Saturday X-tra
X-Sheet: 1
Basic Calculations and Percentages
Key Concepts
In this session, we will focus on summarising what you need to know about:
- Basic Calculations (revision of earlier work)
o Scientific Notation
o Rounding off
o Percentage
o Fractions
o Calculator work
- Working with percentages
o Finding the percentage of a quantity
o Increasing or decreasing a quantity by a given percentage
o Writing one quantity as a percentage of another
o Calculating the percentage profit or loss
o Calculating price and selling price
Terminology & definitions
- Scientific Notation = a way of writing numbers that deals with values too
large or too small to be written in standard decimal form.
- Exponent = the power of a base number like 10. So in 103, 3 is the exponent
and indicates how many times to multiply 10 (instead of writing out 10 x 10
x10 or 1,000).
- Percentage = a way of expressing a number as a fraction of one hundred.
General tips when answering Maths Literacy questions
• Always remember to give the units in your answer.
• Always check that you have rounded off your answer to the correct number of
decimal places or significant figures. If no instruction is given, then round off
to TWO decimal places.
• Use a comma and not a dot to show your decimal places.
• Show your number sentence, substitution and working.
Concept:
Scientific Notation.
•
Scientific notation is the way that scientists work with very large or very
small numbers.
X-ample 1
Express as a number:
• If the exponent is positive, the number is very big and the comma must move
to the right.
• If the exponent is negative the number is very small the comma must move to
the left.
• 5,0056 x 106 (move comma → six places) = 5 005 600
• 4,5 x10-5
(move comma ← five places) = 0, 000045
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X-ercise 1
Express as a number:
a. 3,21 x 105
b. 4,9421 x 103
c. 4,5 x10-5
d. 1,09 x 10-7
X-ample 2
In order to write in Scientific Notation:
•
If the number is a very big number (1whole or bigger) the exponent
must be positive.
•
If the number is a very small number (smaller than 1 whole) the
exponent must be negative.
•
Count the number of places you need to move the decimal comma
so that the number can be written with one digit in front of the
comma, then x 10, and write the number of places you moved the
comma as the exponent of 10.
Write in Scientific Notation:
2. 139 000 000 = 1,39 x 108
• one digit
• comma 1,39
• x 10
1,39 x 10
• +exponent (it’s a very big number)
• exponent 8 (comma moved eight times)
3. 0,000504
(2)
= 5,04 x 10-4
(2)
• one digit
• comma 5,04
• x 10
5,04 x 10
• - exponent (it’s a very small number - it is smaller than 1)
• exponent 4 (comma moved four times)
X-ercise 2
Write in Scientific Notation:
a. 600 500 300 000
b. 0,000000000121
c. The speed of light 299 792,4 km/sec
(2)
(2)
(2)
X-ample 3: Calculating in Scientific Notation
Calculate the following and give your answer in Scientific Notation:
You need to use the scientific mode on your calculator.
1. (5,67 x 1012 ) x (3,45 x 10-8) round off to 2 significant figures
= 5,67 x 1012
= 5,7 x 1012 round off to 2 significant figures
Enter into calculator: ( 5 , 6 7 x10x 1 2 ) x ( 5 , 6 7 x10x (-) 8 ) =
(3)
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X-ercise 3
Calculate the following: Give your answer in Scientific Notation
(3)
a. (1,7 x 104 ) + (2,95 x 105) round off to 2 significant figures
b. (3,74 x 10-3) 4 round off to 3 significant figures
(3)
8
-5
a. There are 2,3 x 10 bees in a hive. If each bee weighs 1,38 x 10 g, what is
the total weight of all the bees? round off to 3 significant figures
(4)
Concept:
Basic calculations with numbers and percentages
X-ample 4
Converting basic numbers into percentages
a. 30/100 – simply multiply by 100 / 1
This gives 30% (revision of grade 10 maths literacy work)
b. 5 / 20 x 100 / 1
= 25%
c. 0.25 x 100 – move the decimal points 2 places to the right, answer is
therefore 25%
X-ample 5
Writing percentages as decimals
Write percentages as decimals
1. 20% = 20 / 100. Dividing by 100 means moving the comma two places to the
left = 0,2
2. 5% = 5/100 move comma two places to the left = 0,05
3. 120% = 120/100 – move comma two places to the left = 1,2 (don’t need to
add the 0 at the end)
Concept:
Revise finding the percentage of a quantity
X-ample 1
Calculate: 35% of R600
= 35 / 100 x 600 or
= 35 % x 600
= R210
= R210
Concept:
(2)
Increase or decrease a quantity by a given percentage.
Initial value I x (1 + %/100)
X-ample 2
1. Increase R365 by 15
R365 + 15 / 100 x R365
=R365 + R54,75
=R419,75
Increase means to make bigger, ADD
R365 + 15 % x R365
or
=R365 + R54,75
=R419,75
2. Decrease 150 by 3
150 – 3/100 x 150
= 150 – 4,5
= 145,5
Decrease means to make smaller, SUBTRACT (2)
or
150 – 3 % x 150
= 350 – 4,5
= 145,5
Concept
(2)
To write one quantity as a percentage of another.
•
write one quantity as a fraction of the other
•
change the fraction to a percentage
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X-ample 3
1. Write the first number as a percentage of the second.
a. 5 and 40
5 / 40 % =
5/40 x 100/1
= 12,5%
b. 45 and 300
45/ 300 x 100/1
= 15%
(2)
(2)
45 / 300 % =
2. In a school 10 out of every 45 learners walk to school.
(give the answer to two decimal places)
a. What percentage walk to school?
10 / 45 % =
= 22,22%
b. What percentage don’t walk to school?
