PERCENTAGES
What does percent mean?
Percentages are just special fractions. The denominator is 100. “Per” means “out of” and
“cent” means 100. So:

10% means 10 out of 100 or
10
1

100 10

25% means 25 out of 100 or
25
1

100
4

50% means 50 out of 100 or
50
1

100
2

45% means 45 out of 100 or
45
9

100
20
12 12

25
1

200
8

12½ % means 12½ out of 100 or

100% means 100 out of 100 or

The fraction

A mark of 100% in an exam means everything correct.

100% certain means absolutely sure, no doubt whatsoever.

100% of the population means everyone.
100
100
1
100
100
is everything. 100% means “all of it”
100
Converting fractions and decimals to percentages
To change a fraction or decimal to a percentage, multiply by 100.
5
100  50%
10
0.5 100  50%
15
100  300%
5
1
 100  5%
20
0.05  100 = 5%
1
 100  25%
4
1
Updated Feb 2013
Converting percentages to fractions
To express a percentage as a fraction, divide by 100
75% 
75
3

100
4
150% 
33 13 
5% 
5
1

100
20
150
 1 12
100
33 13
100

100
1

300
3
12 12 % 
12 12
100
25
1

200
8

It is a good idea to remember the well known fractions and percentages, such as:
1 1 1
1
1
, s, s, s and s .
2 3 4
5
8
Finding the percentage of a quantity
To find a percentage of a quantity, express the percentage as a fraction, then multiply by the
quantity.

25% of $200 =
25
 200 = $50
100

7.5% of $300 =
7.5
 $300 = $22.50
100

A factory employs 500 people and makes 10% redundant. How many people lose
their jobs?
10
 500 = 50 people
100

750 people attend a function, 40% of them are female. How many are female?
40
 750 = 300 females
100


An item costs $296 GST exclusive. How much would the GST be?
(GST is 12½%)
12 12
100

25
 296 = $37 GST
200
2
Updated Feb 2013
Expressing one quantity as a percentage of another
To express one quantity as a percentage of another, divide the one by the other, then
multiply by 100%.

There are 500 pets at a pet show, 200 are dogs. What percentage are dogs?
200
 100 %  40%
500

800 people attend a concert, 500 are male. What percentage are male?
500
 100 %  62.5%
800

What percentage is 7.50 g of 28.6 g?
7.50g
100%  26.2% (3sf)
28.6g

In a class of 28 students, 22 passed the final exam. What percentage passed?
22
 100 %  78.6%
28

65 out of 155 people surveyed supported the candidate. What percentage supported
the candidate?
65
 100 %  41.9%
155

A 2.4 m stud is accidentally cut at 2.35 m. What is the percentage error in the length
of the stud?
The stud is 0.05 m undersize.
Percentage error is
0.05m
 100%  2.1% (2sf)
2.4m
Increasing a quantity by a percentage
To increase a quantity by a certain percentage, multiply the quantity by the percentage
increase then add the increase to the original quantity.

A company’s sales were up by 10% on their previous year’s sales of $20 000. How
much are the current year’s sales?
Increase in sales =
10
 20 000  $2000
100
Current year’s sales = $20 000 + $2000 = $22 000
3
Updated Feb 2013

The population of a city has increased by 3% on last year’s population of 850 000.
What is the size of the population this year?
3
Population increase
=
 850 000  25 500
100
Present population size = 850 000 + 25 500 = 875 500
Decreasing a quantity by a percentage
To decrease a quantity by a certain percentage, multiply the quantity by 100% to find the
percentage decrease, then subtract the decrease from the original amount

A shop assistant gives 15% discount for cash on an item costing $500. How much is
the discounted price?
Discount = $500 
15
 $75
100
Discounted price = $500 - $75 = $425

The number of births in a town has dropped by 8% from the previous year’s total of
1425. How many births are there in the current year?
Decrease in births =
8
 1425  114
100
Number of births in present year = 1425 - 114 = 1311

A piece of machinery depreciates by 25% in its first year of purchase. If it cost $1250,
what will be its value at the start of the second year?
Depreciation =
25
 1250  $312.50
100
Value at start of second year = $1250 - $312.50 = $937.50
Finding the original quantity
Sometimes you have a quantity which includes a percentage that has already been added or
deducted and you need to find the original quantity. For instance, finding the GST exclusive
price, having been given the GST inclusive price.

