business mathematics jerome chapter 03
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3 -1Ratios
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Set up and manipulate ratios
After completing this chapter, you will be able to:
Set up and solve proportions
Use proportions to allocate or prorate an amount on a
proportionate basis
also…
Learning ObjectivesLearning
Objectives
LO 1.LO 1.
LO 2. & 3.LO 2. & 3.
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Use quoted exchange rates
to convert between currencies
Relate currency exchange rate movement
to currency appreciation or
depreciation
Learning ObjectivesLearning
Objectives
LO 4.LO 4.
LO 5.LO 5.
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LO 1.LO 1.
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Colleen, Heather and Mark’s partnership interests in Creative Crafts are in the
ratio of their capital contributions of $7800,
$5200 and $6500 respectively.
What is the ratio of Colleen’s to Heather’s to Marks’s partnership interest?
What is the ratio of Colleen’s to Heather’s to Marks’s partnership interest?
LO 1.LO 1.
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Colleen, Heather and Mark’s partnership interests in Creative Crafts are in the ratio of their capital contributions of
$7800, $5200 and $6500 respectively.
7800 5200 : 6500:
Colleen Heather Mark Expressed In colon
notation format
Expressed In colon
notation format
Equivalent ratio (each term divided by 100)
78 52 65: :Equivalent ratio with lowest terms Divide 52
into each oneDivide 52
into each one
1.5 : 1 : 1.25 1.5 : 1 : 1.25
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The ratio of the sales of Product X to the sales of Product Y is 4:3. The sales of product X in the next month are forecast to be $1800.
What will be the sales of product Y if the sales of the two products maintain
the same ratio?
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The ratio of the sales of Product X to the sales of Product Y is 4:3. The sales of product X in the next month are forecast to be $1800.
Since X : Y = 4 : 3, then $1800 : Y = 4 : 3
$1800Y
=34 Cross - multiply
4Y = 1800 * 3
Y = 1800 * 34
Divide both sides of the equation by 4
= $1350= $1350
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A 560 bed hospital operates with 232 registered nurses and 185 other support staff. The hospital is
about to open a new 86-bed wing.
Assuming comparable staffing levels, how many
more nurses and support staff will need to be hired?
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A 560 bed hospital operates with 232 registered nurses and 185
other support staff. The hospital is about to open a new 86-bed
wing.560 : 232 : 185 = 86 : RN : SS
560 =
86232 RN
R N
560RN = 232*86
560RN = 19952
RN = 19952 / 560
Hire 35.63 or 36
RN’s
Hire 35.63 or 36
RN’s
560 =185
86
SS
560SS = 185*86
560SS = 15910
SS = 15910 / 560
Hire 28.41 or 29
SS
Hire 28.41 or 29
SS
SS
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A punch recipe calls for fruit juice, ginger ale
and vodka in the ratio of 3:2:1.
If you are looking to make 2 litres of punch
for a party,
how much of each ingredient is needed?
LO 2. & 3.LO 2. & 3.
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A punch recipe calls for fruit juice, ginger ale
and vodka in the ratio of 3:2:1.
3+2+1 = 6Total Shares
2 litres / 6 = 333 ml per share
F J G A V
* 3 * 2 * 1= 1 litre
= 667 mls= 333 mls
333 ml per share
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A punch recipe calls for fruit juice, ginger ale
and vodka in the ratio of 3:2:1.
If you have 1.14 litres of vodka, how much
punch can you make?3+2+1 = 6Total Shares
1 1.14Punch6 = Cross - multiply
Punch = 6 * 1.14 litres = 6.84 litres= 6.84 litres
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You check the liquor cabinet and determine that someone has been drinking the vodka.
You have less than half a bottle, about 500 ml.
How much fruit juice and ginger ale do you use if you want to make more punch using the following new
punch recipe?
