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Quick Review Flashcards - Click to flip and test your knowledge!
Question
In the context of compound interest, what is the 'amount' ($A$)?
Answer
The sum of the principal and the interest accumulated over a specific period.
Question
How is compound interest ($C.I.$) fundamentally defined as a process?
Answer
As a repeated simple interest computation where the principal grows in each conversion period.
Question
What makes manual computation of compound interest 'quite tedious' as the number of conversion periods increases?
Answer
The requirement to calculate simple interest repeatedly for every year or half-year.
Question
What is the standard formula for the amount ($A$) when interest is compounded yearly?
Answer
$A = P(1 + \frac{r}{100})^n$
Question
In the compound interest formula, what does the variable $P$ represent?
Answer
The principal, which is the initial sum of money invested or borrowed.
Question
In the compound interest formula, what does the variable $r$ represent?
Answer
The rate of interest compounded yearly.
Question
In the compound interest formula, what does the variable $n$ represent?
Answer
The number of years (or conversion periods).
Question
Formula: Calculate Compound Interest ($C.I.$) using Amount ($A$) and Principal ($P$).
Answer
$C.I. = A - P$
Question
What is the direct formula to calculate Compound Interest ($C.I.$) without finding the amount first?
Answer
$C.I. = P[(1 + \frac{r}{100})^n - 1]$
Question
Formula: What is the amount ($A$) when rates for successive years are different ($r_1, r_2, r_3$)?
Answer
$A = P(1 + \frac{r_1}{100})(1 + \frac{r_2}{100})(1 + \frac{r_3}{100})...$
Question
How is the principal ($P$) calculated if the amount ($A$), rate ($r$), and time ($n$) are known?
Answer
By rearranging the formula to $P = \frac{A}{(1 + \frac{r}{100})^n}$.
Question
To find the rate percent ($r$) when $A$ and $P$ are known, what is the first algebraic step?
Answer
Divide the amount by the principal to isolate the term $(1 + \frac{r}{100})^n$.
Question
If $\frac{A}{P} = (\frac{21}{20})^3$ and the time is $3$ years, what is the value of $(1 + \frac{r}{100})$?
Answer
$\frac{21}{20}$
Question
When finding the number of years ($n$), if $(\frac{11}{10})^n = (\frac{11}{10})^3$, what is the value of $n$?
Answer
$3$ years.
Question
When interest is compounded half-yearly, how is the annual rate ($r$) adjusted in the formula?
Answer
The rate percent is divided by $2$ (i.e., $\frac{r}{2}$).
Question
When interest is compounded half-yearly, how is the number of years ($n$) adjusted in the formula?
Answer
The number of years is multiplied by $2$ (i.e., $n \times 2$).
Question
What is the formula for Amount ($A$) when interest is compounded half-yearly?
Answer
$A = P(1 + \frac{r}{2 \times 100})^{n \times 2}$
Question
Although not in the I.C.S.E. syllabus, what is the formula for Amount ($A$) when interest is compounded quarterly?
Answer
$A = P(1 + \frac{r}{4 \times 100})^{n \times 4}$
Question
How is the amount calculated when the time is not an exact number of years (e.g., $2 \frac{1}{2}$ years) and interest is compounded yearly?
Answer
Calculate the amount for the full years, then use that as the principal for the remaining fractional year.
Question
Formula: Amount ($A$) for $2 \frac{1}{2}$ years at $10\%$ interest compounded yearly.
Answer
$A = P(1 + \frac{10}{100})^2 \times (1 + \frac{10}{2 \times 100})^1$
Question
What is the conversion period for interest compounded half-yearly?
Answer
Six months.
Question
In growth problems, what formula is used to find production after $n$ years given an initial production and growth rate $r$?
Answer
Production after $n$ years = Initial production $\times (1 + \frac{r}{100})^n$
Question
Concept: Depreciation
Answer
Definition: The reduction in the value of an asset (like machinery) over time due to wear and tear or age.
Question
What is the formula for the value of a machine after $n$ years if it depreciates at $r\%$ every year?
Answer
Value after $n$ years = Present value $\times (1 - \frac{r}{100})^n$
Question
How is the present value of a machine calculated if its value $n$ years ago and depreciation rate are known?
Answer
Present value = Value $n$ years ago $\times (1 - \frac{r}{100})^n$
Question
What is 'scrap value' in the context of depreciation?
Answer
The reduced value of an asset at the time it is sold or discarded after several years of use.
Question
Formula: Population after $n$ years given present population ($P$) and growth rate ($r$).
Answer
Population after $n$ years = $P(1 + \frac{r}{100})^n$
Question
How do you find the population $n$ years ago if the present population and growth rate $r$ are given?
Answer
Present population = Population $n$ years ago $\times (1 + \frac{r}{100})^n$
Question
In population problems, which variable in the standard $A = P(1 + \frac{r}{100})^n$ formula represents the population at the earlier point in time?
Answer
The principal ($P$).
