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Ch 01 GM 1&2 YR 11 Page 21 Wednesday, December 15, 1999 5:49 PM
Chapter 1 Arithmetic
21
Ratios
A ratio is a comparison of two or more quantities measured in the same units. Consider
the following extract from a microwave recipe book:
‘. . . Place rice and water in the ratio 1:2 in a microwave-proof dish,
cover and cook on high for 10 minutes. . .’
The ratio 1:2 (which is read as ‘one to two’) of rice and water means that for each
cup of rice we should add two cups of water (that is, the amount of water is twice that
of rice). The numbers 1 and 2 are called the terms of the ratio. Naturally, if we expect a
large number of guests for dinner, 1 cup of rice won’t be enough. If we use 2 (or 3 or
even more) cups of rice, we should then increase the amount of water accordingly. Here
are some possible ratios of rice and water:
1 1--2- :3 2:4 2.5:5 3:6
All of these ratios are equivalent and, of all of them, the ratio 1:2 (the one from the
recipe book) is said to be in its simplest form. It is the most convenient to work with, as
the numbers are small and easy to interpret.
a
Generally the ratio a:b (or --- ) is in its simplest form if both a and b are whole
b
numbers and it has been reduced to its lowest terms.
WORKED Example 15
Express each of the following ratios in simplest form.
a 24:8
b 3.6:8.4
c 1 4--9- :1 2--3THINK
WRITE
a
Write the question.
Divide both terms by the highest common factor
(HCF) of 8.
a 24:8
= 3:1
3
Write the question.
Multiply both terms by 10 to obtain whole numbers.
Divide both terms by the HCF of 12.
b 3.6:8.4
= 36:84
= 3:7
1
Write the question.
c 1 4--9- :1 2--3-
2
Change both terms into improper fractions.
=
3
Multiply both terms by the lowest common denominator
(LCD) of 9 to obtain whole numbers.
= 13:15
1
2
b
1
2
c
13
-----9
: 5--3-
Finding the ratio of two quantities
As was discussed before, a ratio compares quantities that are measured in the same
units. Therefore to find the ratio of two quantities, we first have to make sure that their
units are the same. We then write the two numbers as a ratio and omit the units
altogether, as they are not relevant anymore.
Note that the order of the numbers in the ratio is important. Remember the example
of the ratio of rice to water? When written as 1:2, it means ‘1 cup of rice for 2 cups of
water’; when written as 2:1 however, it would mean ‘2 cups of rice for each 1 cup of
water’. (The rice will most likely burn in this case!)
Ch 01 GM 1&2 YR 11 Page 22 Wednesday, December 15, 1999 5:49 PM
22
General Mathematics
Keeping this in mind, always write the numbers in the same order as they were given
to you in the question when solving problems involving ratios.
WORKED Example 16
Find the ratio of 2 hours to 112 minutes. Write your answer in simplest form.
THINK
1
2
3
WRITE
Convert 2 hours into minutes to make
both units the same.
Omit the units and write the two
quantities as a ratio. (Keep the same
order as in the question.)
Simplify by dividing both terms by 8.
2 hours = 2 × 60 minutes
= 120 minutes
The ratio of 2 h to 112 min is 120:112
= 15:14
Dividing a quantity in a given ratio
Here is a recipe for Chilli prawns.
Ingredients
1
--- cup tomato sauce
4
1 red chilli, chopped
2 gloves garlic, crushed
1 teaspoon finely chopped ginger
6 shallots, chopped
500 g prawns
360 g cooked rice
This recipe poses a problem, as it does not tell us how
many cups of uncooked rice to use, but instead tells us
the amount of the ready-made product, containing two
ingredients (rice and water) in the ratio 1: 2.
WORKED Example 17
Divide 360 g in the ratio 1: 2; hence state the amount of rice and water needed to prepare
360 g of cooked rice.
THINK
WRITE
2
Write the ingredients as a ratio.
Count the number of parts.
Rice:water = 1: 2
Number of parts = 1 + 2
=3
3
Find the size of each part.
Each part =
4
Find the amount of rice needed.
5
Find the amount of water needed.
1
360
--------3
g
= 120 g
We need 1 part of rice; hence the amount of rice
is 120 g.
We need 2 parts, so 2 × 120 g = 240 g of water
is needed.
Ch 01 GM 1&2 YR 11 Page 23 Wednesday, December 15, 1999 5:49 PM
Chapter 1 Arithmetic
23
remember
1. A ratio compares two (or more) quantities in the same units.
2. A ratio itself does not contain any units.
3. The ratio a:b is in the simplest form if both of its terms (a and b) are whole
numbers and have been reduced to their lowest terms.
4. Order of numbers in the ratio is important.
5. To write the ratio of two quantities, convert them to the same units first, then
put them as a ratio in the same order as they were given to you in the question
and omit the units.
