Simple Interest, Ratios and Proportions - Home - …smcmaths.webs.com/Form-4/Simple Interest, Ratios and...Simple interest, Ratios and Proportions Form 4 [email protected] 2

Simpleinterest,RatiosandProportions Form4

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Simple Interest, Ratios and Proportions

SECSyllabuscovered:

1.8.1Money

• Solveproblemsonpersonalandhouseholdfinanceinvolvingearnings(e.g.stocks),simpleinterest,taxandinsurance.

1.6.1Ratio

• Userationotationinpracticalsituations(e.g.inmapsandscaledrawings).

• Recognisetheconnectionbetweenratiosandfractions.

• Reduceratiostotheirsimplestform.

• Divideaquantityinagivenratio.1.6.2Proportion

• Understandandusetheelementaryideasandnotationofdirectandinverseproportion.

• Calculateanunknownquantityfromquantitiesthatvaryindirectorinverseproportion.


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SimpleInterest

Whensomeonelendsmoneytosomeoneelse,theborrowerusuallypaysafeetothelender.

Thisfeeiscalled‘interest’.Thesimpleinterestformulaisasfollows:

where:

I=Interest–thetotalamountofinterestpaid

P=Principal–theamountofmoney

T=Time–thetimeoftheloan

R=Rate–isthepercentagechargedasinteresteachyear.

ThreeotherformulasareusedtofindP,RandT.

𝑃 = !""!!"

𝑇 = !""!!"

𝑅 = !""!!"

Whenever,themoneyisborrowed,thetotalamounttobepaidbackistheprincipalborrowed

plustheinterestcharged.

I=!"#!""

TotalRepayments=Principal+Interest


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Example1:Astudentpurchasesacomputerbyobtainingasimpleinterestloan.Thecomputercosts

$1500andinterestrateontheloan12%.Theloanistobepaidbackin2years.Howmuchinterestis

incurredovertheyears?

I=PTR/100

=1500x2x12/100

=360

TotalPaid=$1500+$360=$1860

Example2:Youborrow$10000for3yearsatasimpleinterestrateof5%.Findthetotalamountpaid.

I=PTR/100

=10000x3x5/100

=1500

Total=$10000+$1500=$11500

Example3:Susanborrows$8650tobuyausedcarandischarged4.5%interest.Ifthetermofher

borrowingis5years,howmuchinterestdoesshepayintotal?

I=PTR/100

=8650x5x4.5/100

=1946.25

Interest=$1946.25


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Example4:Withaninterestrateof10%for18months,theinterestincurredoverthemonthsisof

$1199.47.Findtheamountwhichheborrowed.

18months=1year6months=1.5years

R=10%

𝐼 = 𝑃𝑇𝑅100

𝑃 =100𝐼𝑇𝑅

1199.47 × 1001.5 × 10

= 𝑃

$7996.47=P

SECQuestionsonpercentages

1. Fillinthemissingcellsofthetable.

Fractioninitssimplestform Decimal Percentages(%)

½ 0.5

0.75

5

33.𝟑

(7marks)

(P2B,May2015,no.4)


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2. Stellainvestedtwosumsofmoneyatasimpleinterestworkedoutmanually.SumSwasinvestedinMarch2009whilstSumTwasinvestedinMarch2011.StellakeepstheinterestshecollecteveryMarchathome.StellaisusingaspreadsheettocalculatethetotalsummoneycollectedfromthesetwoinvestmentsbyMarch2015.

I. WhatnumbershouldStellawriteincellC4?

II. IncellB5,writeaformulathatworksoutthesimpleinterestcollectedfromSumSforyears2009to2015.

III. IncellC5,writeaformulathatworksoutthesimpleinterestcollectedfromSumTforyears2011to2015.

IV. IncellD5,writeaformulathataddsthevaluesinthecellsB5andC5.

V. WorkoutthetotalsimpleinterestcollectedfromthesetwoinvestmentsbyMarch2015.

(7marks)

(P1,May2015,no.2)


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Ratios

Section 1 Understanding Ratios

A ratio shows the relative sizes of two or more values.

There are 3 blue squares to 1 yellow square

Ratios can be shown in different ways:

3 : 1 Using the ":" to separate example values

¾ as a fraction, by dividing one value by the total

(3 out of 4 boxes are blue)

0.75 as a decimal

75% as a percentage

Example 1: If there is 1 boy and 4 girls the ratio is 1:4


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Section 2 Simplifying Integer Ratios

We use ratios to make comparisons between two or more things.