Fruit juice: ginger ale: vodka = 3 : 2 : 1.5
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How much fruit juice and ginger ale do you use if you want to make more punch using the following new punch recipe?:
Fruit juice: ginger ale: vodka = 3 : 2 : 1.5
F J G A
1.5 =3 F J
0.5
500 ml500 ml
Cross - multiply
Fruit Juice = 3 * 0.5 /1.5
= 1 litre= 1 litre
21.5
= G A0.5
Ginger Ale = 2 * 0.5 /1.5
= .667 litre = 667 ml.= .667 litre = 667 ml.
Cross - multiply
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If $1000 grows to $2500, find the percent change
10002500
100 1 2.5
1500 150 1.5
1000
=100%
x 2500
X = 2500*100% / 1000 = 250%
Cross - multiply
If $1000…………= 100% or 1What does $2500 = ?If $1000…………= 100% or 1What does $2500 = ?
…Also
250Initial ValueFinal Value
% (/100)
Difference or Change
$
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We can use the following alternative method to calculate the percent change!
We can use the following alternative method to calculate the percent change!
If $1000 grows to $2500, find the percent change
% change = Difference Base
This method is referred to as
the Base Method
This method is referred to as
the Base Method
$1000
$ 1000 2500
$ 1500
% change = $1500
= 1.5 or 150% increase
Initial(Base)ValueFinal Value
Difference
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If 15kg. of fruit shrinks to 3 kg. in the drying process, find the percent change.
15 3
100 1 20 .2
12 80 .8
If 15kg.…………= 100% or 1What does 3kg. = ?If 15kg.…………= 100% or 1What does 3kg. = ?
X = 3*100% / 15 = 20%
Cross - multiply
Initial ValueFinal Value
% (/100)
Difference or Change
Kg.
=
100% x
153
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Converting Percent Differences to ProportionsConverting Percent Differences to Proportions
Two companies, Markham Tire and Unionville Tire, normally offer the same tire prices. Markham
Tire has now marked-down the price of its Bridgestone Blizzard tires by 20%.
What percentage more will you pay if you buy your new tires from Unionville Tire?
Be Aware
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Be Aware…when interpreting certain comparisons!
“A is 40% of B”“A is 40% of B”
Means … A = 40 when B = 100 and AB = 40
100
“A is 40% greater than B”“A is 40% greater than B” or “A is 140% of B”“A is 140% of B”
Mean … A = 100 + 40 = 140 when B = 100 and A
B = 140100
“A is 40% less than B”“A is 40% less than B”
Means … A = 100 - 40 when B = 100 and AB = 60
100
or “A is 60% of B”“A is 60% of B”
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Be Aware
If A and B both represent positive quantities, it is NOT POSSIBLE that “A is 140% Less than
B”
If A and B both represent positive quantities, it is NOT POSSIBLE that “A is 140% Less than
B”
The limit is “A is 100% less than B”
…which makes A = 0…which makes A = 0
Back to the tire question
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Two companies, Markham Tire and Unionville Tire,
normally offer the same tire prices. Markham Tire has
now marked-down the price of its Bridgestone Blizzard
tires by 20%.
What percentage more will you pay if you buy your new tires from Unionville Tire?
Two companies, Markham Tire and Unionville Tire,
normally offer the same tire prices. Markham Tire has
now marked-down the price of its Bridgestone Blizzard
tires by 20%.
What percentage more will you pay if you buy your new tires from Unionville Tire?
…both MT & UT = 100%
MT = 100 – 20 = 80
MTUT
= 80100
To find the ‘price size’ of UT compared with MT,
we invert the proportion…UT
MT= 100
80= 1.25 or 125%
UT charges 125% of MT’s price(25% more than MT)
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Africa
North &
South Americ
a
EuropeEurope
Asia
AustraliaAustralia
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As we just saw, every area in the world uses some form of currency
As we just saw, every area in the world uses some form of currency
1. The currency of one country may not be able to be converted into that of another
on a one-for-one basis.
…some are recognized as having more value than others.
1. The currency of one country may not be able to be converted into that of another
on a one-for-one basis.
…some are recognized as having more value than others.
2. In order to place each currency on an ‘equivalent basis’, it is necessary to
‘exchange currency’ at the prevailing rate in the marketplace.
2. In order to place each currency on an ‘equivalent basis’, it is necessary to
‘exchange currency’ at the prevailing rate in the marketplace.