6. To divide a quantity in a given ratio, find the total number of parts, the size of
one part and then the size of each share.
WORKED
Example
15b
WORKED
Example
15c
2 Express each of the following ratios in simplest form.
a 1.2:0.2
b 3.9:4.5
c 9.6:2.4
e 1.8:3.6
f 4.4:0.66
g 0.9:5.4
i 6:1.2
j 12.1:5.5
k 8.6:4
4
--7
:2
g 5:1 1--2-
h 2 3--4- :1 1--3-
4 multiple choice
The ratio not equivalent to 3:5 is:
C
A 18:30
B 0.6:1
WORKED
Example
16
2
--5
: 4--6-
i
3 5--6- :2 1--2-
D 2.1:3.5
e 3: 2--3j
1 3--5- : 6 4--8-
E
1
--3
Ratios
D 1:60
HEET
1.3
: 1--5-
5 Find the ratio of each pair of quantities and write the answer in the simplest form.
a 5 cm to 20 cm
b 12 mm to 1 cm
c 2 m to 78 cm
d 4.6 km to 400 m
e 250 ml to 3 l
f 504 kg to 1 tonne
g 20 kg to 1050 g
h $12 to 60 c
i 4 month to 5 years
j 18 min to 100 sec
6 multiple choice
The ratio of 36 seconds to 6 hours is:
A 36:6
B 6:1
C 1:6
L Spread
XCE
d 18:3.6
h 0.35:0.21
l 0.07:14
3 Write each of the following ratios in the simplest form.
a 1 1--2- :2
b 2:1 3--4c 1 1--3- :2
d 1 2--5- :1 1--4f
d 14:35
h 144:44
E
15a
1 Express each of the following ratios in the simplest form.
a 12:18
b 8:56
c 9:27
e 88:66
f 16:60
g 200:155
i 32:100
j 800:264
sheet
Example
SkillS
WORKED
Ratios
E 1:600
7 Divide each of the following quantities into the ratio given in brackets.
a 270 kg (4:5)
b 600 m (1:11)
c 215l (2:3)
d 5000 mm (3:5)
e 420 g (4:3)
f 3.6 tonnes (8:1)
g $4000 (2:3:3)
h 250 km (2:3:5)
i 700 ml (2:3:9)
j 48 h (11:3:2)
HEET
SkillS
1E
1.4
Ch 01 GM 1&2 YR 11 Page 24 Wednesday, December 15, 1999 5:49 PM
24
General Mathematics
WORKED
Example
17
8 Michelle is studying for her Maths and Science exams. She decides to divide the 8
hours that she has for study in a ratio 1:3. How many hours will Michelle study for
each exam?
9 A brother and a sister are sharing a packet of chewing gum in the ratio 5:7. If a packet
contains 60 pieces of chewing gum, how much will each receive?
10 Three friends, Lena, Vicky and Margaret, always order household goods in bulk from
the home delivery service to save on costs to suit their families’ needs. They then
divide the goods in the ratio 1:2:3. On one occasion, their order contained a carton of
tissues (36 packets per carton) valued at $39.60 and a box of 240 garbage bags valued
at $18. Find:
a the number of packets of tissues each of the friends received
b the number of garbage bags that each received
c the total amount of money each has to contribute to pay for the order (the delivery
is free).
11 Leon and Igor invested $18 000 and $22 000 respectively in International Independent
shares. Calculate how they should divide a $3000 dividend if they agreed to share it in
the ratio of their investments.
12 The angles of a quadrilateral are in the ratio of 3:4:4:5. Find the size of each angle
and hence name this quadrilateral.
13 To make wholegrain bread with a bread-maker, I must use water, bread mix and yeast
(in that order) in the ratio 35:50:1. If the total weight of the mixture is 0.86 kg, find
the amount of each ingredient.
Work
ET
SHE
1.2
14 The estimated volume of the earth’s salt water is about 1285 600 000 cubic kilometres. The estimated volume of fresh water is about 35 000 000 cubic
kilometres.
a What is the ratio of fresh water to salt water (in simplest
form)?
b Find the value of x, to the nearest whole number,
when the ratio found in a is expressed in the
form 1: x
Ch 01 GM 1&2 YR 11 Page 25 Wednesday, December 15, 1999 5:49 PM
Chapter 1 Arithmetic
25
Proportion
A proportion is a statement about the equality of two ratios. Consider the following
example. To make a cup of coffee a person combines 1 teaspoon of instant coffee
and 2 teaspoons of sugar; hence the ratio of coffee and sugar is 1:2. Clearly, to
make 3 cups of coffee, this person will use 3 times as much coffee and sugar,
namely 3 teaspoons and 6 teaspoons respectively. The ratio in this case is 3:6.
