Numeracy
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18.
Adding Fractions
Subtracting Fractions
Multiplying Fractions
Dividing Fractions (C)
Fractions, Decimals and Percentage
Calculating a Percentage
Finding a Percentage
Decimal Places
Significant Figures (C)
Standard Form
Ratio
Distance, Speed and Time
Simple Interest
Compound Interest (C)
Depreciation and Appreciation (C)
Exchange Rates
Direct Relationships
Finding the Average (mean)
Knightswood©Copyright Kayar publishers 2000
Outline Course Notes
Numeracy
page 1
1)
Adding Fractions
Example (Using the "top heavy" method):
1 23 + 2 34
5 + 11
3 4
~change to "TOP HEAVY" fractions
=
20 + 33
12
~use a COMMON DENOMINATOR
=
53 = 4 5
12
12
~change to a MIXED NUMBER
Answer:
2)
Subtracting Fractions
Example (Using the "top Heavy" method):
3 12 − 1 34
7−7
2 4
~change to "TOP HEAVY" fractions
=
14 − 7
4
~use a COMMON DENOMINATOR
=
7
4
Answer:
3)
=
13
4
~change to a MIXED NUMBER
Multiplying Fractions
Example:
3 13 % 1 45
10 % 9
3 5
Answer:
=
1
=
~change to "TOP HEAVY" fractions
3
2
10 % 9
3 5
6 =
1
~cancel top and bottom numbers
1
using common factors
6
~change into mixed numbers.
Note: Find out how to use the a b/c button (if you are lucky enough to have
one) on your calculator.
Knightswood©Copyright Kayar publishers 2000
Outline Course Notes
Numeracy
page 2
4)
(C)
Dividing Fractions
Example:
2 13 +
5
6
7+5
3 6
=
=
5)
7%6
3 5
1
14
5
~change to "TOP HEAVY " fractions
2
=
~turn last fraction up-side-down
1
PERCENTAGE
(multiply by 100)
DECIMAL
(Divide top by bottom)
2
0.5
50%
4
0.75
75%
8
0.125
12.5%
5 8
0.625
62.5%
7
0.7
70%
1.25
125%
3
1
10
1 14
Calculating a Percentage
Example:
Find 45% of £300.
Answer:
7)
~change to a MIXED NUMBER
Fraction, Decimals and Percentage
FRACTION
6)
and multiply
24
5
0.45 X 300 =
£135
~change the percentage to a
decimal
Finding a Percentage
Example:
Out of a class of 30, 5 people are absent. Express the absences
as a percentage of the total in the class.
Answer:
Amount you are interested in
Total amount
=
X
100%
5 % 100%
30
= 16.67%
Knightswood©Copyright Kayar publishers 2000
Outline Course Notes
Numeracy
page 3
8)
Decimal Places
Rule: Always round up if the next figure is a 5 or greater.
Examples:
9)
3.1415927 = 3.1
3.1415927 = 3.14
3.1415927 = 3.142
3.1415927 = 3.1416
3.1415927 = 3.14159
3.1415927 = 3.141593
to 1 decimal place
to 2 decimal places
to 3 decimal places
to 4 decimal places
to 5 decimal places
to 6 decimal places
4.396
to 2 decimal places
= 4.40
(C)
Significant Figures
The first non-zero digit from the left is the first significant digit. Examples:4890000
8526400
399400
399400
0.00345
0.01009
0.55841
= 4900000
= 8526000
= 400000
= 399000
= 0.0035
= 0.010
= 0.6
to 2 significant figures
to 4 significant figures
to 2 significant figures
to 3 significant figures
to 2 significant figures
to 2 significant figures
to 1 significant figure
10) Standard Form
To express a very large or very small number in the form a % 10 n where n is
an integer and a is a number between 1 and 10.
Examples:
= 4.8 % 10 6
= 2.61 % 10 7
= 1.399 % 10 9
= 2.37 % 10 −4
= 5.62 % 10 −7
4,800,000
26,100,000
1,399,000,000
0.000237
0.000000562
When multiplying numbers together in standard form you add the powers.
(3.4 % 10 5 )(5.8 % 10 7 )
Example:
=
3.4 x 5.8 x 10 5+7
=
19.72 x 10 12
=
1.972 x 10 13
When dividing numbers in standard form you subtract the powers.
