8.2 Percentages, Base Conversions & Successive Changes
Key Takeaways
- Percentages are normalized ratios with denominator 100; mastering unit fraction equivalents (1/2 through 1/20) transforms complex division into instantaneous single-step multiplication.
- The base change theorem dictates that if quantity A is x% greater than B, then B is [x / (100 + x)] * 100% less than A; in fractional steps, an upward shift of +1/n maps to an exact reciprocal downward shift of -1/(n + 1).
- Under product constancy (P * Q = K), an increase in factor P by fraction +a/b requires an exact offsetting contraction in factor Q by -a/(a + b) to maintain a constant total expenditure or budget.
- Successive percentage shifts of +a% and +b% yield a net compound change of [a + b + (ab / 100)]%, where algebraic signs govern net directional expansion or contraction.
- Multi-period compounding across branch CASA deposits, retail loan books, and demographic trends applies the factor (1 +/- r/100)^t, which can be evaluated using binomial expansions for small percentage adjustments.
8.2 Percentages, Base Conversions & Successive Changes
Percentage arithmetic forms the computational core of the State Bank of India (SBI) Junior Associate Quantitative Aptitude curriculum. It is not merely an isolated arithmetic chapter; percentage relationships serve as the mathematical engine for Data Interpretation (DI tables, charts, and caselets), Simple and Compound Interest, Profit and Loss, and Partnership profit distributions. In an SBI Clerk examination, proficiency in percentage calculations directly determines whether a candidate finishes the Numerical Ability section with minutes to spare or runs out of time on multi-step word problems.
The Percentage Multiplier & Fraction Equivalents
A percentage is fundamentally a standardized fraction whose denominator is fixed at 100:
In high-speed banking calculations, calculating $16.67%$ of Rs. 4,800 by multiplying $4800 \times 0.1667$ is computationally disastrous. Recognizing that $16.67% = \frac{1}{6}$ reduces the calculation to a trivial mental division: $4800 / 6 = \text{Rs. } 800$.
Master Fraction-to-Percentage Conversion Table (1/2 to 1/20)
Candidates must commit these unit fractions and their decimal multipliers to memory:
| Unit Fraction | Percentage Form | Exact Mixed Fraction | Decimal Multiplier (+ Growth) | Decimal Multiplier (- Decline) |
|---|---|---|---|---|
| $1/2$ | $50.00%$ | $50%$ | $\times 1.50$ ($= 3/2$) | $\times 0.50$ ($= 1/2$) |
| $1/3$ | $33.33%$ | $33\frac{1}{3}%$ | $\times 1.333$ ($= 4/3$) | $\times 0.667$ ($= 2/3$) |
| $1/4$ | $25.00%$ | $25%$ | $\times 1.25$ ($= 5/4$) | $\times 0.75$ ($= 3/4$) |
| $1/5$ | $20.00%$ | $20%$ | $\times 1.20$ ($= 6/5$) | $\times 0.80$ ($= 4/5$) |
| $1/6$ | $16.67%$ | $16\frac{2}{3}%$ | $\times 1.167$ ($= 7/6$) | $\times 0.833$ ($= 5/6$) |
| $1/7$ | $14.28%$ | $14\frac{2}{7}%$ | $\times 1.143$ ($= 8/7$) | $\times 0.857$ ($= 6/7$) |
| $1/8$ | $12.50%$ | $12\frac{1}{2}%$ | $\times 1.125$ ($= 9/8$) | $\times 0.875$ ($= 7/8$) |
| $1/9$ | $11.11%$ | $11\frac{1}{9}%$ | $\times 1.111$ ($= 10/9$) | $\times 0.889$ ($= 8/9$) |
| $1/10$ | $10.00%$ | $10%$ | $\times 1.10$ ($= 11/10$) | $\times 0.90$ ($= 9/10$) |