A television set costs $675 GST inclusive. What is the GST exclusive amount? The
amount that the GST was added to is the original quantity.
Namely, 12½% was added to 100%.
4
Updated Feb 2013
So the value of $675 is 112½%
It is 100% that we need to find.
112½% = $675
100% =
$675
 100 = $600
112.5
1
In this case where GST is 12½%, 12½% is also .
8
9
1
8
12½%   has been added to the original 100%   giving (112½%)
8
8
8
i.e.
9
= $675
8
So
1
= $675  9 = $75
8
The GST exclusive amount is $675 - 75 = $600

A dress had been marked down 30% on its original value and has a sale price of $84.
What was the original selling price?
Original Value is 100%
i.e.

sale price is 100% – 30% = 70%
70% = $84
100% =
$84
 100 = $120
70
5
Updated Feb 2013
Percentage exercises
1. Change the following fractions and decimals to percentages
a)
3
8
b) 0.35
e)
0.005
f)
4
5
c) 0.2
g)
d) 0.125
7
8
h)
1
12
2. Change the following percentages to fractions:
a)
65%
b) 12.5%
c) 0.5%
d) 200%
3. Find the following:
a)
c)
15% of $450
140% of $270
b) 25% of 136 m
d) 27% of $75.95
4. If GST is 12.5%, find how much GST should be added to the price of apparatus which
retails at:
a) $160
b) $275
c) $345
5. A copper compound contains 80% copper by weight. How much copper is there in a
1.98 g sample?
6. A sales tax of 6% is applied to the purchase of a $75 drill. How much is the tax?
7. A bronze casting weighing 250 kg contains 70% copper and 30% tin. Calculate the
weight of each in the casting.
8. Concentrated hydrochloric acid contains 36% by weight of HCl. How many grams of
HCl are there in 500 g of this solution?
9. A single serving of McDonald’s cheeseburger and chips has a total energy content
1975 kilojoule. 45% of this energy is from fat. Calculate how many kJ of fat will be
consumed from this meal.
10. A balance has a tolerance of 0.5%. If the mass of an object on this balance is 23.80 g,
how large is the error? Between what limits does the mass lie?
11. A gold 4th band on a resistor indicates that the actual value is within  5% of the
indicated value. What is the range of possible values of a 3900 resistor, which has
such a gold band?
12. Common silver solder contains 63% silver by weight. How much solder can be made
with 16 g of silver?
13. A supplier of chemicals informs you that the following discounts are available for the
early payment of purchases: within 10 days: 5.0%, 20 days: 2.5%, 30 days: 1.5%.
You have purchased $25 000 worth of goods from this supplier. How much discount
will you get of payment is made after:
a) 8 days
b) 15 days
c) 27 days
d) 890 days?
6
14. $5000 is invested at 6% per annum for a year. If the interest is added to the account
at the end of the year and the total reinvested, how much will be reinvested?
15. House prices in a suburb increased 22% over a 12 month period. How much would a
house be worth at the end of that year if it was bought for $130 000 at the start of
the year?
16. If the present length of a brass rod is 1.4562 m, what will its length be after 0.3%
expansion caused by heating?
17. What is the correct measurement if a length of 204 cm is found to be 2% too long?
18. If your pay is $367 after an increase of 2.3%, what was it before the increase?
19. In October 1989. GST was increased from 10% to 12.5 %. What would the adjusted
retail price of a wheelbarrow have been that was $68.95 in September 1989?
20. An item is advertised at $124 incl. GST. What was the cost before GST?
21. The label on an item says $96.50 + GST. How much will I have to pay?
22. If the bill is $60 (GST included), how much of this is GST?
23. What was the cost of the following items before GST was added?
