Exhaustive Speed Calculation and Competitive Math Techniques Guide

Essential Foundational Knowledge for Competitive Calculation

To achieve elite speed in competitive exams such as SSC CGL, CHSL, MTS, Shipping, Railway, and Banking, certain mathematical constants and tables must be memorized to the point of instant recall.

• Mathematical Tables:
  • Memorize tables from $2$ to $30$ thoroughly.
  • Special attention must be given to the tables of $36$, $45$, and $72$. Examples of common exam values include $36 \times 3 = 108$ and $36 \times 5 = 106$.
• Squares and Cubes:
  • Squares of numbers should be memorized up to $40$.
  • Cubes of numbers should be memorized up to $30$.
• Roots and Values:
  • Square roots from $1$ to $10$ should be learned.
  • Familiarize yourself with square and cube roots to guess answers in high-pressure, calculative environments.
• Advanced Fundamentals:
  • Fraction-to-percentage conversions must be memorized.
  • Master the Digital Sum concept.
  • Divisibility rules for numbers up to $24$ (specifically including rules for $11$, $3$, $4$, and $5$) are mandatory knowledge for every selected aspirant.

Arithmetic Shortcuts: Multiplication and Division

Multiplication and Division by 5

• Multiplication: Instead of basic multiplication, add a zero to the number and divide by $2$. For example, to calculate $48 \times 5$, treat it as $\frac{480}{2} = 240$. For $995 \times 5$, add a zero and divide by $2$: $\frac{9950}{2}$.
• Division: To divide a number by $5$, multiply the numerator by $2$ and place a decimal point before the last digit. For example, to solve $\frac{94}{5}$, calculate $94 \times 2 = 188$, then place the decimal to get $18.8$. For $\frac{79}{5}$, $7 \times 2 = 14$, $9 \times 2 = 18$, resulting in $15.8$.

Multiplication and Division by 25

• Multiplication: Treat $25$ as $\frac{100}{4}$. To multiply $97 \times 25$, calculate $\frac{9700}{4}$.
• Division: To divide by $25$, multiply the numerator by $4$ and divide by $100$ (move the decimal two places left). For example, $\frac{37}{25}$ becomes $37 \times 4$, which is $148$. Placing the decimal results in $1.48$.

Multiplication and Division by 125

• Multiplication: Use the fact that $125 \times 8 = 1000$. To calculate $132 \times 125$, add three zeros and divide by $8$ ($\frac{132000}{8}$).
• Division: To divide by $125$, multiply the numerator by $8$ and place a decimal three places from the right. For example, to calculate $\frac{962}{125}$, solve $962 \times 8 = 7696$ and place the decimal to get $7.696$.

Squaring Techniques

Numbers Ending in 5

To find the square of a number ending in $5$ (e.g., $n5$):

1. The last two digits will always be $25$ (the square of $5$).
2. Multiply the first digit ($n$) by the next consecutive integer ($n+1$).

• Example 1 ($95^2$): $5^2 = 25$. $9 \times (9+1) = 90$. Result: $9025$.
• Example 2 ($45^2$): $5^2 = 25$. $4 \times 5 = 20$. Result: $2025$.

General 2-Digit Squaring Method

This method uses the unit digit and carry-forward steps. For example, finding $35^2$:

1. Square the unit digit ($5$): $25$. Write $5$ and carry over $2$.
2. Multiply the two digits and the exponent together ($3 \times 5 \times 2 = 30$). Add the carry ($30 + 2 = 32$). Write $2$ and carry over $3$.
3. Square the first digit ($3^2 = 9$). Add the carry ($9 + 3 = 12$).
4. Result: $1225$.

Advanced Multiplication and Proximity Rules

Multiplication by 11

Instead of long-form multiplication, use the sequence addition method. For a hypothetical calculation like $11 \times 197$:

1. Write the last digit: $7$.
2. Add consecutive digits: $9 + 7 = 16$ (write $6$, carry $1$).
3. Add next digits: $1 + 9 = 10$ (add carry: $10 + 1 = 11$, write $1$, carry $1$).
4. Final digit: $1$ (add carry: $1 + 1 = 2$).
5. Result: $2167$.

Numbers Near 100

• Case 1: Both numbers below 100 ($97 \times 96$):
  1. $97$ is $3$ less than $100$. $96$ is $4$ less than $100$.
  2. Multiply the differences: $(-3) \times (-4) = 12$. These are the last two digits.
  3. Subtract cross-wise: $97 - 4 = 93$ (or $96 - 3 = 93$).
  4. Result: $9312$.
• Case 2: One above and one below 100 ($98 \times 105$):
  1. $98$ is $-2$ relative to $100$. $105$ is $+5$ relative to $100$.
  2. Multiply differences: $-10$.
  3. Adjust cross-wise: $105 - 2 = 103$.
  4. Since we have a negative product ($-10$), take one hundred from the hundred's place ($103 \rightarrow 102$) and subtract the product: $100 - 10 = 90$.
  5. Result: $10290$.

