6.2 Fractions, Decimals, Percentages & Profit-Loss Math

Fractions, Decimals, Percentages & Profit-Loss Math

Fractions, decimals, percentages, and commercial profit-loss math form the core of practical quantitative reasoning on the NTS GAT General. These concepts evaluate your numerical agility in switching between equivalence formats and solving multi-step financial problems. Whether computing successive retail discounts, adjusting for inflation rates, or determining compound interest yields, mastering these operational rules ensures high performance on test day.

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1. Fraction Mechanics & Decimal Conversions

A fraction represents a part of a whole, expressed as $\frac{a}{b}$ where $a$ is the numerator and $b \neq 0$ is the denominator.

Classification of Fractions

• Proper Fraction: Numerator is strictly less than denominator ($a < b$), e.g., $\frac{3}{7}$. Value is strictly less than $1$.
• Improper Fraction: Numerator is greater than or equal to denominator ($a \ge b$), e.g., $\frac{11}{4}$. Value is $\ge 1$.
• Mixed Number: Combination of an integer and a proper fraction, e.g., $2 \frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{11}{4}$.

Rapid Fraction Comparison Methods

When comparing two fractions $\frac{a}{b}$ and $\frac{c}{d}$:

1. Cross-Multiplication Shortcut:

   • Compute $a \times d$ (left product) and $b \times c$ (right product).
   • If $a \cdot d > b \cdot c$, then $\frac{a}{b} > \frac{c}{d}$.
   • If $a \cdot d < b \cdot c$, then $\frac{a}{b} < \frac{c}{d}$.
   • Example: Compare $\frac{7}{9}$ and $\frac{4}{5}$. Cross products: $7 \times 5 = 35$ vs $9 \times 4 = 36$. Since $35 < 36$, $\frac{7}{9} < \frac{4}{5}$.

2. Benchmark Comparison: Compare both fractions against common benchmarks like $\frac{1}{2}$ or $1$.

Converting Repeating Decimals to Vulgar Fractions

Decimals with infinitely repeating digit blocks can be converted into vulgar fractions $\frac{p}{q}$ using algebraic shortcuts:

• Pure Repeating Decimal ($0.\overline{a_1 a_2 \dots a_k}$): Write the repeating block in the numerator over $k$ nines ($9$) in the denominator.

  $$0.\overline{37} = \frac{37}{99}$$ $$0.\overline{456} = \frac{456}{999} = \frac{152}{333}$$

• Mixed Repeating Decimal ($0.b_1 b_2 \dots b_m \overline{a_1 \dots a_k}$): Subtract the non-repeating part from the total number, placed over $k$ nines followed by $m$ zeros.

  $$\text{Fraction} = \frac{\text{Entire Number} - \text{Non-Repeating Part}}{99\dots900\dots0}$$

  Example: Convert $0.2\overline{36}$ to a fraction:

  $$0.2\overline{36} = \frac{236 - 2}{990} = \frac{234}{990} = \frac{13}{55}$$

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2. Percentage Fundamentals & Multi-Step Variations

A percentage represents a fraction with a fixed denominator of $100$ ($x\% = \frac{x}{100}$).

Core Percentage Formulas

$$\text{Percentage Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%$$ $$\text{Final Value} = \text{Original Value} \times \left(1 \pm \frac{r}{100}\right)$$

Base Inversion Rule

• If quantity $A$ is $x\%$ more than quantity $B$, then $B$ is less than $A$ by: $$\left(\frac{x}{100 + x} \times 100\right)\%$$
• If quantity $A$ is $x\%$ less than quantity $B$, then $B$ is more than $A$ by: $$\left(\frac{x}{100 - x} \times 100\right)\%$$

Example: If Salary A is $25\%$ more than Salary B, by what percentage is Salary B less than Salary A?

$$\text{Percentage Less} = \left(\frac{25}{100 + 25} \times 100\right)\% = \frac{25}{125} \times 100\% = 20\%$$

Net Successive Percentage Change Formula

If a quantity undergoes two successive percentage changes of $A\%$ and $B\%$, the single equivalent net percentage change is:

$$\text{Net Change \%} = A + B + \frac{A \times B}{100}$$

(Note: For percentage decreases, enter $A$ or $B$ as negative numbers.)

