Do Number Sense (Mental Math)

Number Sense or mental math is the skill of using applied algebra, mathematics technique, brain power and invention to solve math problems. Full details of some of these techniques are described in links to other wikiHow articles.

Prerequisite: Knowing basic addition, subtraction, multiplication and division by memory.

Steps

Addition and Subtraction

  1. Convert difficult to add numbers to easy to add numbers.
    1. Round the number (to be added) up to the next highest multiple of ten.
    2. Add to the other number.
    3. Subtract the amount rounded up.
      • Example 88 + 56 = ? ; Round 88 up to 90.

        Add 90 to 56 = 146

        Subtract the two added to 88 (to round up to 90).

        146 - 2 = 144 ; the answer!
      • This process is simple reframing of the problem as 56 + (90 -2 ). Examples of other usages of this technique: 99 = (100 - 1); 68 = (70 - 2)
      • You can use a similar reframing technique for subtraction, too.
  2. Convert addition to multiplication. Multiplication is addition of multiple occurrences of the same number.
    1. Note how many times a number to be added is repeated.
      • For example:

        7 + 25 + 7 +7 +7 =

        becomes 25 + (4 × 7) =

        25 + 28 = 53
  3. Cancel Additive Opposites. Additive opposites can be +7 - 7.
    Additive opposites can also be 5 - 2 + 4 - 7.[1]
    1. Look for numbers that add or subtract for a total of 0. Using the example above:
      5 + 4 = 9 is the additive opposite of -2 -7 = -9

      Since they are additive opposites, no actual addition of all four numbers is needed; the answer is 0 (zero) by cancelling.
      • Try this:

        4 + 5 - 7 + 8 - 3 + 6 - 9 + 2 =

        becomes:

        (4 + 5) - 9 + ( -7<del> -<del> 3<del>) + (<del>8<del> + <del>2<del>) + 6 =</del></del></del></del></del></del></del> By grouping
        and remember, don't add them; just remove additive opposites from the problem.

        0 + 0 + 6 = 6

Multiplication

  1. Manage numbers ending in 0 (zero). For example, 120 × 120 =[2]
    1. Count the total number of zeros on the end. (In this case, 2).
    2. Do the rest of the problem.

      12 × 12 = 144
    3. Append the number of zeros counted to the end of the number;

      14400
  2. Use the distributive property of multiplication to convert difficult to multiply numbers to easy to multiply numbers. You might then be able to use some of the techniques below.[3]
    • For example:

      Instead of 14 × 6

      break 14 down into 10 and 4, and multiply both by 6, then add them together...

      14 × 6 = = 6×(10 + 4) = (10 × 6) + (4 × 6) = 60 + 24 = 84.
    • For example:

      Instead of: 35 * 37 = ?

      do this: 35 × (35 + 2) =

      = 352 + (2 × 35) = 1225 + 70 = 1295
  3. Square numbers ending in 5 (five).[4]

    Using; 352 = ?
    1. Ignoring the 5 on the end, multiple the number (3) by the next highest number (4).

      3 × 4 = 12
    2. Append 25 to the end of the number.

      1225
  4. Square numbers one less or more than a square you already know.

    Using 412 = ? and 392 = ?
    1. Figure the square you already know.

      402 = 1600
    2. Decide if you need to add or subtract. You will add with a larger square and subtract with a smaller one.
    3. Add the original number that was squared to the next number to be squared.

      40 + 41 = 81

      40 + 39 = 79.
    4. Make the addition or subtraction.

      1600 + 81 = 1,681 ---> 412 = 1,681
      1600 - 79 = 1,521 ----> 392 = 1,521
    • This only works for numbers one unit above or below the original.
  5. Simplify multiplication by using "Difference of Squares".

    Using 39 × 51 = ?
    1. Find the number that is equidistant of both numbers.

      In this case, 45, which is 6 away from both numbers.
    2. Square that number.

      452 = 2025
    3. Square the distance the numbers are from the central number.

      62 = 36
    4. Subtract that number from the first square.

      2025 - 36 = 1989
      • If you have taken algebra, the formula is expressed as:

        51 × 39 =
        (45 + 6)×(45 - 6) = 452 -6 2
        ( x + y )×( x - y ) = x2 - y2
      • For a more complete explanation, see How to Easily Solve Math Problems Using Difference of Squares.
  6. Multiply by 25.

    Using 25 × 12 = ?
    1. Multiply by 100 by appending two zeros to the end of the other (not 25) number.

      25 × 12
      1200
    2. Divide by 4.

      1200 ÷ 4 = 300
      25 × 12 = 300
      • For more details, see How to Multiply by 25 in Your Head.

Relateds

  • Multiply by 25 in Your Head
  • Easily Solve Math Problems Using Difference of Squares

Related Articles

References

Calculator's Cunning: The Art of Quick Reckoning by Karl Menninger

  1. ↑ https://pdst.ie/sites/default/files/Mental%20Maths%20Workshop%201%20Handbook.pdf
  2. ↑ https://www.bbc.co.uk/bitesize/topics/z36tyrd/articles/z2fkwxs
  3. ↑ https://www.mathmammoth.com/lessons/practical_tips_mental_math.php
  4. ↑ https://www.quickanddirtytips.com/education/math/5-tips-for-faster-mental-multiplication
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