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Dividing by 100 produces the multiplier 1.06."},{"text":"0.94","correct":false,"explanation":"Subtracting 0.06 from 1 gives the multiplier for a 6% decrease. For a 6% increase, add 0.06 to 1 to obtain 1.06."},{"text":"1.6","correct":false,"explanation":"Treating 6% as 0.6 misplaces the decimal point. Since 6% is 0.06, the increase multiplier is $1 + 0.06 = 1.06$."},{"text":"1.006","correct":false,"explanation":"This treats 6% as 0.006, which represents 0.6%. Converting 6% correctly gives 0.06, so the required multiplier is 1.06."}]},{"id":"e8743f8d-f702-0a11-7944-08ccfa4680e0","cardType":"basic","points":1,"front":"In percentage calculations, what is the multiplier for a 6% decrease?","back":"0.94","mcqOptions":[{"text":"0.94","correct":true,"explanation":"A 6% decrease leaves 94% of the original value. Converting 94% to a decimal gives the multiplier $94 \\div 100 = 0.94$."},{"text":"1.06","correct":false,"explanation":"Adding 0.06 to 1 gives the multiplier for a 6% increase. A decrease requires subtraction, so $1 - 0.06 = 0.94$."},{"text":"0.6","correct":false,"explanation":"This incorrectly treats 6% as 0.6 and would leave only 60% of the value. A 6% decrease leaves 94%, giving 0.94."},{"text":"0.994","correct":false,"explanation":"This corresponds to a decrease of only 0.6%, caused by placing the decimal incorrectly. A 6% decrease uses $1 - 0.06 = 0.94$."}]},{"id":"cd92281e-2daa-8850-21db-c03b918057b8","cardType":"cloze","points":null,"front":"In the compound interest formula $A = P(1 + \\frac{r}{100})^n$, the power $n$ represents the [ ].","back":"In the compound interest formula $A = P(1 + \\frac{r}{100})^n$, the power $n$ represents the number of years.","mcqOptions":[{"text":"number of years","correct":true,"explanation":"The multiplier is applied once for every year that interest is added, so the exponent $n$ records the number of years in the investment period."},{"text":"annual interest rate","correct":false,"explanation":"The annual interest rate is represented by $r$ inside the multiplier, not by the exponent. The power $n$ gives the number of years."},{"text":"initial investment","correct":false,"explanation":"The initial investment is represented by $P$, because it is the amount multiplied by the growth factor. The power $n$ represents the number of years."},{"text":"final investment value","correct":false,"explanation":"The final investment value is represented by $A$ on the left-hand side. The exponent $n$ instead shows how many years the multiplier is applied."}]},{"id":"47b0afed-c1f7-cc93-b512-418bee8526a3","cardType":"basic","points":1,"front":"What does the term per annum mean?","back":"Each year","mcqOptions":[{"text":"Each year","correct":true,"explanation":"The Latin expression per annum means for each year. An interest rate of 5% per annum is therefore applied at a rate of 5% each year."},{"text":"Each month","correct":false,"explanation":"Confusing annual and monthly rates changes how often the percentage applies. Per annum means each year, while per month would describe a monthly rate."},{"text":"Every two years","correct":false,"explanation":"Per annum describes a rate applying once each year, not once every two years. A two-year period would contain two annual applications."},{"text":"Each week","correct":false,"explanation":"A weekly rate applies each week, whereas per annum refers specifically to a yearly rate. Therefore, the correct meaning is each year."}]},{"id":"7ec4fba5-4135-cf3d-08c5-ea05800f77cb","cardType":"basic","points":2,"front":"Explain the core mechanism of compound interest.","back":"Interest is added to the total, then the next year's interest is calculated on that new, larger amount.","mcqOptions":[{"text":"Interest is added to the total, then the next year's interest is calculated on that new, larger amount.","correct":true,"explanation":"Compound interest adds each period's interest to the balance. Future interest is then calculated on both the original investment and previously earned interest."},{"text":"Interest is calculated from the original total, so every year's interest is based on that same, unchanged amount.","correct":false,"explanation":"Keeping the amount used to calculate interest unchanged describes simple interest. Compound interest uses the new, larger balance after each year's interest is added."},{"text":"Interest is removed from the total, then the next year's interest is calculated on that new, smaller amount.","correct":false,"explanation":"Removing a percentage models depreciation rather than compound interest growth. With compound interest, interest is added and the next calculation uses the new, larger amount."