Percentages


A percentage is a fraction or proportion of some numbers expressed out of 100

$\frac{50}{100}$ is called 50 per cent and written 50%

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Conversions from Fractions or Decimals to Percentages


To convert a fraction or decimal into a percentage, multiply by 100

Examples

\[ \frac{3}{4} \times 100 = \frac{300}{4} = 75\%\]

\[ \ \ \ \ 0.75 \times 100 = 75\% \]

\[ \frac{2}{5} \times 100 = \frac{200}{5} = 40\% \]

\[\ \ \ \ 0.4 \times 100 = 40\% \]


To emphasize the method outlined above

  1. Multiply the fraction by 100

  2. And solve using the rules of multiplication of fractions

    If it is of any help, remember that

    100

    can be expressed as a fraction, in the form

    \[ \frac{100}{1} \]

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Conversions from Percentages to Fractions or Decimals

Conversion of Percentages to Fractions

To convert a percentage into a fraction, express the percentage as a fraction over 100, and then simplify to lowest terms, if possible

Examples

\[45\% = \frac{45}{100} = \frac{9}{20}\]

\[ 7\% = \frac{7}{100} \]

Conversion of Percentages to Decimal Numbers

To convert from a percentage to a decimal, just divide the percentage by 100.

Stated in an alternative but identical way, shift the decimal point two places to the left (remember that a number like 51 can be thought of as 51.0).

Examples

45% = 0.45

7% = 0.07

23% = 0.23

Quick Quiz     Convert to percentages
  1. \[ 1. \ \ 0.235 \]
  2. \[ 2. \ \ \frac{4}{7} \]
  3. \[ 3. \ \ 0.35 \]
  4. \[ 4. \ \ 0.125 \]
  5. \[ 5. \ \ \frac{4}{9} \]
  1. \[ \ \ \ 6. \ \ \frac{7}{8} \]
  2. \[\ \ \ 7. \ \ 0.0455 \]
  3. \[\ \ \ 8. \ \ 0.375 \]
  4. \[\ \ \ 9. \ \ \frac{23}{29} \]

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Finding percentages

Method



  1. Method Number 1 : Convert the percentage into a fraction, and carry out the appropriate calculation



  2. Method Number 2 : Convert the percentage to a decimal and carry out the appropriate calculation

Examples

\[ 10\%\ \mbox{of } 50 = \frac{10}{100} \times 50 = \frac{1}{10} \times 50 = 5 \]


\[ 10\%\ \mbox{of } 50 = 0.1 \times 50 = 5 \]

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\[48\% \ \mbox{of } 200 = \frac{48}{100} \times 200 = \frac{12}{25} \times 200 = 96\]


\[ 48\% \ \mbox{of } 200 = 0.48 \times 200 = 96\]

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\[ 34\% \ \mbox{of } 120 = \frac{34}{100} \times 120= \frac{17}{50} \times 120 = 40.8\]


\[ 34\% \ \mbox{of } 120 = 0.34 \times 120 = 40.8\]

Quick Quiz     Find these percentages
  1. \[1. \ \ 7\% \ \mbox{of } 45 \]
  2. \[ 2. \ \ 20\%\ \mbox{of } 150\]
  3. \[3. \ \ 23\% \ \mbox{of } 345\]
  4. \[ 4. \ \ 34\% \ \mbox{of } 230 \]
  5. \[ 5. \ \ 35\% \ \mbox{of } 45 \]
  1. \[\ \ \ 6.\ \ 15\% \ \mbox{of 600} \]
  2. \[ \ \ \ 7. \ \ 45\% \ \mbox{of } 34.54\ \mbox{pounds} \]
  3. \[ \ \ \ 8. \ \ 57\% \ \mbox{of } 6789 \]
  4. \[ \ \ \ 9. \ \ 37\% \ \mbox{of } 3434 \]
  5. \[ \ \ \ 10. \ \ 23\% \ \mbox{of } 432 \]

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Increasing and Decreasing Percentages

Method

  1. Divide the original amount by 100, thus obtaining 1%
  2. Multiply by the required percentage increase or decrease
  3. Add or subtract this from the original amount

Examples

Increase 150 by 10%

Step 1. Find $1\%$ of the original amount \[ \frac{150}{100} = 1.5 \]

Step 2. Find $10\%$ of the original amount \[ 10\% = 15\]

Step 3. Adding this to the original amount gives 165

Decrease 180 by 5%


Step 1. Find $1\%$ of the original amount \[ \frac{180}{100} =1.8\] Step 2. Find $5\%$ of the original amount

\[ 5\% = 9\]
Step 3. Subtracting this from the original amount gives 171

Quick Quiz    
  1. Increase £ 345 by 10%
  2. Increase 230 by 45%
  3. Increase 45 by 23%
  4. Increase 145 by 75%
  5. Increase 115 by 12%

