do this than dealing with the range from 0.01 – 1.00 (1/100 – 1) We can look at this conversion from fraction to percentage:

7. Percentages are just fractions. They are designed to describe these numbers differently, so that percentages usually fall in the range from 1 – 100. We seem to find it easier to do this than dealing with the range from 0.01 – 1.00 (1/100 – 1) We can look at this conversion from fraction to percentage: FractionPercentage 1/100 = 0.01  1= 0.01 x 100 5/100 = 0.05  5= 0.05 x 100 1/10 = 0.10  10= 0.10 x 100 5/10 = 0.50  50= 0.50 x 100 10/10 = 1.00  100= 1.00 x 100

8. and so you can see that all we need to do to get a percentage is to multiply the fraction by 100.

9. Here are a few examples:

10. 0.47  0.47 x 100  47%

11. Here are a few examples: 0.47  0.47 x 100  47% 14/50 = 0.28  0.28 x 100  28%

12. Here are a few examples: 0.47  0.47 x 100  47% 14/50 = 0.28  0.28 x 100  28% or 14/50 x 100/1 = 28  28%

13. Here are a few examples: 0.47  0.47 x 100  47% 14/50 = 0.28  0.28 x 100  28% or 14/50 x 100/1 = 28  28% 65/87 = 0.701  0.701 x 100  70.1%

14. Here are a few examples: 0.47  0.47 x 100  47% 14/50 = 0.28  0.28 x 100  28% or 14/50 x 100/1 = 28  28% 65/87 = 0.701  0.701 x 100  70.1% 68/25 = 2.72  2.72 x 100  272%

15. Here are a few examples: 0.47  0.47 x 100  47% 14/50 = 0.28  0.28 x 100  28% or 14/50 x 100/1 = 28  28% 65/87 = 0.701  0.701 x 100  70.1% 68/25 = 2.72  2.72 x 100  272% 0.006  0.06 x 100  6%

16. 68/25 = 2.72  2.72 x 100  272% 0.006  0.006 x 100  0.6% These last two show that, although it is not so common, we can talk about percentages of any size, we’re not restricted to the 1 – 100 range.

17. Some examples of how we use this idea of percentages in real life.

18. If we need to find a test result, of 23 marks out of 77, as a percentage, we simply find the fraction of the marks gained,

19. If we need to find a test result, of 43 marks out of 77, as a percentage, we simply find the fraction of the marks gained, i.e. 43/77 or 0.558

20. If we need to find a test result, of 43 marks out of 77, as a percentage, we simply find the fraction of the marks gained, i.e. 43/77 or 0.558, and multiply this by 100  0.558 x 100 = 55.8%

21. If we want 8.5% of 63

22. then we want the fraction (8.5/100) of 63

23. If we want 8.5% of 63 then we want the fraction (8.5/100) of 63 = 8.5/100 x 63 = 0.085 x 63 = 5.36.

24. What % profit do shopkeepers make if they buy pomegranates at 32¢ and sell them at a marked price 58¢?

25. Actual profit is 58–32 = 26¢

26. What % profit do shopkeepers make if they buy pomegranates at 32¢ and sell them at a marked price 58¢? Actual profit is 58–32 = 26¢ and so the fraction of the original price is 26/32 = 0.8125

27. What % profit do shopkeepers make if they buy pomegranates at 32¢ and sell them at a marked price 58¢? Actual profit is 58–32 = 26¢ and so the fraction of the original price is 26/32 = 0.8125 = 0.8125 x100% = 81.25%.

28. Life is more complicated than this, since we might have a sales tax of 7.5%.

29. In order to make this 81% profit, the shopkeeper would need to charge the customer his price plus the sales tax

30. Life is more complicated than this, since we might have a sales tax of 7.5%. In order to make this 81% profit, the shopkeeper would need to charge the customer his price plus the sales tax = (58 + 7.5% of 58)¢

31. Life is more complicated than this, since we might have a sales tax of 7.5%. In order to make this 81% profit, the shopkeeper would need to charge the customer his price plus the sales tax = (58 + 7.5% of 58)¢ = 58 + (0.075 x 58)

32. Life is more complicated than this, since we have a sales tax of 7.5%. In order to make this 81% profit, the shopkeeper would need to charge the customer his price plus the sales tax = (58 + 7.5% of 58)¢ = 58 + (0.075 x 58) = 58 + 4.35 = 62.35¢.