a) $36
b) $500
c) $240
d) $88
24. If my pay is $420 after a 5% rise, what was it before?
25. There is $365 in my bank account after interest is added. If interest is 7%, how much
was in the account before interest was added?
26. A voltmeter reads 120 V. It is known that the voltmeter reading is 6% too high. What
should the correct reading be?
27. The voltage drop in a transmission line is 4.8 V. If this is 2% of the source voltage,
find the voltage at the end of the transmission line.
28. You have a term investment of $20 000 with an annual interest rate of 15%.
a) How much is in the account after one year?
b) If you do not withdraw any money from this account, how much is in it after
2, 3, 4, 5 years?
c) What would it be worth after ten years?
29. A belt designed to transmit 70 kW loses 2% of its energy due to slippage. How many
kilowatts are actually transmitted?
30. A piece of metal shrinks 1.5% on cooling. What length will it shrink to if it is initially
1.2 m long?
31. A metre rod loses 1% of its length through wear and tear. What will a rod, initially
2.5m long, reduce to?
32. Everything is reduced 20% in a sale. If the sale price is $240, what was the old price?
33. 5 m2 of cloth shrink by 2% in area. What is its new area?
7
34. A discount of 25% is allowed on the purchase of parts whose total list price is $235.
What is the sales price?
35. Find the following:
a) What % is $30 of $80?
b) What % is 2.7 m of 5.4 m?
c) What % is $140 of $3000?
d) What % is 7 of 65?
36. If a seminar was attended by 23 females and 14 males, what percentage attendance
was female?
37. Analysis of a 1.435 g sample of iron ore shows that it contains 0.369 of iron. What
percentage is this?
38. Analysis of an iron-sulphur mixture weighing 12.50 g shows it to contain 2.70 g iron.
What is the percentage of iron?
39. A bronze alloy gave an analysis of 0.35 kg of copper, 0.25 kg of zinc and 0.15 kg
aluminum. Calculate the percentage of each of the constituents.
40. A person whose salary is $325 per week is given a raise of $35 per week. What
percentage of her original salary is this?
41. In the production of 576 rods, 27 are found to be faulty. What percentage are faulty?
42. An alloy is made of 140 kg copper, 70 kg nickel and 47 kg zinc. What percentage is
each component?
43. In cooling, a piece of metal shrinks by 18 mm per metre of its length. What is the
percentage shrinkage to the nearest whole number?
44. When the harbour bridge extensions were added in 1966-69, the width of the bridge
increased from 13 m to 35 m. What was the percentage increase?
45. A bolt of cloth measuring 120 m is 107.5 m long after dyeing. What % is lost to
shrinkage?
46. What percentage profit would you make if a product cost you $176 to produce and
you sell it for $235?
47. A shop is currently selling videos at $25.50 (the normal price is $29.99). What
approximate percentage discount is the shop offering on this item?
48. A certain fuse rated 15 A requires an actual current of 16 A to blow. What % of the
rating is this?
49. It costs $1250 in interest each year to finance a new machine in your factory worth
$5000. What is the rate of interest?
50. The radius of a circle is increased from 10 cm to 11 cm. Find the percentage increase
in:
a) its area ( A  r 2 )
b) its circumference ( C  2r )
51. A rectangular sheet of metal 10 m long and 5 m wide has a strip 1 m cut off both the
length and the width. What is the percentage decrease in the original area?
8
52. A part is to be machined to length 147.6 cm. If the finished size is 146.8 cm, what is
the percentage error?
53. A surveyor used 3 17 as an approximation for  . What is the percentage error?