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3. Commercial Arithmetic: Profit, Loss, Markup & Discounts

Commercial arithmetic problems revolve around three core prices: Cost Price (CP), Marked Price (MP) (List Price), and Selling Price (SP).

Fundamental Commercial Formulas

Crucial Distinction: Profit or loss is calculated as a percentage of Cost Price (CP). Discount is calculated as a percentage of Marked Price (MP).

Dishonest Dealer / False Weight Math

If a merchant claims to sell goods at Cost Price but uses a false weight of $W_{\text{false}}$ instead of true weight $W_{\text{true}}$:

$$\text{Profit \%} = \frac{\text{Error}}{\text{True Weight} - \text{Error}} \times 100\% = \frac{W_{\text{true}} - W_{\text{false}}}{W_{\text{false}}} \times 100\%$$

4. Simple Interest & Compound Interest Foundations

Financial interest questions test your ability to compute investment growth over time.

Simple Interest (SI)

Interest computed strictly on the initial principal ($P$) throughout the entire investment tenure ($T$ years) at annual rate ($R\%$):

$$\text{SI} = \frac{P \times R \times T}{100}$$ $$\text{Total Amount } (A) = P + \text{SI} = P \left(1 + \frac{R \cdot T}{100}\right)$$

Compound Interest (CI)

Interest computed on the principal plus all previously accumulated interest. After $T$ compounding cycles:

$$\text{Total Amount } (A) = P \left(1 + \frac{R}{100}\right)^T$$ $$\text{Compound Interest } (\text{CI}) = A - P = P \left[ \left(1 + \frac{R}{100}\right)^T - 1 \right]$$

Fast Shortcut Formulas for Difference Between CI and SI

• For 2 Years ($T = 2$): $$\text{CI} - \text{SI} = P \left(\frac{R}{100}\right)^2$$
• For 3 Years ($T = 3$): $$\text{CI} - \text{SI} = P \left(\frac{R}{100}\right)^2 \left(3 + \frac{R}{100}\right)$$

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5. Comprehensive GAT Worked Examples

Worked Example 1: Successive Retail Discounts

Question: An item listed at PKR $10,000$ is sold after two successive discounts of $20\%$ and $10\%$. Find the final selling price and single equivalent discount percentage.

Solution:

1. Use the net change formula for discounts ($A = -20\%, B = -10\%$): $\text{Net Discount \%} = (-20) + (-10) + \frac{(-20)(-10)}{100} = -30 + 2 = -28\%$
2. Calculate final Selling Price: $\text{SP} = 10,000 \times \left(1 - \frac{28}{100}\right) = 10,000 \times 0.72 = \text{PKR } 7,200$
3. Final Answer: Selling Price = PKR $7,200$; Equivalent Discount = $28\%$.

Worked Example 2: Profit & Markup Computation

Question: A trader marks his goods $40\%$ above the Cost Price and allows a discount of $15\%$ on the Marked Price. What is his net profit percentage?

Solution:

1. Let Cost Price ($CP$) $= 100$.
2. Marked Price ($MP$) $= 100 + 40\% \text{ of } 100 = 140$.
3. Selling Price ($SP$) $= MP \times (1 - 0.15) = 140 \times 0.85 = 119$.
4. Profit $= SP - CP = 119 - 100 = 19$.
5. Profit Percentage $= \frac{19}{100} \times 100\% = 19\%$.
6. Final Answer: $19\%$.

Worked Example 3: CI vs SI Difference Problem

Question: The difference between Compound Interest and Simple Interest on a sum of money for $2$ years at $5\%$ per annum is PKR $25$. Find the principal sum.

Solution:

1. Apply the 2-year shortcut formula: $\text{CI} - \text{SI} = P \left(\frac{R}{100}\right)^2$.
2. Substitute given values: $25 = P \left(\frac{5}{100}\right)^2 = P \left(\frac{1}{20}\right)^2 = P \left(\frac{1}{400}\right)$.
3. Solve for $P$: $P = 25 \times 400 = 10,000$.
4. Final Answer: PKR $10,000$.