35 / 45 % =
= 77,78%
Concept:
(3)
Finding the percentage increase or decrease of quantities.
• the units of the quantities you are comparing must be the same.
• find the change or difference in the two quantities.
• write the difference as a fraction of the original amount/quantity.
• change the fraction to a percentage.
• Other words used for decrease: reduction, discount, less, loss.
• Other words used for increase: more, profit.
X-ample 4:
1. At the beginning of high school your height was 1,09m you have grown by
20cm. What is your percentage increase in height, correct to the nearest
whole number?
(4)
1,09m = 109cm
Difference is given: 20cm
Write as fraction: 20 / 109
Change fraction to percentage: 20 ÷ 109 %
= 18,3486 …
= 18 %
2. The price of a TV is reduced on a sale from R4999 to R3500. What is the
percentage reduction, correct to the nearest whole number?
(4)
Was R4999
Now only R3500
Difference: R4999 – R3500 = R1499
Write as fraction: 1499 / 4999
Change fraction to a percentage: 1499 ÷ 4999 %
= 29,985
= 30 %
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Concept:
A profit or loss can be written as a percentage.
•
It is calculated the same as the previous increase and decrease
examples.
actual loss x 100
cost price
1
actual loss ÷ cost price %
actual profit x 100
cost price
1
actual profit ÷ Cost Price %
X-ample 5
1. A shop keeper makes a R22,50 profit on a pair of jeans. He bought
the jeans for R90 from the manufactures. What is the percentage profit?
Difference is given:
Write as fraction: profit
Cost
(3)
R22,50
=
22,5
90
Change fraction to percentage: 22,50 ÷ 90 %
= 25% profit
Concept:
Calculate cost price & selling price.
X-ample 6
1. A street hawker sells pies at R6,50. He says he is making a 30% profit.
What is the cost price of each pie?
Cost price
=
(4)
selling price
100% + %increase
=
R6,50
(100% + 30 %)
=
R6,50
130%
=
6,50 ÷ 130 % =
R5
2. A car dealer sells a car for R45 000. Because of the poor economy, he
had to sell it at a 10% loss. What was the cost price of the car?
Cost price
=
(4)
selling price
100% - % decrease
= R 45 000
100% - 10%
= R45 000
90%
45 000 ÷ 90 % =
= R50 000
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X-ercise
1. Complete the table by calculating the percentage increase or decrease of the
following quantities. (round off to 2 decimal places)
(6)
Original amount
New amount
a.
250 ml
125ml
b.
1m 125 mm
2m
c.
R400
R550
increase or
decrease
%
2. Complete the table by calculating the percentage profit or loss for the following
amounts.
(12)
Cost
price
Selling
Price
a.
R100
R150
b.
R3 000
R2 500
c.
R25 000
R75 000
d.
R9
R15,50
The difference
between the
cost and
selling price
Profit or loss?
%
3. The price of a motor car was reduced by from R95 000 to R75 000. Calculate the
percentage reduction.
(4)
4. There are 560 learners in a school in 2010. This is 25 % more than 2009. How
many learners where in the school in 2009.
(4)
5. Find the selling price of a shirt which cost R120 to make and was sold at a
loss of 12%.
(3)
6. A shop keeper wants to sells a pair of shoes at a 60% mark up. When they
don’t sell, he discounts them by 10%. How much does he finally sell the shoes
for if he paid R135 for them?
(6)
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7. A bicycle shop buys bicycles for R1 500 from the factory and sells them for
R2625. What percentage profit do they make?
(4)
8. A book shop buys a book from the printers for R500. They sell the book for
35% more. What was the selling price of the book?
(3)
9. A dress shop makes a 15% loss when they sell a matric dance dress that
costs R1 500 to make. How much did they sell the dress for?
(4)
X-ercise Answers
Exercise 1: Scientific Notation – Express as a Number
a. 321000
b. 4942,1
c. 0,000045
d. 0,000000109
Exercise 2: Scientific Notation – Write in Scientific Notation
a. 6,005003 x 106
b. 1,21 x 10-10
(Unit!)
c. 2,997924 x 105 km/sec
Exercise 3: Calculating in Scientific Notation
a. 3,12 x105
→
3,1 x 105
(two significant figures)
b. 1,956530 x 10-10 →
1,96 x 10-10 (round off to 3 significant figures)
-5
c. 2,3 x 108 x 1,38 x 10
= 3,174 x 103g
= 3,17 x 103 g (Unit!)
(round off to 3 significant figures )
Question 1
1.
Original amount
New amount
increase or
decrease
%
a.
250 ml
125ml
Decrease (125ml)
50%
b.
1m 125 mm
1125 mm
2m
2000 mm
Increase (875 mm)
77,78%
c.
R400
R550
Increase (R150)
37,5%
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Question 2
2.
Cost
price
Selling
Price
The difference
between the
cost and
selling price
a.
R100
R150
R50
Profit
50%
b.
R3 000
R2 500
R500
Loss
16,67%
c.
R25 000
R75 000
R50 000
Profit
200%
d.
R9
R15,50
R6,50
Profit
72,22%
Profit or loss?
%
Question 3
21,05%
Question 4
448 Learners in 2009
Question 5
R105,60
Question 6
R135 + 60% of R135 = R216
R216 – 10% of R216 = R194,40
Question 7
Difference is given:
Write as fraction:
R2625 – R1500 = R1125
1125
1500
Change fraction to percentage: 1125 ÷ 1500 %
= 75%
Question 8
Selling Price = 500 + (35% x 500 )
= R675
= R675
Question 9
Selling Price = 1 500 - 15% x 1500
= R1275
or
(100% + 35% ) x 500
= 135% x R500
= R675
or
(100% - 15% ) x 1500
= 85% x R1500
= R1275
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