Root Extraction and Approximation

Extraction of Square Roots (e.g., $\sqrt{6241}$)

1. Reserve the last two digits ($41$). The last digit $1$ means the root ends in either $1$ or $9$.
2. Find the nearest perfect square below $62$. This is $49$ ($7^2$).
3. The root is either $71$ or $79$.
4. Compare with the square of the middle value ($75^2 = 5625$). Since $6241 > 5625$, the answer is $79$.
5. Alternatively, multiply $7$ by the next number ($7 \times 8 = 56$). Since $62 > 56$, choose the larger number ($79$).

Extraction of Cube Roots (e.g., $\sqrt[3]{29791}$)

1. Reserve the last three digits ($791$). Only a cube ending in $1$ results in a unit digit of $1$.
2. Look at the remaining number ($29$). The nearest perfect cube below $29$ is $27$ ($3^3$).
3. Result: $31$.

Approximation for Non-Perfect Square Roots

Use the formula: $\sqrt{x \pm y} \approx \sqrt{x} \pm \frac{y}{2\sqrt{x}}$

• Example ($\sqrt{47}$):
  1. Nearest perfect square is $49$. Write as $49 - 2$.
  2. Apply formula: $7 - \frac{2}{2 \times 7} = 7 - \frac{1}{7}$.
• Example ($\sqrt{33}$):
  1. Nearest perfect square is $36$. Write as $36 - 3$.
  2. Apply formula: $6 - \frac{3}{2 \times 6} = 6 - \frac{3}{12}$.

Digital Sum Calculation Concept

The Digital Sum (DS) is obtained by adding the digits of a number until a single digit remains. This can verify and solve problems quickly.

• Example ($21 \times 16$):
  1. DS of $21 = 3$.
  2. DS of $16 = 7$.
  3. $3 \times 7 = 21 \rightarrow 2+1 = 3$.
  4. The final answer must have a DS of $3$.
• Logic for Perfect Squares: A perfect square will always have a digital sum of $1, 4, 7,$ or $9$. If divided by $8$, the remainder will always be $0, 1,$ or $4$. Numbers ending in $2, 3, 7, 8$ are never perfect squares.

Exam Strategy and Application Case Studies

Successive Percentage Change

For growth or reduction formulas, use: $a + b + \frac{ab}{100}$

• Scenario: A rectangle's length increases by $10\%$ and width by $20\%$.
• Calculation: $10 + 20 + \frac{10 \times 20}{100} = 30 + 2 = 32\%$.

Geometry and Mensuration Shortcuts (The 11-Multiplicity Rule)

When solving for values involving $\pi$ (such as Volume $= \pi r^2 h$), the answer is almost always a multiple of $11$. If given choices for volume using $22/7 \times 21/2 \times 21/2 \times h$, check the options for divisibility by $11$, $7$, and $3$. This eliminates the need for full calculation in $99\%$ of cases.

Rapid LCM Method

Instead of traditional ladders, use prime factors. For numbers like $24, 31, 32$:

1. Start with the largest number: $32$ ($8 \times 4$).
2. Check for $31$: It is prime, so multiply by $31$.
3. Check for $24$: We already have factor $8$ in $32$, but need a factor of $3$. Multiply by $3$.
4. LCM calculation: $32 \times 31 \times 3$. Check units: $2 \times 1 \times 3 = 6$. Look for an answer ending in $6$.

Depreciation and Previous Value Calculation

• Scenario: A value decreases at $20\%$ per year. The current value is $96,000$. Find the value $2$ years ago.
• Setup: Let original value be $x$. $20\% = 1/5$. Reduction means $4/5$.
• Equation: $x \times \frac{4}{5} \times \frac{4}{5} = 96000$
• Calculation: $16x / 25 = 96000$. $16 \times 6 = 96$. $x = 6000 \times 25 = 150,000$.

Questions & Discussion

• Q: What is the square of $36$?
  • A: Required for the comment section/practice.
• Q: What is the result of $75^2$ using the "ending in 5" rule?
  • A: Calculate as $7 \times 8$ followed by $25$, resulting in $5625$.
• Q: Calculate the result of $3375 \times 125$.
  • A: Add three zeros and divide by $8$.
• Q: Solve for $39^2$ using the general method.
  • A: $9^2 = 81$ (carry $8$); $3 \times 9 \times 2 = 54 + 8 = 62$ (carry $6$); $3^2 = 9 + 6 = 15$. Result: $1521$.
• Q: What is the square root of $2049$?
  • A: Apply square root extraction logic discussed above.
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