Maths Revision: N1: Numbers and Arithmetic

1.) Decimal Places:

As simple as what i learnt in primary school.- rounding with a decimal place counting as any number after the decimal point 

2.) Significant figure:

A significant figure, is any number after the first digit that isn't 0.

eg. 34902 has 5 significant figures

and 0.00420 has 3 significant figures

rounding to a significant figure is just just the same as rounding but instead of counting a place as a figure you count it as a significant figure

3.) When estimating ALWAYS do it to the significant figure. round to one significant figure UNLESS IT SAYS OTHERWISE

What do upper and lower bounds tell you?

What a rounded number could be. The range of all the number that would round to a given number

What does the upper bound tell you?

What the highest possible number is that the rounded number could be.

What does the lower bound tell you?

What the lowest possible number is that the rounded number could be.

THE UPPER BOUND IS NOT INCLUDED IN THE RANGE OF POSSIBLE VALUES

THE UPPER BOUND IS NOT INCLUDED IN THE RANGE OF POSSIBLE VALUES

1.) How do you work them out- SINGLE NUMBERS  

• By going half the rounded unit above and below.
• example: upper and lower bound of 8.5 to 1dp. --> UPPER = 8.45 LOWER = 8.55
• example: upper and lower bound of 1500 to 100 km --> UPPER = 1550 LOWER = 1450

2.) How do you work them out- SINGLE NUMBER X ?

• Find the upper and lower bounds of the single number
• Figure out the difference between the single number and one of the bounds--> eg. 5 to 1sf. upper bound is 5.5. --> THE DIFFERENCE IS 0.5
• Times 0.5 by ? (how many of the item there is)
• Times the single number by ? = Subtotal
• To find the Upper Bound ADD the difference to the subtotal
• To find the Lower Bound MINUS the difference to the subtotal

3.) How do you work them out- TWO ROUNDED NUMBERS - AREA

• Figure out the upper bounds and lower bounds for each number
• Times the Upper Bounds together to find the Upper Bound
• Times the Lower Bounds together to find the Lower Bound

• Deviding by a number is the same as multiplying by its reciprocal.
• The reciprocal is one over the orginal number 
• Example: 4 = 1/4

• ROUND TO ONE SIGNIFICANT FIGURE
• you must always put ≈  as an 'approximatly equals' sign

• find common denominators
• multiply to fractions out so that they are the equivalent fraction and the numerator and denominators are equivalent.
• order them
• write down the oringinal fractions in order

• Adding: make the fractions have the same denominator by finding the lowest common multiple then add the numerators.
• Subtracting: make the fractions have the same denominator by finding the lowest common multiple then subtract the numerators.
• Multiplying: Cancel down diagonally then multiply the tops together and multiply the bottoms together
• Deviding: Kiss and flip, then cancel down diagonally them multiply the tops together and multiply the bottoms together

WITH MIXED NUMBERS, CONVERT INTO TOP HEAVY FRACTION BUT REMEMBER TO CONVERT BACK

• Multiplying a fraction is the same as Deviding by its reciprical
• EXAMPLE: x 1/2 = deviding  by 2/1 OR 2
• A number multiplied by its reciprocal will ALWAYS be 1
• EXAMPLE: 2/3 x 3/2 = 1
• You can convert from a mixed number to a top heavy fraction by timesing the big number by the denominator and adding it to the numerator
• EXAMPLE: 4 1/6 = 25/6

• Terminating decimals don't carry on and on
• Recurring decimals carry on and on

How to know if a fraction will be terminating or recurring (if its a percentage, just convert to fraction):

• If the PRIME FACTORS are 2 and/or 5 or combinations of 2 and 5 the fraction will be Terminating
• to find the prime factors do the prime factor tree where you keep deviding it until you get a prime number (which you then circle) at the end of each branch

How to convert a recurring decimal to a fraction (where space for workings and given and workings are required) LONG METHOD:

• Write x =  ? (the recurring number) (eg.0.123123123123123...)
• Multiply to get one 'repeated lump' past the decimal point (eg. 123.123123123123123...)
• Write down how much you needed to times it by to get it there (eg. 10, 100, 1000)- lets call this number 'A'
• Write  'A' x = and then step two (eg. 1000x = 123.123123123123...)
• Then -x (the original recurring number) (eg. 0.123123123123123...)
• Then write (what ever 'A' is - x LETS CALL THIS 'B') = (just the repeated lump) (eg. 123)
• Finally write  x =  just the repeated lump/ 'B'
• CANCELL DOWN

EXAMPLE:

1000r = 234.234234234234234...

-       r = -  0.234234234243234...

---------------------------------------------

 999r = 234.0

       r = 234/999 = 26/111

Fraction --> Decimal:

• devide the numerator by the denominator
• EXAMPLE: 4/5 = 4 % 5 = 0.8

Decimal --> Fraction:

• Put numbers after the decimal point over (eg. 10, 100, 1000, 10000)
• You can decide how many 0's there will be

Decimal --> Percentage:

• Multiply by 100
• EXAMPLE: 0.32 X 100 = 34%

Percentage --> Decimal:

• Devide by 100
• EXAMPLE: 62% / 100 = 0.62

Percentage --> Fraction:

• Write the number that you percent is over 100.
• If the percent is not a whole number, then multiply both top and bottom by 10 for every number after the decimal point. (For example, if there is one number after the decimal, then use 10, if there are two then use 100, etc.)
• Simplify
• EXAMPLE: 0.63 --> 0.63/ 100 -->  X top and bottom by 100 =63/ 10000
• EXAMPLE: 89 --> 89/ 100

Fractions --> Percentage:

• Divide the numerator by the denominator then multiply by 100%
• EXAMPLE: 5/8 = 5 % 8 x 100% = 62.5%

TO REMEMBER:

• 1/2 - 0.5 - 50%
• 1/4 - 0.25 - 25%
• 3/4 - 0.75 - 75%
• 1/3 - 0.33333... OR 0.3 (with recurring sign) - 33 1/3%
• 2/3 - 0.66666... OR 0.6 (with recurring sign) - 66 2/3%
• 1/10 - 0.1 - 10%
• 7/10 - 0.7 - 70%
• 1/5 - 0.2 - 20%
• 2/5 - 0.4 - 40%
• 3/5 - 0.6 - 60%
• 4/5 - 0.8 - 80%

LOOK AT THE END OF THE 'FRACTIONS, DECIMALS, AND PERCETAGES' SECTION OF DVD FOR A QUICK QUIZ ON THEM