},{"text":"Interest is added only at the end, after every year's interest is calculated on the original starting amount.","correct":false,"explanation":"Calculating every year's interest from the starting amount produces simple interest, even if it is added later. Compound interest recalculates using each new balance."}]},{"id":"ea8bbba3-8d8f-fa7a-7526-888a38b6f600","cardType":"basic","points":2,"front":"Why is it incorrect to simply add the annual interest rates together for a multi-year investment?","back":"Each percentage must be calculated from the new value, not the original value.","mcqOptions":[{"text":"Each percentage must be calculated from the new value, not the original value.","correct":true,"explanation":"After each year's percentage change, the base value changes. The next percentage must be applied to that new value, which is why multipliers are compounded."},{"text":"Each percentage must be calculated from the original value, not the new value.","correct":false,"explanation":"Using the original value every year describes simple interest and makes equal cash amounts each year. Compound interest applies each percentage to the new value."},{"text":"Each percentage must be subtracted from the new value, not added to the original value.","correct":false,"explanation":"Whether a percentage is added or subtracted depends on growth or depreciation, not on the number of years. The key issue is using the new value each year."},{"text":"Each percentage must be converted into the same units, not combined as annual rates.","correct":false,"explanation":"Annual rates already use compatible percentage units, so unit conversion is not the problem. Adding them fails because each year's percentage is applied to a new value."}]},{"id":"99237dad-05d9-f713-a7bc-48b5c98a0c5f","cardType":"basic","points":2,"front":"State the calculation to find the final amount for £3600 invested for 4 years at 2.5% compound interest.","back":"$$3600 \\times 1.025^4$","mcqOptions":[{"text":"$$3600 \\times 1.025^4$","correct":true,"explanation":"A 2.5% increase has multiplier $1.025$, and compound interest applies it once in each of four years. Therefore, the multiplier is raised to power 4."},{"text":"$$3600 \\times 1.25^4$","correct":false,"explanation":"Using 1.25 treats 2.5% as 25%, making the annual increase ten times too large. The correct multiplier is 1.025, giving $3600 \\times 1.025^4$."},{"text":"$$3600 \\times 1.025 \\times 4$","correct":false,"explanation":"Multiplying the entire amount by 4 does not represent four years of compound growth. The annual multiplier must be applied four times as $3600 \\times 1.025^4$."},{"text":"$$3600 \\times 0.975^4$","correct":false,"explanation":"The multiplier 0.975 represents a 2.5% decrease, so it models depreciation. Compound interest increases the investment, requiring the multiplier 1.025 for four years."}]},{"id":"e85d83b7-ab7a-5fb5-36a3-b56326755131","cardType":"cloze","points":null,"front":"To find the total interest earned, you subtract the [ ] from the final amount.","back":"To find the total interest earned, you subtract the starting amount from the final amount.","mcqOptions":[{"text":"starting amount","correct":true,"explanation":"The final amount contains both the original investment and the interest earned. Subtracting the starting amount therefore leaves only the total interest."},{"text":"annual interest rate","correct":false,"explanation":"An interest rate is a percentage, so subtracting it directly from a monetary final amount mixes different quantities. Subtract the starting amount to isolate the interest earned."},{"text":"number of years","correct":false,"explanation":"The number of years is used when calculating the final amount, but it cannot be subtracted from a monetary value. The starting amount must be subtracted."},{"text":"interest multiplier","correct":false,"explanation":"The multiplier is used to calculate growth and is not itself a monetary amount. Total interest is found by subtracting the starting amount from the final amount."}]},{"id":"eeb2e707-200f-f04b-9900-0d219e540cfe","cardType":"cloze","points":null,"front":"To find the total interest earned, you subtract the starting amount from the [ ].","back":"To find the total interest earned, you subtract the starting amount from the final amount.","mcqOptions":[{"text":"final amount","correct":true,"explanation":"Total interest is the increase from the starting amount to the final amount, so subtracting the starting amount from the final amount gives the interest earned."},{"text":"starting amount","correct":false,"explanation":"Subtracting the starting amount from itself would give zero. Total interest is found by subtracting the starting amount from the final amount."