FM4 CREDIT AND BORROWING


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1 FM4 CREDIT AND BORROWING Whe you purchase big ticket items such as cars, boats, televisios ad the like, retailers ad fiacial istitutios have various terms ad coditios that are implemeted for the cosumer whe they are payig their items off. The most commo way of purchasig items today is usig a credit card. These hady pieces of plastic eable us essetially to loa from a bak ad repay it (usually with iterest) over a give time period. Credit cards fuctio i two ways: Iterestfree periods whe you have a certai period of time (usually 55 days) before eedig to pay for purchased items. After that a sigificatly high iterest rate kicks i ad iterest is charged daily. No iterestfree periods whe iterest kicks i immediately from the time of purchase. The iterest rate is usually lower tha that of a credit card with a iterestfree period ad oce agai, iterest is calculated daily. The iterest rate o a credit card is geerally expressed as a aual rate of iterest. You will the eed to covert it to a daily rate (ad the to a decimal) as iterest is calculated o a daily basis. The simple iterest formula is used whe workig with credit cards. Alteratively you may purchase items o terms. This is whe you pay a deposit ad borrow the remaider. This amout is the repaid i equal (usually mothly) istalmets with iterest. This is also kow as a flat rate loa as the simple iterest formula is used to calculate the iterest charged. Aother type of loa is a reducig balace loa where the amout of iterest charged each moth is depedat o the outstadig balace o the loa ad ot the iitial amout borrowed as with flat rate loas. Home loas are a classic example of a reducig balace loa. I these cases, the compoud iterest formula is used i calculatios. I questios relatig to reducig balace loas, you will ofte be required to show the ability to track a loa for the first few moths buy fillig i certai missig values i a table. Studets are ofte required to use a repaymet table to make calculatios. I these, the repaymet for loas at varyig iterest rates ad for varyig terms is show. Sometimes the figures give are per $1,000 of borrowigs meaig multiplicatio of the figure i the table may be required. The effective rate of iterest formula is used for the coversio of a compoudig iterest rate to a simple or flat iterest rate. E (1 r) 1 This formula does ot appear o the formula sheet. Loas are looked at i further detail i FM5 The School For Excellece 2011 Trial Exam Preparatio Lectures Geeral Maths Book 2 Page 13
2 EXAMPLE 15 Vic buys a ride o law mower with a cash price of $8,400, o the followig terms. Fid: 15% deposit ad mothly repaymets of $275 per moth over 3 years. The deposit paid by Vic. (b) The amout that Vic borrows. (c) The cost of the law mower o terms. (d) The iterest charged. (e) The flat rate of iterest per aum to 1 decimal place. The School For Excellece 2011 Trial Exam Preparatio Lectures Geeral Maths Book 2 Page 14
3 EXAMPLE 16 Ae s credit card has o aual fee. She is, however, charged 16.95% o ay purchases from ad icludig the day of purchase. Express 16.95% p.a. as a daily iterest rate to 4 decimal places. (b) Ae buys a dress for $420 o April 20. She pays the accout o May 15. Calculate the iterest charged o this purchase. EXAMPLE 17 The table below is used by Motgomery s Bak to calculate home loa repaymets. Mothly Repaymets o a $1000 loa Rate 10 years 12 years 15 years 17 years 20 years 25 years 8.25% $12.27 $10.96 $9.70 $9.13 $8.52 $ % $12.40 $11.10 $9.85 $9.28 $8.68 $ % $12.53 $11.24 $9.99 $9.43 $8.84 $8.22 9% $12.67 $11.38 $10.14 $9.59 $9.00 $ % $12.80 $11.52 $10.29 $9.74 $9.16 $ % $12.94 $11.66 $10.44 $9.90 $9.32 $ % $13.08 $11.80 $10.59 $10.05 $9.49 $ % $13.22 $11.95 $10.75 $10.21 $9.65 $ % $14.35 $13.15 $12.00 $10.55 $11.01 $10.35 Fid Celeste s mothly repaymet if she borrows $ over 20 years at a iterest rate of 8.75%. The School For Excellece 2011 Trial Exam Preparatio Lectures Geeral Maths Book 2 Page 15