| $1/11$ | $9.09%$ | $9\frac{1}{11}%$ | $\times 1.091$ ($= 12/11$) | $\times 0.909$ ($= 10/11$) |
| $1/12$ | $8.33%$ | $8\frac{1}{3}%$ | $\times 1.083$ ($= 13/12$) | $\times 0.917$ ($= 11/12$) |
| $1/13$ | $7.69%$ | $7\frac{9}{13}%$ | $\times 1.077$ ($= 14/13$) | $\times 0.923$ ($= 12/13$) |
| $1/14$ | $7.14%$ | $7\frac{1}{7}%$ | $\times 1.071$ ($= 15/14$) | $\times 0.929$ ($= 13/14$) |
| $1/15$ | $6.67%$ | $6\frac{2}{3}%$ | $\times 1.067$ ($= 16/15$) | $\times 0.933$ ($= 14/15$) |
| $1/16$ | $6.25%$ | $6\frac{1}{4}%$ | $\times 1.0625$ ($= 17/16$) | $\times 0.9375$ ($= 15/16$) |
| $1/20$ | $5.00%$ | $5%$ | $\times 1.05$ ($= 21/20$) | $\times 0.95$ ($= 19/20$) |
The Asymmetric Base Change Theorem
One of the most frequent traps in banking examinations is confusing the direction of percentage comparison. When evaluating two quantities $A$ and $B$, the denominator (base) is always the quantity following the preposition "than" or "as compared to".
Mathematical Formulation
If $A$ is $x%$ more than $B$:
To find by what percentage $B$ is less than $A$, the reference base shifts from $B$ to $A$:
[!WARNING] The Symmetry Fallacy: If candidate salaries or branch deposits state that "A is 20% more than B", untrained candidates incorrectly assume that "B is 20% less than A". This is mathematically false! Since the reference base $A$ is larger than $B$, the identical numerical difference represents a smaller percentage of $A$ ($20/120 = 16.67%$).
The Reciprocal Fraction Shift Ladder
Expressing the upward change as a simple fraction $\frac{1}{n}$ unlocks an instantaneous reciprocal shortcut:
| If $A$ is greater than $B$ by: | Then $B$ is less than $A$ by: | Exact Percentage Shift |
|---|---|---|
| $+1/1$ ($+100%$) | $-1/2$ | $-50.00%$ |
| $+1/2$ ($+50%$) | $-1/3$ | $-33.33%$ |
| $+1/3$ ($+33.33%$) | $-1/4$ | $-25.00%$ |
| $+1/4$ ($+25%$) | $-1/5$ | $-20.00%$ |
| $+1/5$ ($+20%$) | $-1/6$ | $-16.67%$ |
| $+1/6$ ($+16.67%$) | $-1/7$ | $-14.28%$ |
| $+1/7$ ($+14.28%$) | $-1/8$ | $-12.50%$ |
| $+1/8$ ($+12.50%$) | $-1/9$ | $-11.11%$ |
Product Constancy Architecture ($A \times B = K$)
In commercial banking and macroeconomic problems, many governing formulas follow a bivariate multiplicative structure where the product remains invariant:
The Product Constancy Rule
If factor $A$ increases by a fraction $+\frac{a}{b}$, then factor $B$ must decrease by an exact fraction $-\frac{a}{a + b}$ to ensure that the composite product $K$ remains strictly constant:
Application: If the procurement cost of cash-handling machinery at a bank's regional hub increases by $25%$ ($+1/4$), by what percentage must the quantity purchased be reduced to keep total outlay unchanged?
- Upward shift $= +1/4$ (where $a = 1, b = 4$).
- Required downward shift $= -\frac{a}{a + b} = -\frac{1}{1 + 4} = -\frac{1}{5} = -20%$.
- The hub must reduce purchases by exactly $20%$.
Successive Percentage Changes & Compounding
When a financial baseline changes sequentially by $a%$ followed by $b%$, the net compound percentage change cannot be determined by simple addition ($a + b$). Because the second percentage acts upon the inflated or deflated intermediate base, compounding occurs.