},{"text":"annual interest rate","correct":false,"explanation":"The annual interest rate is a percentage used to calculate growth, not the amount from which the starting value is subtracted. The required term is final amount."},{"text":"interest multiplier","correct":false,"explanation":"The interest multiplier determines growth each year but is not a monetary total. Subtract the starting amount from the final amount to obtain total interest."}]},{"id":"b2a26878-5f4d-d8c9-cf65-5a28f0d10e57","cardType":"basic","points":1,"front":"The value of an investment is $V = 6000 \\times 1.018^n$. What is the annual interest rate?","back":"1.8%","mcqOptions":[{"text":"1.8%","correct":true,"explanation":"The multiplier $1.018$ equals $1 + 0.018$, so the investment increases by $0.018 = 1.8\\%$ each year."},{"text":"18%","correct":false,"explanation":"This treats the decimal part $0.018$ as $0.18$. Converting $0.018$ to a percentage requires multiplying by 100, giving an annual rate of $1.8\\%$."},{"text":"101.8%","correct":false,"explanation":"This uses the whole multiplier as the percentage rate. The growth rate is the amount above 1, so $1.018 - 1 = 0.018 = 1.8\\%$."},{"text":"0.18%","correct":false,"explanation":"This misreads the final digit of the multiplier as the percentage. The decimal increase is $0.018$, which corresponds to an annual interest rate of $1.8\\%$."}]},{"id":"8a86dfae-04da-b5aa-8915-7b4e949eae9f","cardType":"basic","points":1,"front":"Define depreciation.","back":"When a value goes down over time.","mcqOptions":[{"text":"When a value goes down over time.","correct":true,"explanation":"Depreciation is the reduction in an asset’s value as time passes, often modelled by multiplying its current value by a multiplier below 1 each year."},{"text":"When a value goes up over time.","correct":false,"explanation":"A value going up over time describes appreciation or growth, not depreciation. Depreciation means that the value goes down over time."},{"text":"When a value stays constant over time.","correct":false,"explanation":"A value remaining constant experiences no growth or depreciation. Depreciation specifically occurs when a value goes down over time."},{"text":"When a value goes up and down over time.","correct":false,"explanation":"Fluctuation includes both increases and decreases, so it does not specifically define depreciation. Depreciation means that a value goes down over time."}]},{"id":"bb1438dc-a5de-f8aa-3350-7c0c08a0f5a1","cardType":"basic","points":1,"front":"What multiplier is used for a 20% depreciation?","back":"0.80","mcqOptions":[{"text":"0.80","correct":true,"explanation":"After a $20\\%$ decrease, $80\\%$ of the value remains. Converting $80\\%$ to a decimal gives the depreciation multiplier $0.80$."},{"text":"0.20","correct":false,"explanation":"This is the proportion lost rather than the proportion remaining. A $20\\%$ depreciation leaves $80\\%$, so the correct multiplier is $0.80$."},{"text":"1.20","correct":false,"explanation":"This multiplier represents a $20\\%$ increase because it is greater than 1. A $20\\%$ depreciation uses $1 - 0.20 = 0.80$."},{"text":"0.98","correct":false,"explanation":"This would represent a $2\\%$ depreciation, not a $20\\%$ depreciation. Subtracting $0.20$ from 1 gives the correct multiplier $0.80$."}]},{"id":"99f2b846-e53a-dd36-a29b-3b551dae0996","cardType":"cloze","points":null,"front":"The formula for the final amount after depreciation is $A = P(1 \\text{[...]} \\frac{r}{100})^n$.","back":"The formula for the final amount after depreciation is $A = P(1 \\boxed{-} \\frac{r}{100})^n$.","mcqOptions":[{"text":"$$-$","correct":true,"explanation":"Depreciation removes a proportion of the current value each period, so its multiplier is $1 - \\frac{r}{100}$. The missing operation is subtraction."},{"text":"$$+$","correct":false,"explanation":"Addition produces the compound growth formula and would increase the value each period. Depreciation requires the multiplier $1 - \\frac{r}{100}$, so subtraction is needed."},{"text":"$$\\times$","correct":false,"explanation":"Multiplication here would not subtract the depreciated percentage from the whole value. The correct multiplier is $1 - \\frac{r}{100}$, requiring a minus sign."},{"text":"$$\\div$","correct":false,"explanation":"Division does not form the proportion of value remaining after depreciation. That proportion is $1 - \\frac{r}{100}$, so the required operation is subtraction."}]},{"id":"07e2c1af-0020-202c-948b-04d49e3a495e","cardType":"basic","points":2,"front":"An item loses 25% value in Year 1, then 10% in Year 2. What are the two multipliers used?","back":"

