Terminology And Concepts

1. Estimation:

Estimation involves finding a value that is close enough to the right answer, usually by rounding numbers. This skill is useful for making quick calculations and decisions without needing exact values.

Example:

You are shopping and want to estimate the total cost of three items priced at R45.50, R78.25, and R120.75.

Steps:

1. Round each price to the nearest whole number.

R45.50 rounds to R46
R78.25 rounds to R78
R120.75 rounds to R121

2. Add the rounded prices.

R46 + R78 + R121 = R245

So, the estimated total cost is approximately R245 (written as R245 ≈ R244.50).

2. Addition (+):

Addition is one of the basic arithmetic operations used to find the total or sum by combining two or more numbers. It is fundamental in mathematics and is often the first operation learned by students.

Example:

The sum of 347 and 289 is 636.

3. Subtraction (-):

Subtraction is a basic arithmetic operation that involves taking away one number from another. It helps determine the difference between two numbers. The symbol for subtraction is a minus sign (-).

Example:

Thus, .

4. Multiplication(×):

Multiplication is a basic arithmetic operation that combines groups of equal size. It is essentially repeated addition.

Example:

if you have 3 groups of 4 apples, multiplication helps you find the total number of apples by calculating meaning you have a total of 12 items when you combine 3 groups of 4 items each.

5. Division (÷):

Division is a mathematical operation where a number, called the dividend, is divided by another number, called the divisor, to find how many times the divisor fits into the dividend. The result of the division is called the quotient. If there is a remainder that does not fit into the quotient, it is often noted separately.

Example:

Let’s take the division of 15 by 5 will be 3.

6. of

In mathematics, the word “of” often signifies multiplication.

Example:

Also used when dealing with percentages and calculations involving quantities like money. Here’s an example:

15% of R200 is R30

Similarly 0.15 × 200 = 30

7. BODMAS

BODMAS is an acronym that stands for

B Brackets (Alaways work inside-out)

O Of(multiplication)

D Division

M Multiplication

A Addition

S Subtraction.

It dictates the order in which mathematical operations should be carried out to ensure accurate results.

8. Power

In mathematics, “power” refers to the operation of raising a number to an exponent. The power operation involves two key components:

Base: The number that is being multiplied.
Exponent: The number that indicates how many times the base is multiplied by itself.

Example:

Let’s take 23 as an example:

\[2^3\;=\;2\;\times\;2\;\times\;2\;=\;8\;\]

Here, 2 is the base and 3 is the exponent. We multiply 2 by itself three times to get 8.

\[Consider\;\;54:=\;54=\;5\;\times\;5\;\times\;5\;\times\;5\;=\;625\]

In this case, 5 is the base and 4 is the exponent. We multiply 5 by itself four times to get 625.

9. Square root

The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the radical symbol (√). For example, the square root of 9 is 3 because 3 × 3 = 9

Example:

\[\sqrt{49}\;=\;7\;because\;7\;\times\;7\;=\;49\;.\]

10. Decimal Places

A “decimal place” refers to the position of a number to the right of the decimal point in a decimal number. Each position represents a fraction of a whole number based on powers of ten. Understanding decimal places is important for precision in calculations and various numerical operations.

Examples:

Consider the number 45.678:

The digit 6 is in the first decimal place, so it represents 6 tenths \[\;(\frac6{10})\\\]

The digit 7 is in the second decimal place, so it represents 7 hundredths \[\;(\frac7{100})\\\]

The digit 8 is in the third decimal place, so it represents 8 thousandths \[\;(\frac8{1000})\\\]

Decimal places:

1st Decimal Place (Tenths):

Example: In the number 3.1, the digit 1 is in the first decimal place, representing 1 tenth (0.1).

2nd Decimal Place (Hundredths):

Example: In the number 3.14, the digit 4 is in the second decimal place, representing 4 hundredths (0.04).

3rd Decimal Place (Thousandths):

Example: In the number 3.142, the digit 2 is in the third decimal place, representing 2 thousandths (0.002).

4th Decimal Place (Ten-thousandths):

Example: In the number 3.1428, the digit 8 is in the fourth decimal place, representing 8 ten-thousandths (0.0008).

5th Decimal Place (Hundred-thousandths):

Example: In the number 3.14285, the digit 5 is in the fifth decimal place, representing 5 hundred-thousandths (0.00005).

11. Rounding

Rounding off is a technique used to simplify a number to make it easier to work with, usually by reducing the number of decimal places. In rounding, you adjust the number to the nearest specified place value, such as the nearest whole number, tenth, hundredth, etc.

Examples:

Rounding to the nearest whole number:

Original amount: 45.67

To round to the nearest whole number, look at the digit in the tenths place (6). Since it’s 5 or greater, round up.

Rounded number: 46

Rounding to the nearest rand:

Original amount: R45.67

To round to the nearest rand, look at the digit in the cents place (6). Since it’s 5 or greater, round up.

Rounded amount: R46.00

11. Ratios

A ratio is a way to compare two or more quantities by showing their relative sizes. It tells us how much of one thing there is compared to another.

Example:

Imagine you have R100 and want to divide it between two people in a ratio of 3:2.

Add the parts of the ratio together: 3 + 2 = 5 parts in total.
Determine the value of one part: R100 ÷ 5 = R20 per part.
Calculate each person’s share:
- The first person gets 3 parts: 3 × R20 = R60.
- The second person gets 2 parts: 2 × R20 = R40.

So, in a ratio of 3:2, the first person receives R60 and the second person receives R40.