LO 4.LO 4.
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Frankfurt
Some major ‘marketplaces’ for prevailing currency rates
Some major ‘marketplaces’ for prevailing currency rates
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Table 3.1 Foreign Exchange Rates (noon, Toronto, April 11, 2001)
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C$ US$ Euro DM F S
Canadian $
= Y
Table 3.2 Currency Cross rates (noon, Toronto, April 11, 2001)
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Using the exchange rates given, calculate the number of yen that C$500 could purchase.
* C$500C$1.00¥79.82 =
¥ y = Cross - multiplyC$500
¥79.82C$1
¥y = ¥39,910Purchase
Canadian $ C$
NoteNote
If a C$.…………… = 79.82What does C$500 = ?If a C$.…………… = 79.82What does C$500 = ?
Y=
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C$
US$ Euro DM Y= =
… instead of this,
Canadian $ C$
we could use the following exchange rate to find the number of Canadian$’s that each yen will buy!
Calculation
1 Y = =
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Using the exchange rates given, calculate the number of yen that C$500 could purchase.
* C$500C$0.012528
¥1 =
= Cross - multiply¥ yC$500
¥y = ¥39,910Purchase
If C$ 0.012528..… = What does C$500 = ?If C$ 0.012528..… = What does C$500 = ?
¥1
C$0.012528¥1
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Using the exchange rates given, calculate the number of Canadian$ that 8000 could buy. Y=
US$ Euro DM Y= =Canadian $ C$
Cross - multiply
=Purchase
C$100.22
¥8000¥1.00=C$0.012528 C$X
* ¥8000¥1.00
C$0.012528=
C$ 0.012528
If a ……………… …. = C$ 0.012528What does 8000 = ?If a ……………… …. = C$ 0.012528What does 8000 = ?
¥¥
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How much will it cost in Canadian dollars to purchase US$500 of currency at a bank that charges
1.5% commission on the transaction?
Assume C$1.5628 = US$1.00Calculate the cost of US$ in Can$
Calculate the commission cost in Can$
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C$1.5628 US$500
C$781.40C$781.40
How much will it cost in Canadian dollars to purchase
US$500 of currency at a bank that charges 1.5% commission on the transaction? Assume C$1.56280 = US$1.00
C$781.40C$781.40* 1.5% = C$11.72C$11.72C$793.12C$793.12
C$ cost to Purchase US$500
C$ cost to Purchase US$500
CommissionUS$1.00
=C$ ?
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Gasoline sold for C$0.659 per litre in Vancouver and US$1.39 per gallon in Seattle. How much cheaper (based
on the Vancouver price) was gas in Seattle? (1 US gallon = 3.785 litres) Assume C$1.5628 = US$1.00
Calculate the cost in Canadian$ for the equivalent of 1 gallon of gas.
Convert US$1.39 to C$.
How much cheaper is the gas in Seattle?
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Gasoline sold for C$0.659 per litre in Vancouver and US$1.39 per gallon in Seattle. How much cheaper (based on the
Vancouver price) was gas in Seattle? (1 US gallon = 3.785 litres) Assume C$1.5628 = US$1.00
3.785 litres C$0.659 per litre = C$2.494
C$2.172C$2.172
US$1.39C$X
US$1.00=
C$1.5628Cross - multiply
C$X = C$1.5628 US$1.00 * US$1.39
C$2.494 - = C$0.322 cheaper
= C$2.172= C$2.172
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If the C$ weakens by 0.5% relative to the DM, what will be the new values
for DM per C$1.00 and C$ per DM1.00?
Canadian $ C$C$1.00 = DM1.4104
C$1.00 =
DM 1.403435 DM 1.403435DM1 = 1 = C$0.7125
DM 1.403435 DM 1.403435
If the Canadian dollar weakens, that means that we can buy fewer DM with each dollar!If the Canadian dollar weakens, that means that we can buy fewer DM with each dollar!
If C$ drops 0.5%, we can buy .995(DM 1.4104)or DM 1.403435
LO 5.LO 5.
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This completes Chapter 3
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