business mathematics jerome chapter 03

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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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