Obviously the two ratios are equivalent and hence can be written as a proportion:
1:2 = 3:6.
This should be read as ‘1 is to 2 as 3 is to 6’.
When three terms of a proportion are known, the fourth term can be found.
WORKED Example 18
b
2
Rewrite each of the two ratios as fractions.
3
Solve for x:
(a) Cross-multiply.
(b) Divide both sides by 18.
1
Write the equation.
2
Rewrite each of the two ratios as fractions.
3
Solve for x:
(a) Cross-multiply.
(b) Divide both sides by 3.
WRITE
a x:4 = 9:18
x
9
--- = -----4 18
18x = 36
-----x = 36
18
x=2
b 3:10 = 7.2:x
3
7.2
------ = ------10
x
3x = 72
-----x = 72
3
x = 24
L Spread
XCE
Ratios
sheet
The Maths Quest Excel file ‘Ratio’ may be used to solve problems like the one in the
previous worked example.
E
Find the value of x in each of the following:
a x:4 = 9:18
b 3:10 = 7.2:x
THINK
a 1 Write the equation.
Ch 01 GM 1&2 YR 11 Page 26 Wednesday, December 15, 1999 5:49 PM
26
General Mathematics
WORKED Example 19
Find the value of x in the proportion (x − 3):(x + 5) = 3:4.
THINK
1
2
3
4
5
6
WRITE
Write the question.
Rewrite the ratios as fractions.
Solve for x: cross-multiply.
Expand.
Subtract 3x from both sides and
simplify.
Add 12 to both sides and simplify.
(x − 3):(x + 5) = 3:4
( x – 3)
3
----------------- = --( x + 5)
4
4(x − 3) = 3(x + 5)
4x − 12 = 3x + 15
4x − 3x − 12 = 3x − 3x + 15
x − 12 = 15
x = 15 + 12
x = 27
WORKED Example 20
If a set of 5 identical books costs $31.50, how much will 3 of these books cost?
THINK
1
2
WRITE
Since we are talking about the same
books, the ratio of the number of books
to the total cost will be equivalent;
therefore a proportion can be formed.
Remember to keep the number of
books and the total cost in the same
order for both ratios.
Rewrite the ratios as fractions.
3
Solve for x. Cross multiply then divide
both sides by 5.
4
Interpret the result.
5:31.50 = 3:x
5
3
------------- = --31.50
x
5x = 31.5 × 3
31.5 × 3
x = ------------------5
x = 18.9
The total cost of three books is $18.90.
remember
1. Proportion is a statement about the equality of two ratios.
2. When solving a proportion:
(a) Write the two ratios as fractions.
(b) Solve for x.
3. When forming a proportion, remember to keep the same order on both sides.
Ch 01 GM 1&2 YR 11 Page 27 Wednesday, December 15, 1999 5:49 PM
27
Chapter 1 Arithmetic
WORKED
Example
1 Find the value of x for each of the following proportions.
a x:2 = 9:4
b x:7 = 24:56
c x:3 = 21:9
e 4:x = 6:8
f 3:x = 9:8
g 5:9 = x:15
i 6:5 = 18:x
j 4:11 = 22:x
L Spread
XCE
d 6:x = 4:3
h 10:3 = x:12
Proportion
2 multiple choice
If 5:70 = 9:x, the expression to use to find x is:
5 × 70
5×9
9 × 70
A --------------B -----------C --------------9
70
5
70
D -----------9×5
5
E --------------70 × 9
3 multiple choice
If 4:x = 5:8, the expression which is not true is:
4
5
8
x
4×8
A --- = --B --- = --C x = -----------x
8
5
4
5
x
4
D --- = --5
8
E x×5=4×8
4 Form a proportion from the following numbers: 12, 3, 32 and 8. How many solutions
are possible?
WORKED
Example
19
WORKED
Example
20
5 Find the value of x in each of the following proportions.
a (x + 6):(x − 2) = 12:5
b (x − 8):x = 3:8
c x:(x + 4) = 7:9
d (x − 5):(x + 7) = 3:10
e (x − 2):(x − 5) = 6:4
f (x + 3):5 = x:3
g (x − 6):4 = (x + 5):3
h (2x − 9):2 = (x − 3):12
i (7x − 3):(2x + 9) = 3:2
j (5 + 3x):9 = (4 − 2x):5
6 If 12 identical pens cost $11.64, how much would 5 such pens cost?
7 Two 100-g cans of tuna in spring water cost $2.70. How many such cans can be
bought for $10.80?
8 A large box contains chocolate biscuits and shortbread biscuits in the ratio of 4:9. If
there are 28 chocolate biscuits, find:
a the number of shortbread biscuits
b the total number of biscuits in the box.