54. A chair whose original selling price was $125 is on sale at 20% discount at one store.
It is possible to buy the same chair at another store for $90. Should you buy the chair
from the first or second store?
55. A dealer paid $7200 for a car and later sold it at a profit of 20%. What profit did he
make?
56. A student paid $450 for a car and later sold it at a price, which netted a $45 profit.
What was the percentage profit?
57. 85 is 62 ½ % of what number?
58. A young man spent 24% of his salary on room and board. If he spent $150 per week
for room and board, what was his yearly salary?
59. A timber merchant offered 15% discount to a man who was building his own house.
If the cost of supplies for his house was $56 700, how much did the man save?
60. A quantity of metal was purchased for $3600. Three quarters of that quantity was
then sold for the cost of the whole. What percentage gain would have resulted had
the entire amount been sold at the same rate as the three-quarters?
61. The population of City A increased from 2 649 000 to 3 110 000 from 1989 to 1999.
In the same period, the population of city B increased from 3 762 000 to 4 200 000.
What city had the largest percentage increase?
62. An estimate indicates that an 18% reduction in traffic deaths in one year would save
72 lives. What were traffic fatalities for the previous year if the estimate was
accurate?
63. An electric oven regularly priced at $1750 is on sale at a reduction of 20%. If for a
cash purchase an additional discount of 5% is allowed after the 20% has been
computed, what is the selling price of the oven?
64. A rough casing weighs 20.475 kg. After having been finished in a lathe it weighs
19.575 kg. The loss in finishing is what percentage of its weight before machining?
65. A steel I-beam expands 0.01% of its length when exposed to the sun. Find the
increase in the length of such a beam 7.58 m long after expansion.
66. A firm increases the wages of its employees by 2 ½ %. Find the new wages of a
person who was earning $156.25 per day.
9
67. An airline carried 31 223 passengers in a week. A year later it carried 47 722
passengers in the equivalent week. What was the percentage increase?
68. The usual allowance made for shrinkage on the casting of an iron pipe is 1.042 cm/m.
What percentage is this?
69. An iron meteorite found in 1908 weighed 1488.6 kg. If 91.63% of its weight was iron
and 7.33% nickel, what was the weight of the iron, nickel and other materials?
70. A dealer sold two used cars for $12 950 each. On one he lost 25% and on the other
he gained 25%. Did he lose or gain on the entire transaction and by how much?
71. A manufacturer sold a suit of clothes to a retailer at a profit of 20% over the cost of
making. The retailer then sold the suit to a customer for $250 and made a profit of
1
33 3 %. How much did the suit cost the retailer and what was the cost of making?
72. If an author receives 10% of the selling price of a book, how many books, selling at
$29.95 each, must be sold to pay the author $8397.90?
73. In a compound the weights of two substances A and B are in ratio 1.3498 to 1. What
is the percentage of each compound?
74. The recorded measurement of a city block is 528 m. However careful measurement
reveals the length to be 527.75 m. What is the percentage error in the recorded
length?
75. The population in New Zealand in 1996 was 3 681 546. 22.6% of the population were
children aged 0 – 14 years. How many children were there?
76. By the year 2051, the number of children is expected to drop to 696000, which will
be 16% of the population. What is the projected population for 2051?
