Basic Calculations and Percentages - Mindset Learn

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

(2)

? 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

= 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

(2)

b. 0,000000000121

(2)

c. The speed of light 299 792,4 km/sec

(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

(3)

= 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 ) =

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X-ercise 3

Calculate the following: Give your answer in Scientific Notation

a. (1,7 x 104 ) + (2,95 x 105) round off to 2 significant figures

(3)

b. (3,74 x 10-3) 4 round off to 3 significant figures

(3)

a. There are 2,3 x 108 bees in a hive. If each bee weighs 1,38 x 10-5g, 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

(2)

= 35 / 100 x 600 or = 35 % x 600

= R210

= R210

Concept: 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

2. Decrease 150 by 3 150 ? 3/100 x 150 = 150 ? 4,5 = 145,5

Increase means to make bigger, ADD

(2)

or

R365 + 15 % x R365

=R365 + R54,75

=R419,75

Decrease means to make smaller, SUBTRACT (2)

or

150 ? 3 % x 150

= 350 ? 4,5

= 145,5

Concept

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

(2)

5/40 x 100/1

5 / 40 % =

= 12,5%

b. 45 and 300

(2)

45/ 300 x 100/1

45 / 300 % =

= 15%

2. In a school 10 out of every 45 learners walk to school.

(give the answer to two decimal places)

(3)

a. What percentage walk to school? 10 / 45 % = = 22,22%

b. What percentage don't walk to school? 35 / 45 % = = 77,78%

Concept:

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 profit x 100

cost price

1

actual profit ? Cost Price %

actual loss x 100

cost price

1

actual loss ? 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? (3)

Difference is given:

R22,50

Write as fraction: profit = 22,5

Cost

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?

(4)

Cost price = 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?

(4)

Cost price = selling price 100% - % decrease

= R 45 000 100% - 10%

= R45 000 90%

45 000 ? 90 % =

= R50 000

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