(3.4 % 10 5 ) + (1.7 % 10 7 )
Example:
3.4 + 1.7 x 10 5−7
=
2.0 % 10 −2
=
Find out how to use the
EXP
key on your scientific calculator.
Knightswood©Copyright Kayar publishers 2000
Outline Course Notes
Numeracy
page 4
11) Ratio
Example:
Divide the sum of £1000 in the ratio of 3:2.
Answer:
3+2=5
1000 + 5 = 200
~add the numbers in the ratio
~divide the amount by this sum
3 X 200 = £600 and 2 X 200 = £400
~find answers by multiplying
12 Distance, Speed and Time
Example:
A train is travelling at 60 km/h. How far will it have travelled
after 3 hours 30 minutes?
Answer:
D = ?, S = 60 km/h,
T = 3.5 hours
D
From the triangle, D = S x T
= 60 x 3.5
= 210 km
Example:
S
T
Calculate the time it would take to travel 400 miles at a speed of
250 mph (miles per hour).
Answer:
D = 400 miles,
S = 250 mph,
T=?
D
From the triangle, T = D
S
T=
400
250
S
= 1.6
T
(0.6 hours is
0.6 x 60 = 36mins)
T = 1 hour 36 minutes
REMEMBER: Always use units of measurement which are consistent.
13) Simple Interest
Example:
A man invests £500 in an account which offers 11% simple
interest on the amount invested. How much will he have after 5
years?
Answer:
11% of £500
= 0.11 X 500 = £55
~find interest for 1 year
5 X £55 = £275
~find interest for 5 years
Amount in bank after 5 years = principal + interest
= £500 + £275
= £775
Knightswood©Copyright Kayar publishers 2000
Outline Course Notes
Numeracy
page 5
14) Compound Interest
(C)
Note: Interest is calculated at the end of every year (or perhaps half year)
and is calculated on whole number of pounds (pence ignored).
Example:
Calculate the compound interest made on a principal sum of
£500 over 5 years with a fixed interest rate of 11%.
Answer:
Year
Principal
Interest(11%) New Amount
1
500
55
555
2
555
61.05
616.05
3
616.05
67.76
683.81
4
683.81
75.22
759.03
5
758.94
83.49
842.52
Total Interest = £842.52 - £500 = £342.52
Note: Compound interest makes more money than simple interest
(compare this example with the previous one).
15) Depreciation and Appreciation
(C)
This is calculated in the same way as compound interest.
Example:
The value of a car depreciates by 12% each year. If the car is
initally worth £5000, how much will it be worth after 3 years?
Answer:
Year
Value
Depreciation (12%)
New Value
1
5,000
600
4,400
2
4,400
528
3,872
3
3,872
464.64
£3407.36
Example:
A valuable gem appreciates in value at the rate of 7% in the first
year, 10% in the next and 8% in the third year. If the original
value was £2000, how much will it be worth after 3 years?
Answer:
Year
Value
% Rate
Appreciation New Value
1
2,000
7
140
2,140
2
2,140
10
214
2,354
3
2,354
8
188.32
£2542.32
Note: The rate of interest (or appreciation/depreciation) may vary from year
to year (as in the last example).
Knightswood©Copyright Kayar publishers 2000
Outline Course Notes
Numeracy
page 6
16) Exchange Rates
Example:
Suppose the exchange rate is £1 = $1.66 (U.S. Dollars).
Answer:
a) How many dollars can you buy for £55.60?
Pounds
£1
£55.60
Dollars
$1.66
$1.66 x 55.6
= $92.30
b) At the same exchange rate, how many pounds could you buy
for $200?
Dollars
$1.66
$1
$200
Pounds
£1
£1 1.66 = £0.602
£0.602 x 200
£120.48
17) Direct Relationships
Example: If 1 litre of fuel cost £1.36 :Answer: a) How much will 15 litres cost?
Litres
1
15
Cost(£)
1.36
1.36 x 15
£20.40
b) How many litres of this fuel could you buy for £30?
Cost (£)
1.36
1
30
Litres
1
1 1.36 = 0.735
0.735 x 30
22.1 Litres
18) Finding the Average (mean)
An average is a typical value from a set of values.
Example:
Find the average of 8, 10, 15, 40, 56 and 111.
Answer:
Average (mean) = 8 + 10 + 15 + 40 + 56 + 111 = 40
6
Knightswood©Copyright Kayar publishers 2000
Outline Course Notes
Numeracy
page 7