9 The diagram at right shows a pair of similar triangles. If the
lengths of the corresponding sides of similar triangles are in
the same ratio, find the values of x and y.
12
9
5
x
*10
Challenge
For each of the following proportions find x, if x is a positive number.
a x:5 = 45:x
b (x + 2):2 = 9:(x − 1)
c 3:(x − 5) = (x + 7):15
y
2.5
sheet
18
Proportion
E
1F
Ch 01 GM 1&2 YR 11 Page 28 Wednesday, December 15, 1999 5:49 PM
28
General Mathematics
Gears and drive belts
Proportions are also used in solving problems which involve gears and drive belts
(or chains).
The proportion used in the case of a gear drive (or a chain drive) is:
Number of teeth on the driver
Speed of the follower
----------------------------------------------------------------------- = --------------------------------------------------Number of teeth on follower
Speed of the driver
Note the inverse relationship between the number of teeth and the speed.
Gear drive
Chain drive
The speeds of the driver and follower are measured in revolutions per minute (rpm).
The proportion used in the case of the belt dive is:
Diameter of the driver
Speed of the follower
----------------------------------------------------------- = --------------------------------------------------Diameter of the follower
Speed of the driver
Belt drives
1 The driving wheel of a chain drive has 42 teeth and rotates at a speed of
600 rpm. Find the number of teeth of the follower if its speed is 400 rpm.
2 The driver of a belt drive has a diameter of 16 cm and rotates at a speed of
120 rpm. Find the speed of the follower if its diameter is 4 cm.
3 The follower of a belt drive has a diameter of 18 cm and rotates at a speed of
360 rpm. Find the speed of the driver if its diameter is 2.4 times larger than that
of the follower.
4 The speed of the driver and the follower in a water pump are 800 rpm and
1200 rpm respectively. If the diameter of the driver is 24 cm, find the diameter
of the follower.
Ch 01 GM 1&2 YR 11 Page 29 Wednesday, December 15, 1999 5:49 PM
Chapter 1 Arithmetic
29
Percentages
The term per cent means per hundred. For instance, 29% means 29 parts out of 100.
Any percentage can be changed into a fraction by dividing it by 100.
WORKED Example 21
Write the following percentages as:
a 12%
b 10.3%
c 76 3--8- %
i common fractions
THINK
a i
WRITE
1
To express 12% as a fraction, put it over a
hundred and remove the % sign.
2
Simplify.
a
1
b
Put 10.3 over 100 and remove the % sign.
ii
Change the fractional part of the percentage  --8-
into a decimal by dividing the numerator by the
denominator (3 ÷ 8).
3
3
-----25
i 10.3% =
=
Multiply the numerator and the denominator
by 10 to get rid of the decimal point.
ii Divide 10.3 by 100 by moving the decimal point 2
places to the left and removing the % sign.
1
12
--------100
ii 12% = 0.12
2
c i
i 12% =
=
ii To write 12% as a decimal, divide it by 100 (that
is, move the decimal point 2 places to the left) and
remove the % sign.
b i
ii decimals.
10.3
---------100
103
--------100
ii 10.3% = 0.103
c
i 76 3--8- % = 76.375%
2
Put this over 100 and remove the % sign.
=
76.375
---------------100
3
Multiply the numerator and the denominator
by 1000 to get rid of the decimal point.
=
76 375
------------------100 000
4
Simplify.
=
611
--------800
1
Change the fractional part of the percentage
into a decimal.
Divide by 100 by moving the decimal point 2
places to the left and dropping the % sign.
2
ii 76 3--8- % = 76.375%
= 0.763 75
Note: As was shown in the preceding example, a percentage can be a whole number
(for example 76%), a common fraction (for example 1--4- %) or a decimal (for example
2.25%). When the percentage contains a fraction (for example 1--4- %) be careful not to
confuse it with the common fraction (in this case, 1--4- , which is 25%).
Ch 01 GM 1&2 YR 11 Page 30 Wednesday, December 15, 1999 5:49 PM
30
General Mathematics
The table below contains percentages and corresponding fractions in everyday use.
Knowing them by heart helps in solving problems as well as in many practical
situations.
%
Fraction
Decimal
1%
10% 12.5% 20%
25%
33 1--3- %
50%
1
--3
1
--2
1
--------100
1
-----10
1
--8
1
--5
1
--4
0.01
0.1
0.125
0.2
0.25
.
0.3̇
0.5
66 2--3- % 100%
2
--3
.
0.6̇
1
1.0
In order to change a percentage into a fraction we had to divide by 100 and remove
the % sign. To change a fraction into a percentage, we need to do the opposite, that is,
multiply the fraction by a 100 and add a % sign.