Year 1: 0.75
Year 2: 0.90

","mcqOptions":[{"text":"

Year 1: 0.75
Year 2: 0.90

","correct":true,"explanation":"Losing $25\\%$ leaves $75\\%$, giving $0.75$ in Year 1. Losing $10\\%$ then leaves $90\\%$, giving $0.90$ in Year 2."},{"text":"

Year 1: 0.25
Year 2: 0.10

","correct":false,"explanation":"These decimals represent the proportions lost rather than those remaining. The correct multipliers are $1 - 0.25 = 0.75$ and $1 - 0.10 = 0.90$."},{"text":"

Year 1: 1.25
Year 2: 1.10

","correct":false,"explanation":"Multipliers greater than 1 represent increases, so these would model appreciation. Depreciation leaves $75\\%$ and then $90\\%$, requiring multipliers $0.75$ and $0.90$."},{"text":"

Year 1: 0.65
Year 2: 0.65

","correct":false,"explanation":"This incorrectly combines the two percentage losses before applying them. Each year needs its own multiplier: $0.75$ for Year 1 and $0.90$ for Year 2."}]},{"id":"5f3a06ec-d949-6d3b-3005-c298e95e6ddd","cardType":"basic","points":2,"front":"A value rises by 2% then falls by 4%. State the calculation to find the final value of original amount $X$.","back":"$$X \\times 1.02 \\times 0.96$","mcqOptions":[{"text":"$$X \\times 1.02 \\times 0.96$","correct":true,"explanation":"A $2\\%$ rise uses multiplier $1.02$, and a subsequent $4\\%$ fall uses multiplier $0.96$. Both multipliers must be applied successively to $X$."},{"text":"$$X \\times 1.02 \\times 1.04$","correct":false,"explanation":"The multiplier $1.04$ models a $4\\%$ increase rather than a decrease. The fall must use $0.96$, giving $X \\times 1.02 \\times 0.96$."},{"text":"$$X \\times 1.06$","correct":false,"explanation":"The percentages cannot simply be added to form one growth multiplier because the changes apply successively to different values. Use $X \\times 1.02 \\times 0.96$."},{"text":"$$X \\times 0.98$","correct":false,"explanation":"Subtracting the percentages gives a net $2\\%$ fall, but this ignores compounding. Applying both changes correctly requires $X \\times 1.02 \\times 0.96$."}]},{"id":"e5c1e1d5-ac11-aac0-7e5a-9689eddfff9e","cardType":"basic","points":1,"front":"What is the standard method for finding the number of years $n$ if you are given the final amount?","back":"Try whole-number powers on a calculator until the value matches or passes the target.","mcqOptions":[{"text":"Try whole-number powers on a calculator until the value matches or passes the target.","correct":true,"explanation":"At GCSE, substitute successive whole-number values of $n$ into the compound formula. Continue until the calculated amount matches or first passes the specified target."},{"text":"Divide the final percentage change by the annual rate to find the target.","correct":false,"explanation":"Using only the percentage change ignores the starting amount and repeated compounding. Instead, test whole-number powers in the full formula until the result reaches the target."},{"text":"Add the first year’s interest repeatedly until the value matches the target.","correct":false,"explanation":"Repeatedly adding the same interest amount models simple interest, not compound change. Test whole-number powers because each year’s change acts on the updated value."},{"text":"Round the annual multiplier to a whole number and divide by the target.","correct":false,"explanation":"Rounding the multiplier removes information and cannot determine the elapsed time. Keep the exact multiplier and test whole-number powers until the calculated value reaches the target."}]},{"id":"547470d1-cec8-558a-2efc-576621c38d0e","cardType":"basic","points":1,"front":"When calculating how many years it takes for an item to be worth less than half its original value, what original value is simplest to use?","back":"1","mcqOptions":[{"text":"1","correct":true,"explanation":"Using 1 normalises the original value, so half is simply 0.5. Repeated depreciation multipliers can then be applied until the value falls below 0.5."},{"text":"0.5","correct":false,"explanation":"Using 0.5 makes the comparison threshold 0.25, adding unnecessary decimals. An original value of 1 is simplest because half of it is directly represented by 0.5."},{"text":"0","correct":false,"explanation":"An original value of 0 cannot model depreciation because every later value would remain 0. Use 1 so the changing value can be compared directly with 0.5."},{"text":"100","correct":false,"explanation":"Using 100 works mathematically but is not the simplest choice, since it introduces larger values. Using 1 turns the calculation into a direct comparison with 0.5."}]},{"id":"d5aeb110-f14d-d9f2-ca49-01480ede9b8e","cardType":"basic","points":3,"front":"Contrast simple interest and compound interest.","back":"

Simple: calculated only on the original amount.
Compound: calculated on the growing amount.

","mcqOptions":[{"text":"

Simple: calculated only on the original amount.
Compound: calculated on the growing amount.

