WHAT IS A RATIO?
A RATIO compares two numbers
FRACTION
RATIO
6/18
6:18
21/7
21:7
Reduce all Ratios to LOWEST TERMS
A Ratio is SIMPLEST FORM or LOWEST TERMS
when the Great Common Factor of the terms is 1.
FRACTION
LOWEST
TERMS
RATIO
LOWEST
TERMS
6/18
2/9
6:18
2:9
EQUIVALENT RATIOS
EQUIVALENT RATIOS or EQUAL RATIOS
make the same comparison
Remember what you do the BOTTOM you must
do the same to the TOP (Either Multiplication or
Division)
EQUIVALENT
FRACTION FRACTIONS
2x
RATIO
EQUIVALENT
RATIOS
3/5 x2
6/10
2x
3:5 x2
6:10
3x
3/5 x3
9/15
3x
3:5 x3
9:15
4x
3/5 x4
12/20
4x
3:5 x4
12:20
25/5
5/1
25:5
5:1
COMPARING RATIOS
The DENOMINATORS MUST BE THE SAME to compare fractions
The SECOND NUMBER in the RATIO MUST BE THE SAME to compare
fractions
QUESTION :
Which fraction or ratio is larger 7:9 (7/9) or 4:6 (4/6)?
FRACTIONS
Numerator
7/9Denominator
RATIOS
Num.
4/6Den.
1st Number
7:92nd Number
1st
4:62nd
Remember to find COMMON DENOMINATOR or SIMILAR
SECOND NUMBER of the Ratio use LOWEST COMMON
MULTIPLE
9 – 9, 18
6- 6, 12, 18
18 is the Common Denominator or Common Second Number
FRACTIONS
2x
7/9x2
14/18
4/6x3
RATIOS
3x
2x
7:9x2
12 /18
14:18
3x
4:6x3
12:18
The Fraction/Ratio 7:9 (7/9) is LARGER than 4:6 (4/6)
FINDING THE MISSING MEASURE IN EQUIVALENT
RATIOS
THE QUESTION
X:12 = 15:10
STEP 1:
The second number of each ratio must be same so you need to find the LCM
or Lowest Common Multiple between 12 and 10
12, 24, 36, 48, 60
10, 20, 30, 40, 50, 60
THEREFORE THE LOWEST COMMON MULTIPLE IS 60
(5 x) X:12 (x5) = (6x) 15:10 (x6)
YOU MUST MULTIPLY EACH SIDE TO GET THE SECOND NUMBER
TO 60
REMEMBER WHAT YOU DO TO ONE SIDE YOU DO TO THE OTHER
NEW RATIO WITH COMMON SECOND NUMBER
(5x) X:60 = 90:60
STEP 2:
TO FIGURE THE MISSING VARIABLE X
YOU MUST USE THE OPPOSITE OF multiply by five WHICH IS dividing
by five
90 divided by 5 = 18
THEREFORE X = 18
UNDERSTANDING SCALES
THE FIRST NUMBER OF THE SCALE IS THE MODEL OF THE OBJECT
THE SECOND NUMBER OF THE SCALE IS THE ACTUAL OBJECT
This scale tell us that the Model is 20times
SMALLER
1cm
:
20cm
Model
(Smaller)
:
Actual
(Bigger)
This scale tell us that the Model is 50times
BIGGER
50cm
Model
(Bigger)
:
:
1cm
Actual
(Smaller)
FINDING THE RATIO OR SCALE
QUESTION:
A set designer builds a model of the stage and the different pieces of
furniture on it. He uses a scale of 5cm to 1m.
a) What is the ratio of a length on the model to a length of the
stage?
THE SCALE
Model
5cm:1mActual
(The scale CANNOT be used because the measurements are
different)
STEP 1: The scale must be all in the same measurement- always
choose the SMALLER MEASUREMENT in this case
CENTIMETERS
THE SCALE
Model
5cm:100cmActual
(The scale CAN be used because measurements are the SAME)
STEP 2: Remember to Reduce the Scale to Lowest Terms SO
THAT 1cm is on ONE SIDE OF THE SCALE
THE SCALE
5cm : 100cm is reduced to Model1cm : 20cm Actual
USING THE SCALE TO DISCOVER
THE MODEL OR ACTUAL OBJECT
QUESTION:
b) The length of a table is 1.4m. What is the length of the model?
Scale
1cm
Model
(Smaller)
: 20cm
: Actual
(Bigger)
This means that the MODEL is 20 times smaller than the ACTUAL
object.