4 (b) Fid the cost of Celeste s loa. (i.e. the total of her repaymets) (c) How much iterest does Celeste pay over the term of the loa? (d) Show that the equivalet flat rate of iterest is 5.608%. EXAMPLE 18 The table show below ca be used to calculate home loa repaymets. Iterest rate Mothly repaymets o a $1000 loa over 10 years 15 years 20 years 25 years 8.25% p.a. $12.27 $9.70 $8.52 $ % p.a. $12.40 $9.85 $8.68 $ % p.a. $12.53 $9.99 $8.84 $ % p.a. $12.67 $10.14 $9.00 $ % p.a. $12.80 $10.29 $9.16 $ % p.a. $12.94 $10.44 $9.32 $8.74 Joae s gross weekly icome is $700. Show that her mothly icome, correct to the earest dollar, is $3033. The School For Excellece 2011 Trial Exam Preparatio Lectures Geeral Maths Book 2 Page 16
5 (b) The bak will ot allow loa repaymets to be more tha 30% of a customer s gross mothly icome. What is the maximum amout Joae ca repay per moth? (c) What is the maximum amout (to the earest $1000) Joae ca borrow at 9.25% p.a. iterest? (d) Joae decides to purchase a uit worth $ She settles o a loa at 9.25% over 25 years. Calculate her mothly repaymet. (e) How much i iterest will Joae pay over the term of the loa? (f) Approximately how may times more tha the amout she borrowed will she have to repay the bak? The School For Excellece 2011 Trial Exam Preparatio Lectures Geeral Maths Book 2 Page 17
6 FM5 ANNUITIES AND LOAN REPAYMENTS A auity is a type of ivestmet where regular equal cotributios are made. The most commo type of auity is a superauatio fud. Whe a sigle deposit is made, the future value of this ivestmet ca be calculated by simply applyig the compoud iterest formula: A P(1 r) I the evet that regular cotributios (M), are to a fud, the future value of a auity is foud by applyig the formula: Where: A (1 r) 1 M r A is the future value, M is the amout of each regular cotributio, r is the iterest rate per period (expressed as a decimal) is the umber of periods. The preset value of a auity ca be foud usig the formula: N A (1 r) This formula ca tell us how much we would eed to deposit today, as a oe off paymet, to achieve the same cumulative result as the auity. The A ca be see as the goal amout. The preset value ca also be foud usig the formula: (1 r) 1 N M r(1 r) This formula is used for calculatig loa repaymets, M, whe the size of the home loa is N. It is also used to fid the amout that must be ivested, N, to produce regular withdrawals of $M per period. Usig preset ad future value tables is also a requiremet of the course. The School For Excellece 2011 Trial Exam Preparatio Lectures Geeral Maths Book 2 Page 18
7 EXAMPLE 19 The table below shows future values of a auity of $1. Future Values of $1 Period Iterest Rate (per period) 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% Do is savig for a holiday. He deposits $2,000 ito a accout at the ed of each year for 5 years. The accout pays 8% per aum compouded aually. Use the table to show that the value of Do s ivestmet at the ed of 5 years is $11,734. (b) Do chages his mid ad decides to take a tour that will cost $20,500. How log will he eed to cotribute $2,000 per year to this accout before he ca afford this tour? Fid the preset value of Do s auity. The School For Excellece 2011 Trial Exam Preparatio Lectures Geeral Maths Book 2 Page 19
8 EXAMPLE 20 Sarah borrowed $ for a home uit. The iterest rate is ow 9% per aum compouded mothly. She repays $2 300 per moth ad expects to repay the loa withi 30 years. Moth Opeig balace Mothly iterest Mothly repaymet Ed of moth balace 1 $ $ x = $ $2 300 $ $ A $2 300 B Explai why the value is used to calculate the iterest for the moth. (b) Calculate the values of A ad B i the table showig all of your workig. EXAMPLE 21 Elizabeth plas to travel i 2 years time ad saves $400 a moth. She ivests the $400 at the ed of each moth i a accout earig 6.5% per aum compouded mothly. She estimates that she will eed $ for her trip. Will she have eough moey to pay for the trip? Justify your aswer with calculatios. The School For Excellece 2011 Trial Exam Preparatio Lectures Geeral Maths Book 2 Page 20