Question
In population problems, which variable in the standard $A = P(1 + \frac{r}{100})^n$ formula represents the population at the later point in time?
Answer
The amount ($A$).
Question
When calculating the difference between $C.I.$ and $S.I.$ for $2$ years, what is the standard formula for Simple Interest ($S.I.$)?
Answer
$S.I. = \frac{P \times r \times 2}{100}$
Question
Concept: Conversion Periods
Answer
Definition: The fixed intervals of time (e.g., year, half-year) after which interest is added to the principal.
Question
Under what condition does a sum of money double itself in $n$ years in a compound interest scenario?
Answer
When $(1 + \frac{r}{100})^n = 2$.
Question
If the rate of interest is $20\%$ per annum, what is the rate used for half-yearly compounding calculations?
Answer
$10\%$ per half-year.
Question
In a $2 \frac{1}{2}$ year period, how many half-yearly conversion periods are there?
Answer
$5$
Question
If a sum of money is borrowed and repaid in two yearly instalments, how is the total sum borrowed calculated?
Answer
By finding the present value (Principal) for each instalment separately and adding them together.
Question
Why is the interest for the second year in compound interest higher than the interest for the first year?
Answer
Because the interest from the first year is added to the principal, increasing the sum on which interest is calculated.
Question
Formula: Find the rate ($r$) when the amount in $1$ year and amount in $2$ years are given (yearly compounding).
Answer
$r = \frac{\text{Amount in 2nd year} - \text{Amount in 1st year}}{\text{Amount in 1st year}} \times 100$
Question
In growth problems, which formula applies to 'the rapid growth of plants' or 'inflation'?
Answer
The standard compound interest formula: $A = P(1 + \frac{r}{100})^n$.
Question
If the question asks for 'the rate of interest per annum' and the calculation for a half-year yields $5\%$, what is the final answer?
Answer
$10\%$ per annum ($5\% \times 2$).
Question
True or False: The principal remains constant throughout the entire duration in compound interest.
Answer
False; the principal increases after every conversion period.
Question
When comparing yearly and half-yearly compounding for the same rate and time, which yields a higher amount?
Answer
Half-yearly compounding.
Question
What is the value of $(1 + \frac{10}{100})^2$ expressed as a decimal?
Answer
$1.21$
Question
If $(1 + \frac{r}{100})^n = 1.44$ for $n=2$, what is the value of $1 + \frac{r}{100}$?
Answer
$1.2$ (since $\sqrt{1.44} = 1.2$).
Question
In the formula for depreciation, why is a minus sign used inside the bracket?
Answer
To represent the decrease in value over time.
Question
When $n = 1 \frac{1}{2}$ years and interest is compounded half-yearly, what value of $n$ is used in the power of the formula?
Answer
$3$ (conversion periods).
Question
How do you calculate the $C.I.$ earned in the $3^{rd}$ year only?
Answer
Subtract the Amount at the end of $2$ years from the Amount at the end of $3$ years ($A_3 - A_2$).
Question
If an asset depreciates by $10\%$ in year 1 and $12\%$ in year 2, what is the formula for final value $V$?
Answer
$V = P(1 - \frac{10}{100})(1 - \frac{12}{100})$
Question
In the equation $\frac{9261}{8000} = (1 + \frac{r}{100})^3$, what is the cubical root of $\frac{9261}{8000}$?
Answer
$\frac{21}{20}$
Question
If a sum of money doubles in $5$ years, how many years will it take to become eight times itself ($2^3$) at the same rate?
Answer
$15$ years ($5 \times 3$).
Question
What is the relationship between Amount ($A$) and Principal ($P$) for the first year if interest is compounded annually?
Answer
They are the same as in simple interest for one year.
Question
In the 'Direct Method' formula for $C.I.$, what does the term $(1 + \frac{r}{100})^n - 1$ represent?
Answer
The total interest earned on a principal of $1$ unit.
Question
How is the 'rate of growth' usually expressed in population or industrial problems?
Answer
As a percentage ($r\%$) per year.
Question
If $18,000$ amounts to $23,805$ in $2$ years, what is the total compound interest earned?
Answer
$5,805$ ($23,805 - 18,000$).
Question
Cloze: In compound interest, the interest for the first year becomes _____ for the second year.
Answer
Part of the principal
Question
Formula: Calculate simple interest ($I$) to find principal ($P$) when $I$, $R$, and $T$ are known.
Answer
$P = \frac{I \times 100}{R \times T}$
Question
If interest is reckoned half-yearly, what is the amount formula for $n=1$ year?
Answer
$A = P(1 + \frac{r}{200})^2$
Question
What is the result of $1,600(1 + \frac{20}{100})^2$?
Answer
$2,304$
Question
To find the rate $r$ from $(1 + \frac{r}{100}) = \frac{11}{10}$, what is the final percentage?
Answer
$10\%$
Question
When solving for $n$ in $(\frac{21}{20})^n = \frac{9261}{8000}$, what power is $9261$ of $21$?
Answer
The $3^{rd}$ power (cube).