12. Equivalent ratios

Equivalent ratios are two or more ratios that express the same relationship between numbers. To determine if ratios are equivalent, you can simplify them to their lowest terms or cross-multiply to see if they are equal.

Example:

Suppose you have two ratios in rands:

Ratio 1: 10:15
Ratio 2: 20:30

13. Simplified ratios

Simplified ratios represent a way of expressing the relationship between two quantities in its most basic form. To simplify a ratio, you divide both numbers by their highest common factor (HCF), also known as the greatest common divisor (GCD).

Example:

Consider the ratio 24:36.

Find the HCF of 24 and 36:
- The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
- The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.
- The common factors are 1, 2, 3, 4, 6, and 12. The highest common factor is 12.
Divide both numbers by their HCF (12):
- 24 ÷ 12 = 2
- 36 ÷ 12 = 3
The simplified ratio is 2:3.

So, 24:36 simplifies to 2:3.

14. Proportion Of Ratios

two ratios are equivalent if their cross-products are equal.

Example 1:

This is equivalent to saying: 𝑎 × 𝑑 = 𝑏 × 𝑐

\[Let’s\;verify\;if\;the\;ratios\;\;\frac23\;\;\;and\;\;\frac46\;\;\;are\;in\;proportion.\\\]

Compute the cross-products:
2 × 6 = 12
3 × 4 = 12

Since the cross-products are equal (12 = 12), the ratios are in proportion.

Example 2:

\[Let’s\;verify\;if\;the\;ratios\;\;\frac58\;\;\;and\;\;\frac{10}{15}\;\;\;are\;in\;proportion.\\\]

Compute the cross-products:
5 × 15 = 75
8 × 10 = 80

Since the cross-products are not equal (75 ≠ 80), the ratios are not in proportion.

15. Direct proportion of ratios

refers to a relationship between two quantities where the ratio of one quantity to another remains constant.

Example:

\[\frac62=3\\\] \[\frac{15}3=3\\\] \[\frac{30}{10}=3\\\]

16. Indirect or inverse proportion

Is the ratio or relationship between two quantities, often used to express things like speed, price per unit, or interest. Here are a few key points:

Example:

6×15 = 9×10 = 90 (constant product)

8×12.5 = 10×10 = 100 (constant product)

17. Rates

A rate compares two quantities with different units, such as kilometers per hour (speed) or rands per kilogram (price). It can also be understood as the ratio or relationship between two quantities, often used to express things like speed, price per unit, or interest.

Example:

Unit Price: To find the unit price of an item, divide the total cost by the number of units. For example, if a 2-liter bottle of soda costs 30 rands, the unit price is 30 rands / 2 liters = 15 rands per liter.

Interest Rates: If you deposit 1000 rands in a bank at an interest rate of 5% per year, after one year, you’ll earn 1000 rands * 0.05 = 50 rands in interest.

Exchange Rates: If the exchange rate is 15 rands per US dollar, to convert 150 rands to US dollars, divide 150 rands by 15 rands per US dollar = 10 US dollars.

18. Speed

Speed is a measure of how quickly an object moves from one place to another. It is calculated using the formula

\[Speed=\frac{Dis\tan ce\;}{Time}\]

This formula means that speed is equal to the distance traveled divided by the time it takes to travel that distance.

you can actually remember it as a pyramid where:

D= Distance
S= Speed
T= Time

and can be summarized as:

\[D\;=\;S\;\times\;T\;(dis\tan ce\;=\;speed\;\times\;time)\] \[S=\frac DT\;(Speed=\frac{dis\tan ce}{time})\] \[T\;=\frac DS\;(time\;=\;\frac{dis\tan ce}{speed})\]

Example:

Suppose you walk 5 kilometers in 1 hour.

\[Speed=\frac{5\;km}{1\;hour}=5\;km/h\]

Your walking speed is 5 kilometers per hour.

19. Percentage

A percentage is a way of expressing a number as a fraction of 100. It’s denoted by the % symbol. To calculate a percentage, you multiply the number by the percentage and then divide by 100.

Example:

\[\frac{25}{100}=0.25\;\times\;100\;=\;25\%\]

20. Percentage increase

Measures how much something has grown relative to its original value.

Example:

using this formula

\[percentage\;increase=\frac{increase}{starting\;value}\times100\%\]

Starting value: R50
New value: R70

\[Increase:\;R70\;-\;R50\;=\;R20\] \[Percentage\;increase:\;\frac{20}{50}\times100\%\;=\;0.4\] \[Convert\;to\;percentage:\;0.4\;\times\;100\%\;=\;40\%\]

21. Percentage decrease

Measures how much a quantity has reduced compared to its original value.

Example:

using this formula

\[percentage\;decrease=\frac{decrease}{starting\;value}\times100\%\]

Starting value: R80
New value: R60

\[Increase:\;R80\;-\;R60\;=\;R20\] \[Percentage\;increase:\;\frac{20}{80}\times100\%\;=\;0.25\] \[Convert\;to\;percentage:\;0.25\;\times\;100\%\;=\;25\%\]

22. Discount

A discount is a reduction from the original price of a product or service. It is usually expressed as a percentage.

Example:

If a jacket costs R500 and the discount is 20%, the discount amount is 𝑅 500 × 0.20 = 𝑅 100

23. Sale price

A Sale price is the final price after a discount has been applied to the original price. To calculate it, you subtract the discount amount from the original price.

Example:

Original Price: R500
Discount: 20% (which is R100)

Calculation:

Find the discount amount: 20% of R500 = R100
Subtract the discount from the original price: R500 – R100 = R400

Sale Price: R400