","correct":true,"explanation":"Simple interest repeatedly uses the initial principal, whereas compound interest includes previously earned interest. Therefore, the amount used to calculate compound interest grows each period."},{"text":"

Simple: calculated on the growing amount.
Compound: calculated only on the original amount.

","correct":false,"explanation":"This reverses the two definitions. Simple interest is based only on the original amount, while compound interest is based on the growing amount after earlier interest is added."},{"text":"

Simple: calculated only on the original amount.
Compound: calculated only on the original amount.

","correct":false,"explanation":"Only the simple-interest statement is accurate. Compound interest is calculated on the growing amount, including interest already added, rather than repeatedly using just the original amount."},{"text":"

Simple: calculated on the growing amount.
Compound: calculated on the growing amount.

","correct":false,"explanation":"Only the compound-interest statement is accurate. Simple interest is calculated solely from the original amount, so the interest added each period remains constant."}]},{"id":"93f60320-407f-3b24-8281-42400cda6d43","cardType":"cloze","points":null,"front":"When working with money in compound interest problems, round the final answer to [ ] decimal places.","back":"When working with money in compound interest problems, round the final answer to 2 decimal places.","mcqOptions":[{"text":"2","correct":true,"explanation":"Money is normally written to the nearest penny, and one pound contains 100 pence. The final answer should therefore be rounded to 2 decimal places."},{"text":"1","correct":false,"explanation":"Rounding to 1 decimal place gives values only to the nearest ten pence, not the nearest penny. Monetary answers should be rounded to 2 decimal places."},{"text":"3","correct":false,"explanation":"Three decimal places would represent tenths of a penny, which are not normally used in monetary answers. The final amount should be rounded to 2 decimal places."},{"text":"0","correct":false,"explanation":"Rounding to 0 decimal places gives the nearest whole pound and discards the pence. Money calculations should normally be rounded to 2 decimal places."}]},{"id":"f34b61ba-6993-9864-afd6-c90dbb83ad1d","cardType":"basic","points":1,"front":"Why should you avoid rounding values during the intermediate steps of a compound interest calculation?","back":"Rounding too early can change the final penny (accuracy is lost).","mcqOptions":[{"text":"Rounding too early can change the final penny (accuracy is lost).","correct":true,"explanation":"Small rounding errors can be compounded across several calculation stages. Keeping full calculator values until the end preserves accuracy and prevents the final monetary answer changing by a penny."},{"text":"Rounding too early can change the annual interest rate (the rate becomes inaccurate).","correct":false,"explanation":"Rounding intermediate values does not alter the stated annual interest rate; it alters the calculated balance. Accumulated rounding error can change the final penny, so round only at the end."},{"text":"Rounding too early will always increase the final value (the answer is overestimated).","correct":false,"explanation":"Rounding can increase or decrease an intermediate value, so it does not always cause an overestimate. The real risk is accumulated inaccuracy changing the final penny."},{"text":"Rounding too early can stop the multiplier being used (the method no longer works).","correct":false,"explanation":"A multiplier can still be applied to a rounded number, but the resulting value may be inaccurate. Keeping unrounded intermediate values prevents the final penny from changing."}]},{"id":"ee001b06-a8d8-c92e-9dd8-ba8c9b12df40","cardType":"basic","points":1,"front":"A car depreciates by 7% per year. What multiplier is used to find its value after $n$ years?","back":"0.93","mcqOptions":[{"text":"0.93","correct":true,"explanation":"A 7% decrease leaves 93% of the value each year. Converting 93% to a decimal gives the multiplier 0.93, used as $0.93^n$ after $n$ years."},{"text":"0.07","correct":false,"explanation":"This is the decimal form of the 7% depreciation rate, not the proportion remaining. Subtracting it from 1 gives the correct annual multiplier, $1-0.07=0.93$."},{"text":"1.07","correct":false,"explanation":"A multiplier greater than 1 represents growth, so this would increase the car’s value by 7% annually. Depreciation requires $1-0.07=0.93$."},{"text":"0.97","correct":false,"explanation":"This multiplier represents a 3% decrease because it leaves 97% of the value. A 7% decrease leaves 93%, giving the correct multiplier 0.93."}]}]}],"examCountdown":null,"heading":"Compound Interest and Depreciation","title":"Compound Interest and Depreciation Flashcards","topbar":{"backHref":"/gcse/maths/ccea/flashcards"},"topicSidebar":"$L39"}]}],"topicId":"dd2f9fac-c35f-7f46-1c0a-de6b2c632aae","walled":"$L3a"}],"$L3b","$L3c"]