The Universal Net Formula
Sign Convention:
- If a change is an increase / appreciation, assign a positive sign ($+a$).
- If a change is a decrease / depreciation, assign a negative sign ($-a$).
| Scenario | Formula Variant | Numerical Example ($a = 20%, b = 10%$) |
|---|---|---|
| Two Successive Increases | $+a + b + \frac{ab}{100}$ | $20 + 10 + \frac{200}{100} = +32%$ |
| Increase followed by Decrease | $+a - b - \frac{ab}{100}$ | $20 - 10 - \frac{200}{100} = +8%$ |
| Decrease followed by Increase | $-a + b - \frac{ab}{100}$ | $-20 + 10 - \frac{200}{100} = -12%$ |
| Two Successive Decreases | $-a - b + \frac{ab}{100}$ | $-20 - 10 + \frac{200}{100} = -28%$ |
Decimal Multiplier Method for Multi-Stage Changes
When more than two successive shifts occur (e.g., three consecutive quarters of portfolio expansion: $+10%$, $-5%$, and $+20%$), the net factor is computed by multiplying successive decimal factors:
The composite growth over the three periods is $(1.254 - 1) \times 100% = +25.4%$.
Population Dynamics & Asset Depreciation Compounding
Problems modeling municipal population growth or computer hardware depreciation in bank branches utilize discrete compound interest formulas:
where $V_0$ is the initial base, $r$ is the annual rate of change, and $n$ is the number of elapsed years.
[!TIP] Binomial Approximation for Small Percentage Rates: When the annual rate $r$ is very small ($r \le 4%$) and $n \le 3$, candidates can approximate the compound multiplier using the first two terms of the binomial expansion: $\left(1 + \frac{r}{100}\right)^n \approx 1 + \frac{nr}{100}$. For example, $(1.03)^2 = 1.0609 \approx 1 + 2(0.03) = 1.06$ ($6%$ net growth).
Step-by-Step Worked Banking Examination Problems
Problem 1: Branch Target Base Change
Problem: In a fiscal performance review, SBI Branch X mobilized Rs. 72 Crore in Current Account Savings Account (CASA) deposits, which is $20%$ higher than the CASA deposits mobilized by Branch Y. By what percentage is Branch Y's mobilization lower than Branch X's?
Solution:
- Let Branch Y's mobilization be $B$. Then Branch X's mobilization is $A = B \times (1 + 0.20) = 1.20 B$.
- We are given $A = 72$ Crore. Thus, $B = 72 / 1.20 = 60$ Crore.
- We must find by what percentage $B$ is less than $A$. The base is $A$:
- Fraction Shortcut: Branch X is $+1/5$ ($+20%$) higher than Y. By the reciprocal ladder, Branch Y is $-1/(5+1) = -1/6 = -16.67%$ lower than X. Solved in 5 seconds.
Problem 2: Portfolio Multi-Stage Expansion
Problem: An SBI regional loan hub manages an agricultural credit portfolio of Rs. 250 Crore. The portfolio grows by $12%$ in Year 1. In Year 2, due to severe regional drought, the portfolio contracts by $10%$. What is the net value of the loan portfolio at the conclusion of Year 2?
Solution:
- Compute the net successive percentage change:
- The net change is a modest expansion of $+0.80%$.
- Calculate the monetary value on the base of Rs. 250 Crore:
- The portfolio closes at exactly Rs. 252 Crore.
If the total CASA deposit volume of SBI Branch Alpha is 25% higher than that of Branch Beta, by what percentage is Branch Beta's deposit volume lower than Branch Alpha's?
Due to market fluctuations, the price of branch stationery paper increases by 20%. By what percentage must the branch administration decrease its paper consumption so that the total monthly expenditure on stationery remains strictly constant?
An SBI rural credit branch reports an active micro-finance portfolio of Rs. 40 Crore. If the portfolio expands by 10% during the first financial year and subsequently expands by 20% during the second financial year, what is the composite percentage expansion of the portfolio over the two-year span?