Meaning: By one more than the previous one.

Application: Finding squares of numbers ending in 5.

Problem: Find 35².

Step-by-step solution:

1. Take the digit before 5: 3.

2. Add 1 to it: 3 + 1 = 4.

3. Multiply: 3 × 4 = 12.

35² = 3 × 4 | 25
= 12 | 25
= 1225

Meaning: All from 9 and the last from 10.

Application: Multiplication of numbers close to 10, 100 or 1000.

Problem: Multiply 98 × 97.

Step-by-step solution:

1. Take the base: 100.

2. Find the deficiencies:

   • 100 − 98 = 2

   • 100 − 97 = 3

3. Cross-subtract: 98 − 3 = 95.

4. Multiply deficiencies: 2 × 3 = 06.

5. Join both parts.

Calculation:

  98   -2
  97   -3
 -----------
  95 | 06
 -----------

Answer: 9506

Meaning: Vertically and Crosswise.

Application: Multiplication using vertical and crosswise operations.

Problem: Multiply 23 × 14.

Step-by-step solution:

1. Multiply the right digits: 3 × 4 = 12. Write 2 and carry 1.

2. Cross-multiply: (2 × 4) + (3 × 1) = 11. Add carry 1 = 12. Write 2 and carry 1.

3. Multiply the left digits: 2 × 1 = 2. Add carry 1 = 3.

Calculation:

       2   3
       1   4
     ---------
       3   2   2

Answer: 322

Meaning: Transpose and Apply.

Application: Solving suitable algebraic equations by transposing terms.

Problem: Solve 5x + 3 = 23.

Step-by-step solution:

1. Move +3 to the right side.

2. 5x = 23 − 3

3. 5x = 20

4. Divide both sides by 5.

Calculation:

5x + 3 = 23
5x      = 23 - 3
5x      = 20
x       = 20 / 5
x       = 4

Answer: x = 4

Meaning: When the matching totals are equal, the corresponding sum is zero.

Application: Simplifying certain algebraic equations.

Problem: Solve:

1/(x+3) + 1/(x+5)
=
1/(x+1) + 1/(x+7)

Step-by-step solution:

1. Add the constants in the denominators on the left: 3 + 5 = 8.

2. Add the constants on the right: 1 + 7 = 8.

3. Since the sums match, the corresponding numerator sum is zero:

(x+3) + (x+5) = 0
2x + 8 = 0
2x = -8
x = -8 / 2
x = -4

Answer: x = −4

Meaning: If one is in proportion, the other is zero.

Application: Special simultaneous linear equations.

Problem: Solve:

6x + 7y = 8
19x + 14y = 16

Step-by-step solution:

1. Compare the coefficients of y:

7 : 14 = 1 : 2

2. Compare the right-side values:

8 : 16 = 1 : 2

1. Both ratios are equal. Therefore, x = 0.

2. Substitute x = 0 into the first equation:

6(0) + 7y = 8
7y = 8
y = 8/7

Answer: x = 0, y = 8/7

Meaning: By Addition and Subtraction.

Application: Solving simultaneous equations.

Problem: Solve:

x + y = 10
x - y = 2

Step-by-step solution:

1. Add both equations:

 x + y = 10
 x - y =  2
-------------
 2x     = 12

 x = 12/2
 x = 6

2. Subtract the second equation from the first:

(x + y) - (x - y) = 10 - 2
2y = 8
y = 4

Answer: x = 6, y = 4

Meaning: By Completion or Non-Completion.

Application: Simplifying calculations by completing numbers to convenient values.

Problem: Calculate 1000 − 678.

Step-by-step solution:

1. Increase 678 to 680 by adding 2.

2. Find the difference between 680 and 1000: 320.

3. Add the two differences.

Calculation:

1000 - 678

678 + 2 = 680
1000 - 680 = 320

320 + 2 = 322

Answer: 322

Meaning: Differences and Similarities.

Application: Certain calculations involving differences and related expressions.

Problem: Calculate 52² − 48².

Step-by-step solution:

Use the identity:

a² - b² = (a + b) × (a - b)

1. Add the numbers: 52 + 48 = 100.

2. Subtract the numbers: 52 − 48 = 4.

3. Multiply the results: 100 × 4 = 400.

Calculation:

52² - 48²
= (52 + 48) × (52 - 48)
= 100 × 4
= 400

Answer: 400

Meaning: Part and Whole.

Application: Breaking numbers into smaller parts to simplify calculations.

Problem: Calculate 25 × 12.

Step-by-step solution:

1. Break 12 into 10 + 2.

2. Multiply 25 by each part.

3. Add both results.

Calculation:

25 × 12
= 25 × (10 + 2)
= (25 × 10) + (25 × 2)
= 250 + 50
= 300

Answer: 300

Meaning: The Remainder by the Last Digit.

Application: Special remainder and recurring decimal techniques.

Problem: Find the decimal value of 1 ÷ 7.

Step-by-step solution:

1. Divide 1 by 7.

2. Continue the division to obtain the repeating remainders.

3. The remainder sequence is 1, 3, 2, 6, 4, 5 and then repeats.

4. Multiplying these remainders by 7 gives the last digits 7, 1, 4, 2, 8, 5. Arrange the digits from the actual long-division process.

Calculation:

1 ÷ 7 = 0.142857142857...

1/7 = 0.(142857)

Answer: 0.142857142857…

Meaning: The Ultimate and Twice the Penultimate.

Application: A special technique for selected multiplication problems, including multiplication by 12.

Problem: Calculate 23 × 12.

Step-by-step solution:

1. Write 23 with a zero at the beginning and end: 0230.

2. Starting from the right, add each digit to twice the digit immediately before it.

3. The resulting digits are 6, 7 and 2.

Calculation:

0230

0 + (2 × 3) = 6
3 + (2 × 2) = 7
2 + (2 × 0) = 2

Answer = 276

Answer: 276

Meaning: By One Less Than the Previous One.

Application: Multiplication by 9, 99, 999 and similar numbers.

Problem: Calculate 43 × 99.

Step-by-step solution:

1. Subtract 1 from 43: 43 − 1 = 42.

2. Subtract 43 from 100: 100 − 43 = 57.

3. Join the two results.

Calculation:

43 × 99

43 - 1 = 42
100 - 43 = 57

Answer = 42 | 57
       = 4257

Answer: 4257

Meaning: The Product of the Sum.

Application: Checking certain algebraic multiplication identities.

Problem: Verify:

(x + 2)(x + 3) = x² + 5x + 6

Step-by-step solution:

1. Add the coefficients in the first bracket: 1 + 2 = 3.

2. Add the coefficients in the second bracket: 1 + 3 = 4.

3. Multiply the sums: 3 × 4 = 12.

4. Add the coefficients on the right side: 1 + 5 + 6 = 12.

5. Both totals are equal.

Calculation:

Left side:
3 × 4 = 12

Right side:
1 + 5 + 6 = 12

12 = 12

Answer: Verified.

Meaning: The Factors of the Sum.

Application: Certain algebraic factorisation techniques.

Problem: Factorise x² + 5x + 6.

Step-by-step solution:

1. Find two numbers whose product is 6.

2. Their sum must be 5.

3. The required numbers are 2 and 3.

4. Split the middle term:

x² + 5x + 6

= x² + 2x + 3x + 6

= x(x + 2) + 3(x + 2)

= (x + 2)(x + 3)

Answer: (x + 2)(x + 3)