simple interest, ratios and proportions - webs interest, ratios and... · simple interest, ratios...
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Simpleinterest,RatiosandProportions Form4
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.
Simpleinterest,RatiosandProportions Form4
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
Simpleinterest,RatiosandProportions Form4
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
Simpleinterest,RatiosandProportions Form4
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)
Simpleinterest,RatiosandProportions Form4
2. Stellainvestedtwosumsofmoneyatasimpleinterestworkedoutmanually.SumSwasinvestedinMarch2009whilstSumTwasinvestedinMarch2011.StellakeepstheinterestshecollecteveryMarchathome.StellaisusingaspreadsheettocalculatethetotalsummoneycollectedfromthesetwoinvestmentsbyMarch2015.
I. WhatnumbershouldStellawriteincellC4?
II. IncellB5,writeaformulathatworksoutthesimpleinterestcollectedfromSumSforyears2009to2015.
III. IncellC5,writeaformulathatworksoutthesimpleinterestcollectedfromSumTforyears2011to2015.
IV. IncellD5,writeaformulathataddsthevaluesinthecellsB5andC5.
V. WorkoutthetotalsimpleinterestcollectedfromthesetwoinvestmentsbyMarch2015.
(7marks)
(P1,May2015,no.2)
Simpleinterest,RatiosandProportions Form4
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
Simpleinterest,RatiosandProportions Form4
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
Simpleinterest,RatiosandProportions Form4
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
Simpleinterest,RatiosandProportions Form4
€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
Simpleinterest,RatiosandProportions Form4
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
Example 4: Express in its simplest form.
Example 5: Express in its simplest form.
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
Simpleinterest,RatiosandProportions Form4
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
Simpleinterest,RatiosandProportions Form4
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
Simpleinterest,RatiosandProportions Form4
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.
Simpleinterest,RatiosandProportions Form4
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?
Simpleinterest,RatiosandProportions Form4
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
Simpleinterest,RatiosandProportions Form4
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?
Simpleinterest,RatiosandProportions Form4
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?
Simpleinterest,RatiosandProportions Form4
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?
Simpleinterest,RatiosandProportions Form4
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
Simpleinterest,RatiosandProportions Form4
Direct and Inverse Proportions
Directly Proportional
If 1 pencil costs 15 cents, then:
2 pencils cost 30 cents (2 x 15)
3 pencils cost 45 cents (3 x 15)
4 pencils cost 60 cents (4 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.
Simpleinterest,RatiosandProportions Form4
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
Simpleinterest,RatiosandProportions Form4
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.
Simpleinterest,RatiosandProportions Form4
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