Math: No Calculator

These questions must be solved WITHOUT a calculator. The key is knowing

the tricks that make mental math manageable.

INTEGERS AND NEGATIVES

Two rules that handle every question:

  Subtracting a negative = adding a positive
    5 - (-3) = 5 + 3 = 8

  Adding a negative = subtracting
    5 + (-3) = 5 - 3 = 2

  Example: 1 + (-2) - (-3)
    Step 1: 1 + (-2) = 1 - 2 = -1
    Step 2: -1 - (-3) = -1 + 3 = 2

EXPONENT RULES

  a^m × a^n = a^(m+n)     (same base: ADD exponents)
  a^m ÷ a^n = a^(m-n)     (same base: SUBTRACT exponents)
  (a^m)^n = a^(m×n)       (power of a power: MULTIPLY exponents)
  a^0 = 1                 (anything to the zero = 1)
  a^1 = a                 (anything to the one = itself)

  Example: 3^4 ÷ 3 = 3^(4-1) = 3^3 = 27
  Or just: 3^4 = 81, then 81 ÷ 3 = 27

PLACE VALUE

For any number, each digit has a value based on its position:

  Number: 425,380
  4 = hundred-thousands = 400,000
  2 = ten-thousands = 20,000
  5 = thousands = 5,000
  3 = hundreds = 300
  8 = tens = 80
  0 = ones = 0

  How to find it: cover the digits to the right, read what's left,
  then take just that digit's contribution.

  "What is the value of the 2 in 425,380?"
  The 2 is in the ten-thousands place → 2 × 10,000 = 20,000

DIVIDING BY A FRACTION

THE RULE: Dividing by a fraction = multiplying by its reciprocal.

  The reciprocal of a/b is b/a (flip it upside down).

  Example: 24 ÷ (2/3)
    = 24 × (3/2)
    = 72/2
    = 36

  Why this works: "How many 2/3 fit into 24?" is the same as
  "24 groups of 3/2."

DIVISION WITH ROUNDING UP

Some word problems ask "what's the MINIMUM number of X to reach Y?"

  Example: How many times must you add $15 to get at least $100?
    100 ÷ 15 = 6.67
    You can't add $15 a fraction of a time.
    Round UP to 7. (7 × $15 = $105, which is at least $100)

  ALWAYS round UP in "minimum needed" problems, even if the decimal
  is small (e.g., 6.01 still rounds up to 7).

DECIMAL ADDITION

Line up the decimal points, then add normally. Add trailing zeros

if needed to make all numbers the same length.

  Example: 9.05 + 6.5 + 8.05 + 5 + 1.5
  Rewrite as:
    9.05
    6.50
    8.05
    5.00

1.50

   30.10 = 30.1

DECIMAL DIVISION

THE TRICK: Move both decimal points the same number of places to

make the divisor a whole number.

  Example: 17.86 ÷ 0.19
  Move both decimals 2 places right: 1786 ÷ 19
  Now it's a whole number division: 19 × 94 = 1786
  Answer: 94

  Quick check: 19 × 90 = 1710, 19 × 4 = 76, total = 1786 ✓

EXPANDED FORM WITH POWERS OF 10

Each place value can be written as a power of 10:

  10^0 = 1 (ones)
  10^1 = 10 (tens)
  10^2 = 100 (hundreds)
  10^3 = 1,000 (thousands)
  10^4 = 10,000 (ten-thousands)
  10^5 = 100,000 (hundred-thousands)
  10^6 = 1,000,000 (millions)

  Example: (3 × 10^6) + (2 × 10^4) + (5 × 10^3) + (4 × 10^2) + (7 × 10^1) + (1 × 10^0)
  = 3,000,000 + 20,000 + 5,000 + 400 + 70 + 1
  = 3,025,471

  WATCH OUT: If a power is missing (like 10^5 above), that place
  value is 0. Don't skip over it — write 3,025,471 not 325,471.

ORDER OF OPERATIONS (BEDMAS)

  B = Brackets first
  E = Exponents second
  D/M = Division and Multiplication (left to right)
  A/S = Addition and Subtraction (left to right)

  Example: (7 + 2.3) + 61 × 10^3
  Step 1 — Brackets: 7 + 2.3 = 9.3
  Step 2 — Exponents: 10^3 = 1,000
  Step 3 — Multiply: 61 × 1,000 = 61,000
  Step 4 — Add: 9.3 + 61,000 = 61,009.3

  COMMON MISTAKE: Doing addition before multiplication. In 2 + 2 × 4,
  you must do 2 × 4 = 8 first, then 2 + 8 = 10. NOT (2+2) × 4 = 16.

KEY FORMULAS TO KNOW (NO CALCULATOR)

  Subtracting a negative: a - (-b) = a + b
  Dividing by a fraction: a ÷ (b/c) = a × (c/b)
  Exponent division: a^m ÷ a^n = a^(m-n)
  Decimal division: move both decimals to make whole numbers
  Powers of 10: 10^0=1, 10^1=10, 10^2=100, 10^3=1000, etc.
  BEDMAS: Brackets, Exponents, Division/Multiplication, Addition/Subtraction