STEP 1:
MODEL :
?
ACTUAL
1.4m
(This must be changed to centimeters to match the scale)
MODEL :
?
(Smaller)
ACTUAL
140 cm
(Bigger)
STEP 2:
Object or MODEL is getting SMALLER so you must use DIVISION
140cm divided by 20cm = 7cm
Must convert 7cm back to meters because the ACTUAL object was originally
in meters
THE MODEL IS 7cm OR 0.07m
WHAT IS UNIT RATE OR
UNIT PRICING?
UNIT means ONE
So there must be ONE on one side of the ratio ie. $1.00: 1m or $1.00/m (which
means $1.00 per metre)
EXAMPLE:
Ribbon costs $1.44 for 3m.
a) What is the cost per metre?
$1.44:3m or $1:44/3m
The UNIT RATE is ALWAYS THE SECOND NUMBER IN THE RATIO
STEP:The unit rate is METRES. So we must get 3 METRES TO 1 METRE
$1:44:3m
DIVIDED BY THREE = 0.48 : 1metre of 48 /m
EXAMPLE 2:
b) How much would 5m of ribbon cost?
UNIT RATE
0.48 : 1metre
STEP: To get to 5 METRES YOU MUST MULTIPLY BOTH SIDES BY
FIVE
5X
0.48
: 1metreX5
$2.40 : 5metres
WHAT IS A PERCENT?
PERCENT means “out of 100”. To write a number as a percent
express it as a fraction with a denominator of 100.
37/100 = 37%
THE PERCENT SYMBOL % COMES FROM THE NUMBER 100
FRACTION AS A PERCENT
¾
¾ can be made out of a fraction of 100 by multiplying the
denominator and numerator by 25.
25x
3/4
x25
75/100
75%
MORE FRACTIONS AND DECIMALS AS
PERCENTS
Remember that once you convert a decimal to a fraction you must
Reduce to Lowest Terms
0.45 = 5 by divide45/100divide by 5 or 9/20
ANOTHER EXAMPLE
Remember the difference between 7% and 70% when putting it into
a decimal
7% is 0.07
Whereas
70% is 0.7
Decimal
Fraction
Before
Reducing
Lowest
Terms
Fraction
Percent
0.60
0.30
0.15
0.58
0.38
60/100
30/100
15/100
58/100
38/100
3/5
3/10
3/20
29/50
19/50
60%
30%
15%
58%
38%
PERCENTS AS FRACTIONS AND
DECIMALS
6/25
Remember that Percents can be changed to decimals by
writing a fraction with a denominator of 100.
4x
6/25 x4
24/100
Remember that you can find the decimal by DIVIDING
THE NUMERATOR BY THE DENOMINATOR
24 divided by 100
is
0.24
MULTIPLYING TO FIND PERCENTS
3% of 340
Remember to CHANGE ALL PERCENTS TO A DECIMAL and
then MULTIPLY BY THE NUMBER
0.03 x 340 = 10.2
Remember that the ANSWER IS 10.2 NOT 10.2 %
DRAWING A CIRCLE GRAPH
First, you must find the angles you must draw in the circle
graph by:
Change Percent to a decimal and multiply by 360o (a
complete circle is 360o)
9% is 0.09 x 360o = 32.40 or 320
12% is 0.12 x 360o = 43.20 or 430
37% is 0.37 x 360o = 133.20 or 1330
25% is 0.25 x 360o = 900
17% is 0.17 x 360o = 61.20 or 610
DRAWING A CIRCLE GRAPH PART II
Draw a circle. Find the Radius (Half of the circle)
Radius
Start with the largest angle
Then take all your angles and BEGIN WITH YOUR LARGEST ANGLE FIRST. You
draw your angles FROM BIGGEST TO SMALLEST on the circle.
WHERE CANADIANS LIVE
Other
17%
Ontario 133o
Quebec 90o
Other 61o
BC 43o
Alberta 32o
Alberta
9%
BC
12%
Quebec
25%
Ontario
37%
DIVIDING TO FIND PERCENTS
19 out of 25
Remember this means the same as 19 divided by 25
or
19/25
19 divided by 25 = 0.76 or 76%
EXAMPLE #2
45% of 13.5
What is 100% of the Number?
Remember to FIND THE VALUE OF 1%
13.5 divide by 45
or
13.5/45 = 0.3
Therefore 0.3 is THE VALUE OF 1%
0.3 x 100 (%) = 30
To find 100% of the number MULTIPLY THE VALUE OF 1% by 100
Therefore 100% of the Number is 30
MORE DIVIDING TO FIND PERCENTS
You don’t have to place it in a denominator out of 100.
