4.1 Fundamental Arithmetic & Percentage Applications

4.1 Fundamental Arithmetic & Percentage Applications

Arithmetic and number properties form the fundamental bedrock of the Quantitative Reasoning section in the Higher Education Aptitude Test (HAT). Whether you are taking HAT 1 (Engineering/STEM), HAT 2 (Management Sciences), HAT 3 (Social Sciences/Law), HAT 4 (Biological/Medical Sciences), or HAT-General, computational speed, numerical fluency, and conceptual mastery of number systems are critical. The current HEC content-weight document does not establish a permanent delivery-format or calculator rule. Build mental and written computational fluency, then follow the permitted-materials instructions for the live administration.

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Classification of the Real Number System

All numerical quantities evaluated on the HEC HAT belong to the Real Number System (denoted by $\mathbb{R}$). Understanding the hierarchical subset relations among real numbers is essential for questions involving variables, constraints, and set definitions.

                     REAL NUMBERS (R)
                       /        \
          Rational (Q)            Irrational (I)
          /          \             (e.g., √2, π, e)
    Integers (Z)   Fractions/Decimals
      /      \
 Negative  Whole (W)
             /     \
           Zero   Natural (N)

Subset Hierarchy

1. Natural Numbers ($\mathbb{N}$): The set of positive counting numbers: $\mathbb{N} = \{1, 2, 3, 4, 5, \dots\}$.
2. Whole Numbers ($\mathbb{W}$): Natural numbers including zero: $\mathbb{W} = \{0, 1, 2, 3, 4, \dots\}$.
3. Integers ($\mathbb{Z}$): All positive numbers, negative whole numbers, and zero: $\mathbb{Z} = \{\dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.
4. Rational Numbers ($\mathbb{Q}$): Any number that can be expressed as a quotient $\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \neq 0$. Rational numbers either terminate (e.g., $0.75 = \frac{3}{4}$) or recur infinitely (e.g., $0.333\dots = \frac{1}{3}$).
5. Irrational Numbers ($\mathbb{I}$ or $\mathbb{R} \setminus \mathbb{Q}$): Numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are non-terminating and non-recurring (e.g., $\sqrt{2} \approx 1.4142$, $\sqrt{3}$, $\pi \approx 3.14159$, $e \approx 2.71828$).

Prime and Composite Numbers

• Prime Numbers: Integers strictly greater than $1$ that have exactly two distinct positive divisors: $1$ and the number itself. The smallest prime number is $2$, which is also the only even prime number. The primes below $30$ are: $2, 3, 5, 7, 11, 13, 17, 19, 23, 29$.
• Composite Numbers: Integers strictly greater than $1$ that have more than two positive divisors (e.g., $4, 6, 8, 9, 10, 12$).
• Important Rules: The number $1$ is neither prime nor composite because it has only one positive divisor ($1$). The number $0$ is neither positive nor negative, but it is an even integer.
• Co-Prime (Relatively Prime) Numbers: Two positive integers $a$ and $b$ are co-prime if their Greatest Common Divisor is $1$ (i.e., $\gcd(a,b) = 1$). For example, $8$ and $15$ are co-prime, even though neither number is individually prime.

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Divisibility Rules (Integers 2 through 11)

Divisibility shortcuts allow test takers to factor large numbers rapidly without performing tedious long division on paper.

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Prime Factorization, LCM & GCD

The Fundamental Theorem of Arithmetic

Every composite integer greater than $1$ can be uniquely expressed as a product of prime factors, up to the order of the factors.

$$\text{Example: } 360 = 2^3 \times 3^2 \times 5^1$$

Greatest Common Divisor (GCD) and Least Common Multiple (LCM)

• GCD (or HCF): The largest positive integer that divides two or more integers without a remainder. Obtained by taking the lowest power of common prime factors.
• LCM: The smallest positive integer that is divisible by two or more integers. Obtained by taking the highest power of all prime factors.

Fundamental LCM-GCD Identity

For any two positive integers $a$ and $b$:

$$\gcd(a,b) \times \text{lcm}(a,b) = a \times b$$

Worked Example: Find the GCD and LCM of $48$ and $72$, and verify the product identity.

1. Prime Factorization: $48 = 2^4 \times 3^1$, and $72 = 2^3 \times 3^2$.
2. $\gcd(48, 72) = 2^{\min(4,3)} \times 3^{\min(1,2)} = 2^3 \times 3^1 = 8 \times 3 = 24$.
3. $\text{lcm}(48, 72) = 2^{\max(4,3)} \times 3^{\max(1,2)} = 2^4 \times 3^2 = 16 \times 9 = 144$.
4. Verification: $\gcd \times \text{lcm} = 24 \times 144 = 3,456$. Product $a \times b = 48 \times 72 = 3,456$. Identical!

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Converting Recurring Decimals to Vulgar Fractions

Recurring decimals represent rational numbers. On the HEC HAT, questions frequently test conversion of repeating decimals into reduced fractions $\frac{P}{Q}$.