WORKED Example 22
Express each of the following as a percentage.
a
3
--7
b 1.02
THINK
WRITE
a
a
1
2
Multiply the fraction by 100 and add
a % sign.
Change the improper fraction into a
mixed number.
3
--7
=
=
3
--- × 100%
7
300
--------- %
7
= 42 6--7- %
b Multiply by 100 by moving the decimal
b 1.02 = 1.02 × 100%
point 2 places to the right and add a % sign.
= 102%
To find a percentage of a quantity, express the percentage as a fraction (either common,
or a decimal) and multiply by that quantity. Remember that a percentage represents a
certain part of a quantity, so the answer should be in the same units as the quantity in
question. (That is, when finding a percentage of a certain amount in dollars, the result
is in dollars; when the given quantity is in kilograms, the percentage of that quantity is
also in kilograms.)
WORKED Example 23
Find 23% of $72.
THINK
1
2
WRITE
Change 23% into a decimal fraction.
Multiply by $72 and add a $ sign to the
answer.
23% of $72 = 0.23 × 72
= $16.56
To express one quantity as a percentage of another, form a fraction from the two quantities and multiply by 100%. Remember to keep both quantities in the same units.
Ch 01 GM 1&2 YR 11 Page 31 Wednesday, December 15, 1999 5:49 PM
Chapter 1 Arithmetic
31
WORKED Example 24
Express 3 months as a percentage of 2 years.
THINK
WRITE
2 years = 2 × 12 months
1 Change years into months.
= 24 months
2
Form a fraction from the two quantities
and multiply it by 100%.
3
Evaluate.
3
-----24
× 100%
=
300
--------- %
24
= 12.5%
Career profile
DA RREN BLOOD — Account Executive
Qualifications: Completing Advanced Diploma in
Business Marketing
Employer: De Bortoli Wines Pty Ltd
Company website: http://www.debortoli.com.au
A
s I have a passion for wines, I wanted to work in
the wine industry and wrote letters to companies
outlining my experience. A position was available at
De Bortoli Wines and I have been there for about 5
years. I didn’t want an office job, and sales seemed to
provide variety, challenge and opportunity.
A typical day includes travelling to liquor stores to
sell products, check stock and provide merchandise
(T-shirts, caps, prints, aprons and so on). I conduct
wine tastings for store customers and set up window
displays. At the end of the day, sales are added up and
faxed through to the warehouse. Calculations I need
to perform which relate to using percentages include:
1. percentage discounts of wholesale prices of wine
sold to liquor outlets — discounts apply with orders
above a certain amount
2. percentage increases or decreases in monthly and
yearly sales
3. setting yearly budgets based on the preceding
year’s sales
4. calculating rebates — once a store has sold a
certain amount, they receive a 2% rebate of
monthly sales
5. using Excel spreadsheets to calculate which stores
stock particular products
6. calculating sales tax (41%) and adding it to the
wholesale price.
Other calculations involve:
1. rounding off customers’ change to the nearest 5c
2. measurement — weights of bottles, size (dimensions) of pallets (64 cartons in a pallet and 12
bottles in a carton), different sized bottles — a magnum is 1.5 L and a jeraboam is 3L — and conversion of volume from millilitres (mL) to litres (L).
I once took an order from a customer for 400
cartons when the customer meant to order 400 bottles.
He was a little surprised when a semi-trailer delivered
the stock the next day.
Mathematical calculations are also very useful in
analysing my total sales. I can check that I have been
paid the right commission.
Questions
1. List three ways Darren uses percentages
in his work.
2. Calculate the sales tax on a particular
carton of wine if the wholesale cost of
each bottle is $8.50.
3. Find the rebate calculated by Darren for
a store that has sold over its monthly
quota with sales of $11 224.00.
4. Where could you complete an Advanced
Diploma in Business Marketing?
Ch 01 GM 1&2 YR 11 Page 32 Wednesday, December 15, 1999 5:49 PM
32
General Mathematics
remember
1. To change a percentage into a fraction, divide it by 100 and omit the % sign.
2. To change a fraction into a percentage multiply it by 100%.
3. To find a percentage of a quantity, express the percentage as a fraction and
multiply it by the given quantity.
4. To express one quantity as a percentage of another, form a fraction and
multiply it by 100%.