You can simply divide the numerator by denominator.
6/25
means
6 divided by 25
which is
0.24
or
24%
PERCENTS GREATER THAN 100%
Percent Greater Than 100%
131%
1100%
5000%
Decimal Form
1.31
11.00
50.00
EXAMPLE
230% of 75
Remember to CHANGE PERCENT TO A DECIMAL and
MULTIPLY by the Number
2.30 x 75 = 172.5
Remember your FINAL ANSWER is 172.5 not 172.5%
FINDING COMMISSION
COMMISSION on an item is the percent of the selling price that a
salesperson earns when he or she sells and item
Find how much money a salesperson can make with the following commission
of 5%?
5% of $350.00
Change PERCENT TO A DECIMAL and MULTIPLY BY THE DOLLAR
AMOUNT ($)
0.05 x $350.00 = $17.50
Therefore the salesperson would make $17.50 on 5% commission on a $350.00
sale
SIMPLE INTEREST
SIMPLE INTEREST is an amount paid on money borrowed or
invested
Calculate the amount of interest on each deposit for one year
$500.00 at 7%
Change PERCENT TO A DECIMAL and MULTIPLY by the
DOLLAR AMOUNT
$500.00 x 0.07 = $35
The person has to pay $35 in interest.
Amt.
Borrowed ($)
175.00
2750.00
Interest Rate
(%)
10%
13%
Interest
Owed ($)
.10 x 175.00
.13 x 2750.00
Total Owed
($)
$17.50
$357.50
MORE SIMPLE INTEREST PROBLEMS
$500.00 at 2% paid over 7 months
TO SOLVE THIS QUESTION YOU NEED TO UNDERSTAND
THAT 2% IS INTEREST RATE FOR 12 MONTHS
STEP 1:
YOU NEED TO KNOW THE INTEREST FOR 1 MONTH
1 MONTH INTEREST RATE:
2% divided by 12 months is
0.167%
STEP 2:
Take 1 Month Interest of 0.167% and multiply it by 7 months
7 month Interest Rate is 1.17%
STEP 3:
TO FIGURE OUT 7 MONTH INTEREST RATE ON $500
CHANGE 1.17% TO A DECIMAL 0.0117
THEN MULTIPLY BY $500
0.0117 x $500 = $5.85
EVEN MORE SIMPLE INTEREST
$750 at 12% for 2 years
STEP 1:
REMEMBER THAT 12% IS THE 12 MONTH RATE SO IF YOU
ARE DOING IT OVER 2 YEARS YOU MUST MULTIPLY 12%
BY 2 YEARS
1212% x 2 years = 24% interest rate
STEP 2:
TURN 24% INTO A DECIMAL (0.24) AND MULTIPLY BY $750
0.24 x $750 = $180
DISCOUNT AND SALE PRICE
A DISCOUNT is a reduction in the cost of an item
15% off a sweatshirt at $39.95
To find the DISCOUNT OFF AN ITEM you need to change the
PERCENT to a DECIMAL and MULTIPLY by the DOLLAR
AMOUNT ($)
0.15 x $39.95 = 5.9925
Rounded off is
$5.99
THE DISCOUNT IS $5.99
SALE PRICE of an item is its regular price minus the discount
To find the SALE PRICE one must subtract the DISCOUNT from
the ORIGINAL PRICE
$39.95 - $5.99 = $33.96
The SALE PRICE of the item with 15% OFF is $33.96
Regular Price ($)
14.99
119.99
Discount ($)
1.50
48
Sale Price ($)
14.99-1.50 = 13.49
119.99-48 = 71.99
HST (Harmonized Sales Tax)
HARMONIZED SALES TAX is made up of PST (7%) (Provincial
Sales Tax) and GST (5%) (Government Sales Tax).
Therefore HST is 13%
Calculating HST
A Computer is $1849.00. How much does it cost including sales tax or
the HST.
Example #1
$1849.00 x 13%
$1849.00 x 0.13 = $240.37
Next you need to ADD $240.37 to $1849.00
$1849.00 + $240.37 = $2089.37
Therefore the Final Price of the Computer with Sales Tax is $2089.37
Example #2
There is a MUCH FASTER WAY TO ADD THE HST on the Price
of the Computer
$1849.00 x 1.13 = $2089.37
With this example you skip the step of having to add the HST
on yourself