How to do mental calculations

2024 Author: Samantha Chapman | [email protected]. Last modified: 2023-12-17 09:13

Mental mathematics is the ability to use applied algebra, mathematical technique, brain power and inventiveness to solve mathematical problems. More precise details of some of these techniques are also described in other wikiHow articles.

Prerequisite: basic knowledge of addition, subtraction, multiplication and division by heart.

Steps

Method 1 of 2: Addition and Subtraction

Step 1. Transform numbers that are difficult to manage in mind with others that are easier to add up

1. Round the number (to be added) to the next multiple of ten.
2. Add the other number.

3. Subtract the rounded amount.

   • Example 88 + 56 =?; Rounded 88 becomes 90.

     Add 90 to 56 = 146

     Subtract the two units you added to 88 (to round to 90).

     146 - 2 = 144: here is the answer!

   • This procedure is a simple reformulation of the type 56 + (90 - 2) problem. Examples of other uses of this technique: 99 = (100 - 1); 68 = (70 - 2)
   • A similar technique can also be used for subtraction.

   

   Step 2. Convert addition to multiplication

   Multiplication is the addition of multiple occurrences of the same number.

   1. Note how many times a number to add is repeated.

      • For instance:

        7 + 25 + 7 + 7 + 7 + 7 =

        becomes 25 + (5 × 7) =

        25 + 35 = 60

   

   Step 3. Cancel opposites in algebraic additions

   For example, they can be + 7 - 7. The additive opposites can also be 5 - 2 + 4 - 7.

   1. Look for numbers to add or subtract for a total of 0. Using the above example: (Note: the image above is wrong. It shows 5 + 9 = 9 -2 -7 = 9 while it should be 5 + 4 = 9 - 2 - 7 = - 9)

      5 + 4 = 9 is the additive opposite of - 2 - 7 = - 9

      Since they are additive opposites, it is not necessary to add up all four numbers; the answer is 0 (zero) for cancellation.

      • Try this:

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

        it becomes:

        (4 + 5) - 9 + (-7 - 3) + (8 + 2) + 6 = Group them

        and remember not to add them; just remove the additive opposites from the problem.

        0 + 0 + 6 = 6

   Method 2 of 2: Multiplication

   

   Step 1. Learn to handle numbers ending in 0 (zero)

   For example 120 × 120 =

   1. Count the total number of zeros at the bottom (in this case 2).

   2. Do the rest of the problem.

      12 × 12 = 144

   3. Add the number of zeros you counted to the end of the result;

      14.400

      

      Step 2. Use the distributive property of multiplication to convert hard-to-multiply numbers into simpler ones

      You may then be able to use some of the techniques below.

      • For instance:

        Instead of 14 × 6

        break the 14 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 instance:

        Instead of: 35 × 37 =?

        do this: 35 × (35 + 2) =

        = 35² + (2 × 35) = 1225 + 70 = 1295

      

      Step 3. Square of numbers ending in 5 (five)

      Suppose 35² = ?

      1. Ignoring the 5 at the end, we multiply the number (3) by the next highest number (4).

         3 × 4 = 12

      2. Let's add 25 to the end of the number.

         1225

         

         Step 4. Square numbers that differ by one from the number you already know

         We calculate 41² =? and 39² = ?

         1. We calculate the already known square.

            40² = 1600

         2. Decide if you need to add or subtract. It is added with a larger square and subtracted with a smaller one.

         3. Add the original number to the next or previous.

            40 + 41 = 81

            40 + 39 = 79.

         4. Do the addition or subtraction.

            1600 + 81 = 1.681 --> 41² = 1.681

            1600 - 79 = 1.521 --> 39² = 1.521

            It only works with numbers one unit lower or higher than the original

            

            Step 5. Simplify the multiplication by using the "difference of squares" rule

            We calculate 39 × 51 =?

            1. Find the number that is equidistant from both numbers.

               In this case, 45, which is 6 units away from both numbers.

            2. Square that number.

               45² = 2025

            3. Square the "distance" of the numbers from the central one.

               6² = 36

            4. Subtract that number from the first square.

               2025 - 36 = 1989

               • If you have studied algebra, the formula is expressed as:

                 51 × 39 =

                 (45 + 6)×(45 - 6) = 45² - 6²

                 (x + y) × (x - y) = x² - y²

               • For a more complete explanation, read an article on how to easily solve math problems using the difference of squares.

               

               Step 6. Multiply by 25

               We calculate 25 × 12 =?

               1. Multiply by 100 by adding two zeros to the end of the other number (not 25).

                  25 × 12

                  1200

               2. Divide by 4.

                  1200 ÷ 4 = 300

                  25 × 12 = 300