How can we write the ratio of squares is to circles?

How can we write it as a ratio?

3 : 6

The ratio of squares to circles is:

Squares : Circles

3 : 6

When we have a ratio we can still simplify it by dividing BOTH sides with the same number.

Squares : Circles

3 : 6 ÷3

1 : 2

Example 1: Simplify the following ratios:

18 : 21

12 : 32

4 : 16

SupportExercisePg318Exercise20ANos1

3 squares

6 circles


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Section 3 Simplifying Ratios with Different Units

Whenhavingdifferentunitswemustbecareful.

40m:3m

Step1:Getallunitstobethesame

3m=3x100=300cm

Step2:Simplifyasmuchaspossible

40cm : 300cm ÷10

4cm : 30cm ÷2

2cm : 15cm

Example1:Giveeachratioinitssimplestform.

€4.60

: 240cents

6cm5mm : 15mm


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€5 : 250cents

SupportExercisePg318Exercise20ANo2

Section 4 Simplifying Ratios with decimals, fractions and Mixed Numbers

Ratios with Decimals

Example 1: Give each ratio in its simplest form.

0.8 : 1.2 To remove the decimal point we must multiply both ratios by 10.

WHAT IS DONE ON THE LEFT HAND SIDE MUST BE DONE ON THE RIGHT HAND SIDE.

8 : 12 ÷ 4

2 : 3

Example 2: Give each ratio in its simplest form.

0.6 : 2

0.2 : 0.24 : 3


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Ratios with Fractions and Mixed Numbers

When it comes to fractions we must change both fractions in the ratio to whole numbers.

This is done by multiplying throughout with the LCM.

Example 3: Express in its simplest form.

Multiply both sides by 15.

Simplify



When we have mixed numbers in the ratios, in order to simplify these ratios we must first convert all mixed numbers into improper fractions.

Multiply the entire ratio by 4

Simplify

3 : 4 : 6



Example 6: Simplify

1 31 :2 8

2 32 :33 4

Support Exercise Pg 318 Exercise 20A Nos 4

Section 5 Ratio Problems

Example 1: In a clothes shop there are 8 shirts and 12 tops on a shelf. Find the ratio of shirts to tops.

Shirts : Tops

8 : 12

2 : 3



Example 2: John weighs 60 kg whilst Paul weighs 120kg. Find the ratio of their weights.

Example 3: In a sports team there are 13 boys. In all there are 25 children. Find the ratio of boys to girls.

Example 4: Nathan is 100 cm tall and Isaac is 95 cm tall. Find the ratio of Nathan’s height to Isaac’s.

Example 5: A farmer has a flock of 50 sheep. 10 of them are white and the rest are black. Find the ratio of black sheep to white sheep.

Support Exercise Pg 318 Ex 20A Nos 6 - 10



Section 6 Sharing by Ratio

A ratio is there to show the sharing of a quantity.

Bart and Meggie share a bag of 16 sweets in the ratio 3:1.

This means that for every 3 sweets Bart takes Meggie takes 1.

Meggie takes 4

Bart takes 12

3 : 1 is in shares.

Bart gets 3 shares and Meggie gets 1 share.

There are 4 shares in all

Therefore:

16 ÷ 4 = 4 sweets per share

If Meggie has 1 share

1 x 4 = 4 sweets

If Bart has 3 shares

3 x 4 = 12 sweets

In order to share a quantity in a given ratio we must follow the following steps:

Step 1: Add up the shares.

Step 2: Find the amount represented by each share.

Step 3: Multiply the amount by the individual shares.



Example 1: Mark and Luke share €35 in the ratio 4 : 3.

Work out how much each boy gets.

4 : 3

4 + 3 = 7 shares. Add 4 and 3 to get the number of shares

35 ÷ 7 = €5 Divide to work out how much each share is worth

Mark = 4 x 5 = €20

Luke = 3 x 5 = € 15

To check:

€20 + €15 = €35

The amount matches the total therefore it is correct.

Example 2

A bag contains blue and red beads. Their ratio is 4 : 1. There are a total of 35 beads in the bag. How many are blue and how many are red?



Example 3: Andy, Gary and Alan share $80 in the ratio 2 : 3 : 5. How much money does each one receive?