1G
WORKED
Example
21
Mat
d
hca
Percentages,
fractions
and
decimals
1 Write the following percentages as:
i common fractions
ii decimals.
a 15%
b 27%
e 7.3%
f 19.8%
i
WORKED
Example
22a
WORKED
Example
22b
Percentages
10 3--5- %
44 1--2- %
j
c 58%
g 42.06%
d 112%
h 100.75%
k 67 1--2- %
l
2 Change each of the following into percentages.
7
a 4--9b ----c 5--810
e
5
-----12
f
29
-----50
i
1 4--5-
j
2 3--4-
g
7
--8
3 Change each of the following into percentages.
a 0.63
b 0.29
c 0.51
e 0.05
f 0.003
g 0.6
i 1.2
j 4.03
95 3--4- %
d
2
--3
h
19
-----20
d 0.723
h 0.1998
4 Copy and complete the following table.
Decimal
Fraction
Percentage
a 0.18
b
7
--8
c
46%
d 0.9
e
1 2--3-
f
7.4%
g 1.003
h
i
32
-----40
3
15 -----%
20
Ch 01 GM 1&2 YR 11 Page 33 Wednesday, December 15, 1999 5:49 PM
Chapter 1 Arithmetic
WORKED
Example
23
WORKED
Example
24
5 multiple choice
Which of the following is the same as 23 1--3- %?
7
C -------A 0.23
B 23.3
300
D
70
-----3
E
7
-----30
6 multiple choice
Which of the following is the same as 1.06?
6
C 1 ----A 10.6%
B 106%
10
D 160%
E
6
--------100
7 Find the following, correct to 2 decimal places.
a 12% of $400
b 8% of $75
d 67.5% of $46.50
e 17% of 22 m
g 10.25% of 4060 kg
h 128% of 3.5L
j 0.7% of 74 mm
c
f
i
8 Express the first quantity as a percentage of the second.
a 5 of 20
b 7 of 35
c
d 4 months of 1 year
e 7 cm of 2 m
f
g 750 mL of 3 L
h 12 g of 540 g
i
j 75 m of 2.5 km
33
45% of $2318
3% of 7 tonnes
54% of 765 mL
60 of 2000
245 kg of 3 tonnes
40 mm of 20 cm
9 Rachael scored 87 points on her French test out of a possible 90 points. What
percentage is that?
10 In a kindergarten group of 24 children, 16 are boys. What percentage of the group
are girls?
11 A secondhand-furniture shopkeeper purchased a coffee table at a street market for
$40 and later sold it for $95.
a Find the profit that she made.
b Express the profit as a percentage of
the purchase price.
c Express the profit as a percentage
of the selling price.
Ch 01 GM 1&2 YR 11 Page 34 Wednesday, December 15, 1999 5:49 PM
34
General Mathematics
Percentage change
When a quantity is increased or decreased, we can compare the change with the original quantity by calculating the percentage change:
change in quantity
Percentage change = -------------------------------------------- × 100%
original quantity
WORKED Example 25
Over a period of 11 months, the price of Phonestra shares increased from $1.95 to $4.80
per share. Express the increase as a percentage of the original price.
THINK
1
2
WRITE
Increase = new price − original price
= $4.80 − $1.95
= $2.85
in price
Write the formula for percentage change. Percentage increase = increase
--------------------------------------- × 100%
original
price
(Since it is known that the price has increased,
Find the increase in price per share in
dollars.
use ‘increase’ instead of ‘change’.)
3
Substitute $2.85 for increase and $1.95 for
original price into the formula and evaluate.
2.85
= ---------- × 100%
1.95
= 146.15%
WORKED Example 26
At the beginning of the school year a student purchased a new Maths book for $39.95 and
sold it at the end of the same year for $15.00. Express the decrease in price as a percentage
of the original price.
THINK
WRITE
1
Find the decrease in price of the book
in dollars.
2
Write the formula for percentage
change. (Since the book lessened in
value, use ‘decrease’ instead of
‘change’.)
3
Substitute 24.95 for decrease and
39.95 for the original price into the
formula and evaluate.
Decrease = $39.95 − $15
= $24.95
decrease in price
Percentage decrease = ---------------------------------------- × 100%
original price
24.95
= ------------- × 100%
39.95
= 62.45%
Sometimes we know the percentage by which the original quantity has to be changed
and need to find the new quantity. In such cases the following simple algorithm should
be used:
1. Find the increase (or decrease) in the required units.
2. Add the increase to (or subtract the decrease from) the original quantity.
Ch 01 GM 1&2 YR 11 Page 35 Wednesday, December 15, 1999 5:49 PM
Chapter 1 Arithmetic
35
WORKED Example 27
Find the new amount if $52 were to be:
a decreased by 15%
b increased by 10%.
THINK
WRITE
a
a Decrease = 15% of $52
1
Find the decrease in dollars.
=
2
b
1
Subtract the decrease from the
original quantity to find the new
amount.
Find the increase in dollars.
Add the increase to the original
quantity to find the new amount.
× 52
= $7.80
New amount = original amount − decrease
= $52 − $7.80
= $44.20
b Increase = 10% of $52
=
2
15
--------100
10
--------100
× 52
= $5.20
New amount = original amount + increase
= $52 + $5.20
= $57.20
There is an alternative method to solve this type of problem. If the original quantity
is treated as 100%, then decreasing it by 15% (part a) gives (100 − 15)% = 85%.
