8.3 Simple & Compound Interest Calculations

Interest items on GMAT Quantitative Reasoning are percent word problems from Official Guide Math Review 3.3, not finance-calculator drills. The section is 21 questions in 45 minutes with no calculator, so stems use clean rates ($5\%$, $8\%$, $10\%$, $12\%$) and a small number of compounding periods. You are expected to multiply, not to take logarithms or interpolate a compound-interest table.

Simple Interest

Simple interest is computed only on the original principal $P$. If the annual rate is the decimal $r$ and time is $t$ years,

$$I = Prt, \qquad A = P + I = P(1 + rt)$$

Each year adds the same dollar increment $Pr$. Year 10 pays the same simple interest as Year 1. Simple interest does not care how many times a bank says it compounds: the base never changes, so frequency $n$ is irrelevant.

Compound Interest

Compound interest adds each period's interest to the balance before the next period is calculated. With $n$ compounding periods per year,

$$A = P\left(1 + \frac{r}{n}\right)^{nt}, \qquad CI = A - P$$

Common GMAT frequencies:

When $nt$ is $2$, $3$, or $4$, ignore the closed exponent and multiply period by period:

• Write the periodic rate $r/n$.
• Multiply the current balance by $(1 + r/n)$ once per period.
• Stop after $nt$ multiplications; do not take a logarithm.

Example: $\text{\textdollar}2{,}000$ at $10\%$ compounded annually for $3$ years is $2{,}000 \times 1.1 = 2{,}200$, then $2{,}200 \times 1.1 = 2{,}420$, then $2{,}420 \times 1.1 = 2{,}662$. That three-step product is the official computation path; a logarithm would be the wrong tool.

Two-Year Gap Between Compound and Simple Interest

At the same annual rate $r$, the first year of annual compounding matches simple interest: both credit $Pr$. In the second year, simple interest again credits $Pr$, while compound interest credits $r$ times the new balance $P + Pr$, which is $Pr + Pr^2$. The extra $Pr^2$ is interest earned on the first year's interest. After two years:

$$SI_2 = 2Pr, \qquad CI_2 = 2Pr + Pr^2, \qquad CI_2 - SI_2 = Pr^2$$

If a stem gives both two-year totals, recover the rate from $r = (CI_2 - SI_2)/(Pr) = (CI_2 - SI_2)/(SI_2/2)$, then recover $P = (SI_2)/(2r)$. This identity is exact for two annual periods at a constant rate. It is not a general $t$-year formula.

Effective Annual Rate versus Nominal Rate

The nominal rate $r$ is the quoted annual percent. The effective annual rate accounts for intra-year compounding:

$$\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1$$

Whenever $n > 1$, EAR exceeds the nominal rate. Ranking by yield at a fixed nominal $r$: annual $<$ semiannual $<$ quarterly $<$ monthly. GMAT questions that mention quarterly compounding for six months are really two-period problems: $nt = 4 \times 1/2 = 2$ multiplications at rate $r/4$.

The Rule of 72 Is a Shortcut, Not a Scored Identity

A mental-math estimate for the years needed to double a balance at annual compound rate $r\%$ is

$$T_{\text{double}} \approx \frac{72}{r\%}$$

Examples: $6\%$ $\approx 12$ years, $8\%$ $\approx 9$ years, $9\%$ $\approx 8$ years, $12\%$ $\approx 6$ years. Two doublings estimate a fourfold increase; three doublings estimate an eightfold increase.

Label this correctly on test day. The Rule of 72 is a convenient approximation used in business arithmetic. It is not an official GMAC formula, it does not appear as a scored identity in the Official Guide Math Review, and it will not match $P(1+r)^t = 2P$ except by accident. Use it only when the stem asks for an approximate doubling time, or when you need a quick check that $8\%$ for about $9$ years is in the doubling neighborhood. If the stem demands an exact amount after a stated number of periods, multiply $(1 + r/n)$ instead.

A coarser linear estimate $T = 100/r\%$ is the simple-interest doubling time ($A = P(1 + rt) = 2P$). Compounding always doubles sooner than simple interest at the same rate, which is why $72$ is smaller than $100$.

Binomial Expansion for a Short Power

When you must estimate $(1 + r)^t$ for a small $r$ without a calculator, the first three binomial terms are enough for GMAT precision:

$$(1 + r)^t \approx 1 + tr + \frac{t(t-1)}{2}r^2$$

For $(1.04)^3$: $1 + 3(0.04) + 3(0.04)^2 = 1.12 + 0.0048 = 1.1248$, against the exact $1.124864$.

Comparison Table

Worked Example: Two-Year Difference

A deposit earns $\text{\textdollar}2{,}100$ of compound interest in two years, but would have earned $\text{\textdollar}2{,}000$ of simple interest at the same annual rate. Then $2Pr = 2{,}000$ so $Pr = 1{,}000$, and $Pr^2 = 100$. Thus $r = 100/1{,}000 = 0.10$ and $P = 1{,}000 / 0.10 = 10{,}000$.

Worked Example: Two Quarterly Periods

Principal $\text{\textdollar}8{,}000$ at a $12\%$ nominal annual rate compounded quarterly, held for $6$ months, is two periods at $3\%$ each: $8{,}000 \times 1.03 = 8{,}240$, then $8{,}240 \times 1.03 = 8{,}487.20$. The one-shot $8\%$ simple interest for half a year ($8{,}000 \times 1.06 = 8{,}480$) understates the compound result.