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Discounts

In general, to find a percentage of a quantity, we could

  1. Divide the quantity by 100, to find 1%

  2. Multiply this 1% by whatever percentage we want to find, e.g. to find 65%, we would multiply by 65.

For example, to find 65% of £ 200

  1. Divide £ 200 by 100 to find that 1% of £ 200 is £ 2

  2. Multiply £ 2 by 65 to find that 65% of £ 200 is 130

In practise we can conflate these two operations into one operation, i.e.

\[ 65\% \ \mbox{of } £ 200 = 200 \times \frac{65}{100} \]

Examples

Suppose we want to find 70% off a price of £ 235. The reduction would be given by

\[235 \times \frac{70}{100} \]

= £ 164.50

Subtract this from the original price

£ 235 - 164.50 = £ 70.50


Question 1. What would be 70% off   £ 534 ?

Suppose we want to find 6% off a price of £ 685. The reduction would be given by

\[ 685 \times \frac{6}{100} \]

= £ 41.10

Subtract this from the original price

£ 685 - 41.40 = £ 643.60


Question 2. What would be 8% off   £ 893 ?

Suppose we want to find 65% off a price of £ 174. The reduction would be given by

\[ 174 \times \frac{65}{100} \]

= £ 113.10

Subtract this from the original price

£ 174 - 113.10 = £ 60.90


Question 3. What would be 65% off   £ 132 ?

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Questions from Past Papers



Kay has been given 4 labels to put on the sale rails at the clothes shop where she works.


1.   Which label should she put on the trousers that are down from £ 24 to £ 16?

  • A     Label 1
  • B     Label 2
  • C     Label 3
  • D     Label 4

2.   Which label should she put on the coats that are down from £ 50 to £ 35?

  • A     Label 1
  • B     Label 2
  • C     Label 3
  • D     Label 4





Liz runs an ice-cream stall. To check how popular each flavour is, she records the number of each flavour sold.

1. Which ice-cream flavour sold only two-thirds as many as toffee?

A Banana

B Strawberry

C Mint Choc Chip

D Tutti Frutti

2. What percentage of all the sales was the chocolate flavour?

A 7%

B 14%

C 28%

D 30%

3. Liz used 1 litre of vanilla ice-cream. How much chocolate ice-cream did she use if all the ice-creams were the same size?

4. Additional data was recorded from a group of children who all had tutti frutti ice-cream. This made tutti frutti the unique mode for all the data. What is the minimum number of children in the group to give this statistic?

A 3

B 11

C 12

D 17


1.   120 people are invited to the Christmas Party at work. 10 were not able to come because they were doing other things. 20% of those left did not want to come. Of those who said they would 5 come had to cancel at the last minute.

How many people came to the party?

  • A    81
  • B    82
  • C    83
  • D    84






This year, a company makes a profit of £ 552 374

  1. In its report the company rounds this figure to the nearest £ 100 000, The rounded figure is

    • A   £ 500 000
    • B   £ 550 000
    • C   £ 552 000
    • D   £ 600 000

  2. Next year the company expects to make 10% more profit than this year. Rounding to the nearest £ 100 000 what is its expected profit for next year ?

    • A   £ 600 000
    • B   £ 607 000
    • C   £ 610 000
    • D   £ 660 000






40 000 people watched a football match. The total receipts for the match were £ 640 000.
  1. 6500 of the people at the match were visiting supporters. What percentage of the people watching the match were visiting supporters ?

    • A   1.625%
    • B   6.5%
    • C   16.25%
    • D   65%

  2. The visiting club is paid $\frac{3}{8}$ of the match receipts. How much is the visiting club paid ?

    • A   £ 120 000
    • B   £ 240 000
    • C   £ 320 000
    • D   £ 400 000






Stage 3

A new clothing manufacturing company makes 32 000 items in its first year. The company plans to increase production by 8% per year.

a)   How many items will be produced in the third year of operation?

b)   One of the employees says that the initial production figure will be doubled in the fifth year. Is this statement true?

Show how you would check the statement by using the formula

Where
F = Final number of items produced
I = Initial number of items
p = percentage increase
t = time in years

c)   The company is offered a contract to produce 60 000 items in the third year and needs to know the annual percentage increase required to achieve this target. Use the following formula to calculate the required annual percentage increase, p.








Stage 3

The population of Kenya has been increasing by 4.5% per year.

1)   After how many years could you expect its population to have doubled ?

2)   Give two factors which could affect your forecast in part a






Stage 3

A community charity is allowed to re-claim VAT on some of the items it buys.

The charity buys 15 tables at £ 63.00 each, including VAT at 17.5%.

  How much VAT might the charity reclaim altogether ?






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