Increasing it by 10% (part b), gives (100 + 10)% = 110% of the original quantity. So
instead of finding the decrease/increase first and then subtracting/adding it to the
original amount, we can find 85% and 110% of it respectively and thus obtain the
answers:
85
110
--------- × 52 = $44.20 and --------- × 52 = $57.20 as before.
100
100
WORKED Example 28
After spending 15% of her money on magazines, a student has $25.50 left. Find the initial
amount of money that she had.
THINK
1
2
Identify the unknown.
Express the remaining amount of
money as the percentage of the initial
amount in terms of x.
4
Form an equation by equating the
expression for the remaining amount with
the money that the student had left.
Solve for x.
5
Write the answer.
3
WRITE
Let the initial amount of money be x.
The remaining amount of money
= (100 − 15)%
= 85% of the initial amount.
∴ Remaining amount = 85% of x or 0.85x.
0.85x = 25.5
x = 25.5 ÷ 0.85
= 30
Initially the student had $30.00.
Ch 01 GM 1&2 YR 11 Page 36 Wednesday, December 15, 1999 5:49 PM
36
General Mathematics
remember
change in quantity
1. Percentage change = -------------------------------------------- × 100%
original quantity
2. To find the new quantity when the percentage change (increase or decrease) is
known, find the change in the required units and add to (or subtract from) the
original quantity.
3. Always treat the original amount as 100%.
1H
WORKED
Example
d
hca
Mat
25
Percentage change
1 Over a period of 5 years the price of a packet of long-life milk increased from 73c to
$1.17. Express the increase as a percentage of the original price.
2 The amount of water in a tank increased from 250 L (before rain) to 400 L (after
rain). Find the percentage increase.
Percentage
change
WORKED
Example
26
3 After 2 years, the price of a car purchased for $5700 had dropped to $1800. Express
the decrease as a percentage of the original price.
4 The attendance at a meditation class decreased from 26 people at the first session to
18 people at the second session. Find the percentage decrease.
5 The table below shows the share prices of different companies on the first day of two
consecutive months. For each company, identify whether there was an increase or
decrease in price, find the amount of increase/decrease in dollars and express it as a
percentage of the original price (to 2 decimal places).
Price on
1.9.98
Price on
1.10.98
a BHT
$12.82
$ 8.19
b Super Cole
$ 5.14
$ 7.25
c
$21.35
$19.00
d AMB
$ 4.70
$ 5.76
e ANX Bank
$ 9.52
$ 9.80
f
Pronto Co
$ 0.45
$ 0.61
g EIK Gold
$25.40
$29.12
h Motors
International
$ 7.80
$ 7.06
i
Optocom au
$ 5.70
$ 4.98
j
National
Metro
$18.28
$19.15
Company name
SuNatCo
Increase (I)
or decrease
(D)?
Change of
price ($)
Percentage
change
Ch 01 GM 1&2 YR 11 Page 37 Wednesday, December 15, 1999 5:49 PM
Chapter 1 Arithmetic
WORKED
Example
27
37
6 Find the new amount (correct to the nearest cent) for each of the following.
a $27 increased by 12%
b $45 decreased by 4%
c $12.15 increased by 60%
d $192.83 decreased by 37%
e $586 decreased by 12.5%
f $105.60 increased by 7.2%
g $59 increased by 22 2--3- %
h $1017.50 decreased by 20.46%
7 In country X the average lifespans for males and females increased from 72 years and
76 years by 2% and 3.5% respectively. Find the new average lifespans for males and
females in that country.
8 multiple choice
One style of refrigerator normally retails for $995.00, but during a ‘scratch and dent’
sale floor stock is reduced in price by 15%. A customer requests a further 10% discount on the sale price for a particular scratched refrigerator. The price the customer
wishes to pay (to the nearest 5 cents) is:
A $646.75
B $746.25
C $761.20
D $845.75
E $895.50
9 Lana is planning to buy a washing machine. She is given the following quotes for a
certain model:
a from Freddy Sparks: $780, less 10% if paying cash
b from Beaufort Bulk: $730 when paying by cheque, or 5% less if paying cash.
Which would be a better buy and by how much? (Assume that Lana is paying cash.)
WORKED
Example
28
10 After several weeks on a diet, a man’s weight decreased by 15% to 66.3 kg. Find the
weight of the man before the diet.
11 Over a 12-month period the rental payments for a one-bedroom flat increased by 20%
to $144 per week. Find the rental price before the increase.