Support Exercise Pg 321 Ex 20C Nos 1 - 10

Section 7 Finding Missing Numbers using Ratios

Ratios can be used in order to find missing values. Let us look at this example and see how ratios can be used.

Example 1: To do the sweet dough the ratio of flour to sugar is of 4 : 2. There are 400g of flour. How much sugar is needed?

Find the value of 1 share.

4 shares = 400g

1 share = 400 ÷ 4 = 100g

Sugar is 2 shares

1 share = 100g

2 shares = 2 x 100 = 200g



Example 2: Mary and Alexia receive their pocket money in the ratio of 4 : 6. Alexia receives €60, how much does Mary receive?

Example 3: Three friends, John, Matt and Mike, were given a number of sweets in the ratio of 2:4:8. Mike received 64 sweets. How many sweets did John and Matt receive?



Example 4: Elena and Paul have a number of books shared in the ratio 5:7. Elena had 25 books. How many books did Paul get?

Support Exercise Pg 319 Ex 20B Nos 1 – 3, 6, 9 – 12, 14, 15

Section 8 Map Ratios

Maps scales can be written in ratios and tell you how many units of land, sea etc are equal to one unit on the map.

If you are travelling from Manchester to Newcastle, for example, and need to know how far it is, it would be very difficult to work this out if the map does not have a scale.

Example

The scale of a map is 1:50 000.

A distance is measured as 3cm on the map. How many cm is this equivalent to in real life?



Ratio

1 : 50 000

This means 1 cm is to 50 000 cm

If 1 : 50 000

3 : ?

1 to get 3 we MULTIPLIED by 3

Therefore:

50 000 x 3 = 150 000 cm

3 cm represents 150 000 cm

Example 1: Lucy measured a mall and made a scale drawing. The scale of the drawing was 1 : 4000. In the drawing, a shop in the mall is 4 millimeters wide. What is the width of the actual shop?



In order to write a ratio both sides have to have the SAME units. Both sides have to be in cm, m, km, etc.

Example 2

Write each of these scales in the form 1 : n

a) 1 mm represents 1 cm

To work these out we must make sure that both sides have the same units.

a) Both to mm

1 cm = 1 x 10 = 10 mm

1 mm : 10 mm

1 : 10

b) 1 cm represents 1 km

c) 1 cm represents 5m

d) 1 cm represents 0.25km

Support Exercise Pg 318 Ex 20A Nos 5, 8 - 10



Direct and Inverse Proportions

Directly Proportional

If 1 pencil costs 15 cents, then:

2 pencils cost 30 cents (2 x 15)



The cost depends on the number of pencils. As the number of pencils increases, the cost increases.

The cost is said to increase in the same proportion as the number of pencils increase the cost increases proportionally.

The two quantities of the cost and price are said to be Directly Proportional.

Real Life Situations:

• The amount of petrol bought is directly proportional to the size of the petrol tank

• The number of Euros exchanged to Dollars is directly proportional to the number of Euros

Example 1: 5 buns cost €1.50. Work out the cost of 7 of these buns.



Example 2: 3 pencils cost 96 cents. Work out the cost of 5 of these pencils

Example 3: Karen is paid € 48 for 6 hours of work. How much is she paid for 4 hours of work?

Example 4: The height of a pile of 6 identical books is 15cm. Work out the height of 8 identical books.

Support Exercise Pg 323 Ex 20D Nos 1 – 18



Inversely Proportional

A car travelling at a steady speed of 50 km/hr travels 200km in 4 hours.

A car travelling at a steady speed of 100 km/hr travels 200km in 2 hours.

As the speed increases, the time decreases.

As the speed is multiplied by 2, the time is divided by 2.

Two quantities are said to be in Inverse Proportion if one quantity increases at the same rate as the other quantity decreases.

Example 5: It takes 5 cleaners 6 hours to clean a school. Work out how long it would take 15 cleaners to clean the school.

Example 6: It takes 3 men 4 days to build a wall. Work out how long it will take 2 men to build the wall.



Example 7: If it takes 4 days for 10 men to dig a trench, how long will it take 8 men?

Example 8: 6 pipes are required to fill a tank in 100 minutes. How long will it take if only 5 pipes are used?

Support Exercise Pg 325 Ex 20E Nos 1 – 12

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Simple Interest, Ratios and Proportions - Home - …smcmaths.webs.com/Form-4/Simple Interest, Ratios and...Simple interest, Ratios and Proportions Form 4 [email protected] 2