Algebraic Procedure

1. Let $x$ equal the recurring decimal.
2. Multiply $x$ by $10^m$ to shift non-repeating digits past the decimal point.
3. Multiply $x$ by $10^{m+k}$ to shift one full repeating block past the decimal point (where $k$ is the repeating block length).
4. Subtract the equations to eliminate the infinitely repeating decimal tail.
5. Solve for $x$ and reduce to lowest terms.

Worked Example: Convert $x = 0.2\overline{45} = 0.2454545\dots$ into a simplified fraction.

• Let $x = 0.2454545\dots$
• Multiply by $10$: $10x = 2.454545\dots$ (Equation 1)
• Multiply by $1000$: $1000x = 245.454545\dots$ (Equation 2)
• Subtract Eq 1 from Eq 2: $1000x - 10x = 245.4545\dots - 2.4545\dots \Rightarrow 990x = 243$.
• Simplify: $x = \frac{243}{990}$. Dividing numerator and denominator by $9$: $x = \frac{27}{110}$.

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Order of Operations (PEMDAS / BODMAS)

When evaluating compound numerical expressions, strict precedence must be followed:

1. P / B: Parentheses / Brackets — innermost grouping symbols first: (), [], {}.
2. E / O: Exponents / Orders (powers, roots).
3. MD: Multiplication and Division — evaluated left to right in order of appearance.
4. AS: Addition and Subtraction — evaluated left to right in order of appearance.

$$\text{Evaluate: } 18 - \left[ 6 + 2 \times \left( 5 - 3^2 \div 3 \right) \right]$$

1. Inner exponent: $3^2 = 9 \rightarrow (5 - 9 \div 3)$.
2. Inner division: $9 \div 3 = 3 \rightarrow (5 - 3) = 2$.
3. Multiplication in brackets: $2 \times 2 = 4 \rightarrow [6 + 4] = 10$.
4. Final subtraction: $18 - 10 = 8$.

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Percentages, Profit, Loss & Discount Applications

A percentage expresses a fraction with a denominator of $100$.

Key Percentage Formulas

• Percentage Change: $$\% \text{ Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%$$
• Restoring Original Value: If a quantity increases by $r\%$, the percentage reduction required to restore the original value is: $$\text{Percentage Reduction} = \left( \frac{r}{100 + r} \right) \times 100\%$$

Profit, Loss & Markup Definitions

• Cost Price (CP): The total acquisition cost incurred by the seller.
• Selling Price (SP): The final transaction price paid by the buyer.
• Marked Price (MP): The list price printed on the label before discounts.
• Profit %: $\text{Profit \%} = \frac{SP - CP}{CP} \times 100\%$
• Loss %: $\text{Loss \%} = \frac{CP - SP}{CP} \times 100\%$
• Discount %: $\text{Discount \%} = \frac{MP - SP}{MP} \times 100\%$

Successive Discounts Formula

If an item is subjected to two consecutive discounts of $d_1\%$ and $d_2\%$, the single net equivalent discount percentage $D_{\text{net}}$ is:

$$D_{\text{net}} = d_1 + d_2 - \frac{d_1 \times d_2}{100}$$

Worked Example: An article marked at PKR $8,000$ is offered at successive discounts of $20\%$ and $10\%$. Find the final selling price.

• Net Discount $= 20 + 10 - \frac{20 \times 10}{100} = 30 - 2 = 28\%$.
• Final $SP = \text{MP} \times (1 - 0.28) = 8,000 \times 0.72 = \text{PKR } 5,760$.

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Simple vs. Compound Interest

Simple Interest (SI)

Interest is calculated strictly on the initial principal amount $P$ over time $t$ years at interest rate $r\%$ per annum:

$$I = \frac{P \times r \times t}{100}, \quad A = P + I = P \left(1 + \frac{r \cdot t}{100}\right)$$

Compound Interest (CI)

Interest is added to the principal at the end of each compounding period, so interest earns interest:

$$A = P \left(1 + \frac{r}{100n}\right)^{n \cdot t}, \quad CI = A - P$$

where $n$ is the compounding frequency per year ($n=1$ for annual, $n=2$ for semi-annual, $n=4$ for quarterly).

Two-Year Difference Formula

For $t = 2$ years compounded annually, the difference between Compound Interest and Simple Interest is:

$$CI - SI = P \left( \frac{r}{100} \right)^2$$

Worked Example: PKR $100,000$ is invested for $2$ years at $10\%$ per annum. Calculate SI, CI (annual), and their difference.

• $SI = \frac{100,000 \times 10 \times 2}{100} = \text{PKR } 20,000$.
• $A = 100,000 \times (1 + 0.10)^2 = 100,000 \times 1.21 = \text{PKR } 121,000 \Rightarrow CI = \text{PKR } 21,000$.
• Difference $= CI - SI = 21,000 - 20,000 = \text{PKR } 1,000$.
• Check via shortcut formula: $100,000 \times (10/100)^2 = 100,000 \times 0.01 = \text{PKR } 1,000$. Exact match!