9 (b) The followig table gives the mothly repaymet for each $1000 borrowed Term of Loa 6.00% 6.25% 6.50% 6.75% 7.00% 7.25% 7.50% (years) 5 $19.33 $19.45 $19.57 $19.68 $19.80 $19.92 $ $11.10 $11.23 $11.35 $11.48 $11.61 $11.74 $ $8.44 $8.57 $8.71 $8.85 $8.99 $9.13 $ $7.16 $7.31 $7.46 $7.60 $7.75 $7.90 $ $6.44 $6.60 $6.75 $6.91 $7.07 $7.23 $7.39 (i) Elizabeth s fried Sophia decides to joi her to travel but she has o savigs ad istead borrows the $ at 7.25% p.a. ad will repay over 5 years. Use the table to fid her mothly repaymet. (ii) How much iterest does Sophia pay o the loa? EXAMPLE 22 $2540 is ivested for 3 years at 9% p.a. with iterest compouded mothly. Which of the followig expressios will give the value of the ivestmet after 3 years? A B 2540( ) 36 C D 36 ( ) ( ) ( ) 36 The School For Excellece 2011 Trial Exam Preparatio Lectures Geeral Maths Book 2 Page 21
10 EXAMPLE 23 If Amada eeds $500 per moth while she completes her 4 year uiversity course to become a dace istructor. What amout of moey ca her father ivest i a accout which ears 8% pa compouded mothly for the legth of her course? Solutio EXAMPLE 24 A ew Holde Commodore was advertised at $ Toy bought the car with the coditios of makig mothly repaymets of $585 for 10 years. What is the flat iterest rate that Toy has agreed to pay? A 1.625% B 9.5% C 16.25% D 19.5% The School For Excellece 2011 Trial Exam Preparatio Lectures Geeral Maths Book 2 Page 22
11 FM6 DEPRECIATION There are two methods of calculatig the curret value of a asset which has depreciated over time. The straight lie method The asset has a iitial value of V 0 ad if it has bee losig value at a rate of $D per period for periods, the the curret or salvage value of the asset, S, is foud by usig the formula, S V0 D The decliig balace method Whe this method is applied, the asset is assumed to be losig a fixed percetage of its curret value each year. If the asset had a iitial value of V 0 ad it has bee losig value at a rate of r% per period for periods, the its salvage value, S, is foud by usig the formula, S V (1 r) 0 Notice the similarity to the compoud iterest formula. You may be required to complete a table of values trackig the value of the asset over time ad the graph S agaist. It is very commo to see graphs i depreciatio questios, eve if it was give to you. Examiers are more iterested i your ability to iterpret the graph rather tha create it. You eed to also be familiar with the use of either method for the purpose of calculatig tax deductios. Do t make the assumptio that if the questio quotes a percetage rate of depreciatio, that it is automatically a decliig balace questio. Read the questio carefully. The School For Excellece 2011 Trial Exam Preparatio Lectures Geeral Maths Book 2 Page 23
12 EXAMPLE 25 A photocopyig machie cost $24,000 ew. Use the straight lie method of depreciatio ad a depreciatio rate of 12% per year to fid: The amout by which its value drops each year. (b) The salvage value after 5 years. (c) Whe the photocopier will be sold. The value must be below $4,000 before it ca be sold. EXAMPLE 26 The furiture i the foyer of the Hilto Hotel cost $65,000. It is expected that the furiture will have a life of 8 years after which time they will be sold for $23,400. Fid: The aual rate of depreciatio. (b) The aual depreciatio as a percetage of the cost price. EXAMPLE 27 A va was bought 6 years ago ad has bee depreciatig usig the decliig balace method at 12% per aum. Its salvage value is $12,540. Its purchase price is closest to: A $25,080 B $27,000 C $23,750 D $33,250 The School For Excellece 2011 Trial Exam Preparatio Lectures Geeral Maths Book 2 Page 24
13 EXAMPLE 28 A laptop was bought ad depreciated over time as show i the graph below Value ($) Number of Years What was the cost of the computer whe it was ew? (b) By how much did the computer depreciate each year? EXAMPLE 29 A compay car was purchased for $42,400 ad has a salvage value of $18,400 after years. Fid the aual rate of depreciatio if the decliig balace method was used. Give your aswer to 1 decimal place. Solutio The School For Excellece 2011 Trial Exam Preparatio Lectures Geeral Maths Book 2 Page 25
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