12 After walking 20% of the way, a hiker is 14.4 km from
his destination. Find the total length of the trip.
answers
778
Answers
-----3 a 8 23
24
-----b 2 61
72
1
d 3 ----23
27
f 57 ----35
21
e 8 ----31
4 a
e
5 B
8 a
9 a
10 a
b
c 55 1--5-
12
0.062
Exercise 1F — Proportion
1 a 4.5
b 3
2 2--3-
c 7
e 5 1--3-
d 4.5
8 1--3-
f
g
h 40
i 15
j 60.5
2 C
3 D
4 4 solutions: 3:12 = 8:32, 12:3 = 32:8; 3:8 = 12:32,
8:3 = 32:12
b 9700
c 10
d 73
f 450
6 A
7 E
50
b 40
$800
b $1033.55
35c × 30 = $10.50, so $10 is not enough
It is really a bit over $1 for every 3 lollies, so it
will be over $10 for 30 lollies.
11 a $24; $20 is not enough
b 7 m 27 cm, which is 73 cm less than the length
needed.
6
8
9
10
Investigation — Estimation errors
Investigation — Gears and drive belts
Shape
a
Approxi- Exact
Estimate
area
mation
area
(to 4 dp.) error
130
183.4362 53.4362
471
497.6597 26.6597
5.4%
3.2 cm
10 cm
c
16.9 cm
20 cm
196
209.7563 13.7563
f 4.5
g −38
6.6%
a Underestimates, since error is positive
b Estimates for b and c are very good but rounding
to 1 significant figure in a has given a much lower
approximate area.
Exercise 1E — Ratios
1 a 2:3
b 1:7
c 1:3
d 2:5
e 4:3
f 4:15 g 40:31 h 36:11 i 8:25 j 100:33
2 a 6:1
b 13:15 c 4:1
d 5:1
e 1:2
f 20:3 g 1:6
h 5:3
i 5:1
j 11:5
k 43:20 l 1:200
b 8:7
c 2:3
d 28:25 e 9:2
3 a 3:4
f 2:7
g 10:3 h 33:16 i 23:15 j 16:65
4 E
5 a 1:4
b 6:5
c 100:39 d 23:2 e 1:12
f 63:125 g 400:21 h 20:1 i 1:15 j 54:5
6 E
7 a 120, 150
b 50, 550
c 86, 129
d 1875, 3125 e 240, 180
f 3.2, 0.4
g 1000, 1500, 1500
h 50, 75, 125
i 100, 150, 450 j 33, 9, 6
8 Maths 2 and Science 6
9 Brother 25, sister 35
10 a 6, 12, 18 b 40, 80, 120 c 9.60, 19.20, 28.80
11 1350, 1650
12 67.5, 90, 90, 112.5; Trapezium
13 350 g; 500 g; 10 g 14 a 175:6428
b 37
c 14
h
$4.85
7 8
a 63
b 91
x = 3.75; y = 6
a 15
b 4
2 480 rpm
7
4 ----11
d 10 1--7-
e 11
i 4.125 j
1
--3
c 8
3 150 rpm
4 16 cm
Exercise 1G — Percentages
1 a i
3
-----20
ii 0.15
b i
c i
29
-----50
ii 0.58
3
- ii 1.12
d i 1 ----25
e i
73
-----------1000
ii 0.073
f i
g i
2103
-----------5000
ii 0.4206
3
- ii 1.0075
h i 1 -------400
i i
53
--------500
k i
27
-----40
29%
12.12mm
7.3 cm
b 12.8
1 63
% estimation
error
18.15mm
b
5 a 7 5--7
ii 0.106
ii 0.675
27
--------100
99
--------500
ii 0.27
ii 0.198
j i
89
--------200
ii 0.445
l i
383
--------400
ii 0.9575
2 a 44 4--9- %
b 70%
c 62 1--2- %
d 66 2--3- %
e 41 2--3- %
f 58%
g 87 --12- %
h 95%
j
b
f
j
c 51%
g 60%
d 72.3%
h 19.98%
i
3 a
e
i
180%
63%
5%
120%
275%
29%
0.3%
403%
4 Decimal
a 0.18
b 0.875
c 0.46
d 0.9
e 1.6
f
0.074
g 1.003
h 0.8
i
0.1515
Fraction
Percentage
9
-----50
7
--8
23
-----50
9
-----10
1 2--3
37
--------500
3
1 ----------1000
32
-----40
303
-----------2000
18%
87 1--- %
2
46%
90%
166 2--- %
3
7.4%
100.3%
80%
3
-%
15 ----20
5 E
7 a 48
f 0.21
6 B
b 6
c 1043.1 d 31.39 e 3.74
g 416.15 h 4.48 i 413.1 j 0.52
8 a 25%
b 20%
c 3%
d 33 1--3- % e 3 1--2- %
f 8 1--6- %
g 25%
h 2 2--9- %
i 20%
j 3%