77. In 1998, 117 out of 541 road accident deaths in New Zealand occurred in the
Auckland region. What percentage is this?
78. In 1996, 165 921 people out of 823 887 in Auckland were smokers. What percentage
is this?
79. The population of Wellington in 1996 was 322 584. 21.6% smoked regularly. How
many people in Wellington smoked regularly?
80. The average size of the New Zealand household declined from 2.6 people in 1991 to
2.5 in 2006. What percentage decrease is this?
81. The annual median income for women in 2012 is $23,4000. The median annual
income for men in 2012 is $23,000. How many times greater, by percentage, is the
male median income than the female median income?
10
82. In 2012, 333 000 households out of 1 455 200 households in New Zealand had a
weekly income of $1, 307 or more.
a) What percentage of households was this?
b) 8% of households had a weekly income of under $1,328. How many
households?
83. There were 176 284 new registrations of cars in New Zealand in 1999. 63.4% of these
registrations were ex-overseas cars.
a) How many cars registered were from overseas?
b) The number of registrations of ex-overseas cars increased from 4863 in 1975
to 80 976 in 1995. What was the percentage increase?
84. In March 2012 there were 8187 males and 511 females in prison in New Zealand. In
1996 the equivalent numbers were 4561 and 175. Compare the percentage increase
over this period for males and females.
85. In 2010, 150 000 people were unemployed in New Zealand. 39.4% of the
unemployed had no formal educational qualifications. How many is this?
86. In Jan-July 2011 the average Auckland salary grew 3.4% to $73,355. What was the
average salary in January 2011?
87. In 1991 the Maori population of New Zealand was 434 847 and the non-Maori
population was 2 939 082. The Maori population had increased to 523 374 by 1996.
a) What was the percentage increase?
b) The non-Maori population increased 5.3% in the same period. What was the
non-Maori population in 1996?
88. In 1926, 448 501 people out of 1401674 lived in rural areas in New Zealand.
a) What percentage is this?
b) By 1996, the percentage of the population living in a rural area fell by 53%.
What percentage lived in a rural area in 1996?
11
Percentage exercises solutions
1. a) 37.5%
e) 0.5%
2. a)
b) 35%
f) 80%
13
20
b)
1
8
c) 20%
g) 87.5%
c)
d) 12.5%
h) 8.3%
1
200
d) 2
3. a) $67.50
b) 34 m
c) $378
4. a) $20
b) $34.38
c) $43.13
5. 1.58 g
d) $20.51
28. a)$23 000
6. $4.50
b)$26 450, $30 417.50,
7. 175 kg copper, 75 kg zinc
8. 180 g
$40 227.14
c) $80 911.16
9. 888.8 kJ
29. 68.6 kW
10. 0.119 g
30. 1.18 m
11. 3705 - 4095
31. 2.475 m
12. 25.4 g
32. $300
13. a) $1250 b) $625
33. 4.9 m2
c) $375 d) $0
34. $176.25
14. $5300
35. a) 37.5%
15. $158 600
b) 50%
c) 4.67% d) 10.77%
16. 1.4606 m
36. 62.16%
17. 200 cm
37. 25.7%
18. $358.75
38. 21.6%
19. $70.52
39. 47%, 33%, 20%
20. $110.22
40. 10.8%
21. $108.56
41. 4.7%
22. $6.67
42. 53%, 27%, 18%
23. a) $32
b) $444.44
c) 213.33 d) $78.22
43. 1.8%
44. 169.2%
24. $400
45. 10.4%
25. $314.12
46. 33.5%
26. 113 V
47. 15%
27. 240 V
48. 106.7%
12
49. 25%
50. a) 21%
81. 157.11%
b) 10%
82. a) 22.88%
b)116 416
51. 28%
83. a) 111 765
b) 1565%
52. 0.542%
84. Males 79.50%, Females 192%
53. 0.04%
85. 59, 100
54. $100, second store!
86. $70 942.94
55. $1440
87. a)20.36% b) 3 094 853
56. 10%
88. a)32%
b) 15.04%
57. 136
58. $625
59. $10 006
60. 33%
61. City 1 – 17.4%, City 2 – 11.6%
62. 400
63. $1330
64. 4.3%
65. 7.66 m
66. $160.16
67. 52.8%
68. 1.042%
69. 1473.1 kg
70. Equal, no loss or gain
71. $156.25
72. 2804
73. A 57.4%, B 42.6%
74. 0.047%
75. 832 029
76. 4 350 000
77. 21.63%
78. 20.14%
79. 69 678
80. 3.85%
13