33. Life is more complicated than this, since we have a sales tax of 7.5%. In order to make this 81% profit, the shopkeeper would need to charge the customer his price plus the sales tax = (58 + 7.5% of 58)¢ = 58 + (0.075 x 58) = 58 + 4.35 = 62.35¢. This is probably rounded down to 62¢.

34. So now what is the actual % profit made by the shop? The rounding has produced a slightly different profit. Original purchase price 32¢ Selling price 62¢ Sales tax 4.35¢

35. So now what is the actual % profit made by the shop? The rounding has produced a slightly different profit. Customer’s price – sales tax = 62 – 4.35 = 57.65¢ which is the amount kept by the shop Original purchase price 32¢ Selling price 62¢ Sales tax 4.35¢ Kept by shop 62 – 4.35 = 57.65¢

36. So now what is the actual % profit made by the shop? The rounding has produced a slightly different profit. Customer’s price – sales tax = 62 – 4.35 = 57.65¢ which is the amount kept by the shop and this is an actual profit of 57.65 – 32 = 25.65¢ Original purchase price 32¢ Selling price 62¢ Sales tax 4.35¢ Kept by shop 62 – 4.35 = 57.65¢ Profit 57.65 – 32 = 25.65¢

37. So now what is the actual % profit made by the shop? The rounding has produced a slightly different profit. Customer’s price – sales tax = 62 – 4.35 = 57.65¢ which is the amount kept by the shop and this is an actual profit of 57.65 – 32 = 25.65¢ giving a final % profit of (25.65/32) x 100 % = 80.1% Original purchase price 32¢Original profit calculation = 81.25% Selling price 62¢ Sales tax 4.35¢ Kept by shop 62 – 4.35 = 57.65¢ Profit 57.65 – 32 = 25.65¢ % profit (25.65/32) x 100 % = 80.1%

38. It is a little more tricky to work these problems backwards from the selling price.

39. If you buy an item which costs you $16.79 in a state with a sales tax of 8.5%, how much does the store receive?

40. In this case the price you paid ($16.79) is 108.5% of the store’s price and we want to know the value of 100%.

41. $16.79 ÷ 108.5 will give us the number of dollars per percentage point,

42. In this case the price you paid ($16.79) is 108.5% of the store’s price and we want to know the value of 100%. $16.79 ÷ 108.5 will give us the number of dollars per percentage point, this = $0.1547 and so 100% will be $0.1547x100 = $15.47 which is the amount the store receives.

43. The store wants to make a 15% profit on its sales. How much does it need to buy this item for?

44. This time $15.47 is equal to 115% of the cost and the cost/percentage point is $15.47 ÷ 115

45. The store wants to make a 15% profit on its sales. How much does it need to buy this item for? This time $15.47 is equal to 115% of the cost and the cost/percentage point is $15.47 ÷ 115 and the needed cost price is ($15.47 ÷ 115)x100 = $13.45

46. We have calculated a sale where the customer paid $16.79 and the store received $15.47.

47. The difference between these two figures (16.79 - 15.47 = $1.32) is the sales tax which goes to the state.

48. We have calculated a sale where the customer paid $16.79 and the store received $15.47. The difference between these two figures (16.79 - 15.47 = $1.32) is the sales tax which goes to the state. What % of the sale price goes to the state?

49. We have calculated a sale where the customer paid $16.79 and the store received $15.47. The difference between these two figures (16.79 - 15.47 = $1.32) is the sales tax which goes to the state. What % of the sale price goes to the state? The tax as a fraction of the sale price is 1.32 ÷ 16.79 = 0.079 and so the % is 0.079x100 = 7.9%, slightly less than the 8.5% quoted rate.

50. We have calculated a sale where the customer paid $16.79 and the store received $15.47. The difference between these two figures (16.79 - 15.47 = $1.32) is the sales tax which goes to the state. What % of the sale price goes to the state? The tax as a fraction of the sale price is 1.32 ÷ 16.79 = 0.079 and so the % is 0.079x100 = 7.9%, slightly less than the 8.5% quoted rate. Is this a bargain?

51. Original cost price 13.45 Selling price 16.79 Tax 1.32 Store keeps 16.79 - 1.32 = 15.47 Profit 15.47 - 13.45 = 2.02 Tax as % of store’s income = 1.32 ÷ 15.47 x 100 = 8.5% Tax as % of selling price = 1.32 ÷ 16.79 x 100 = 7.9%