Tag Archives: Math short cuts

MATH-Inic conducts FREE Mental Math Training for Teachers

MATH-Inic Philippines started the series of Mental Math Training seminars for teachers on November 16, 2014.

A total of 23 teachers, mostly from the San Francisco District attended the training at the green school campus of VYP-MSC Institute of Technology at Sitio IX, San Gabriel, San Pablo City.

MATH-Inic President, Virgilio “Sir Ike” Prudente personally conducted the training on mental math techniques and methods to make teaching math easier and fun.

Here are some comments from the trainees:


“Thank you so much. We learned a lot from you. The knowledge you shared with us is of big help if we can share this one also to our students.”


“Thank you very much Mr. Prudente for sharing us some of the techniques in solving math equations. This will be of great help not only for us teachers but especially for our learners.


May God continue to bless you so that you can continue blessing others too.”


“Thank you po sa lahat ng techniques, malaking tulong poi to hindi lang sa akin higit sa lahat para sa aming mga bata.”


“Thank you po sa lahat ng nai-share n’yo. Natutuwa po ako sa mga techniques na kahit po ako ay malaki ang natutunan.


The next training session is scheduled on November 23, 2014 and starts at 8:00 am and ends at 12:00 noon.  This is free and certificates of attendance are given to all participants.

To register, visit or contact us at:

VYP – MSC Institute of Technology
Sitio IX, San Gabriel, San Pablo City
Telephone number: 562-6006
Cellphone numbers:
0917 853 5069
0922 854 3244
0939 939 9702

Conversion: Yards to Meters

One yard is 3 feet or 36 inches, while one meter is 39.37 inches. This makes one yard to be about 0.9155 meters. But for practical use, we can approximate and use the rate 1 yard = 0.9 meters.

We can easily do that if we apply MSC-8 multiplying by 9.

In this case we can use the fact that 0.9 is 1 – 0.1.   Let us see some examples:

  • 10 yard = 10 -1 = 9 meters
  • 20 yards = 20 -2 = 18 meters
  • 35 yards = 35 – 3.5 = 31.5 meters
  • 67.5 yards = 67.5 – 6.75 = 60.75 meters

yards to meters

              yards to meters

From a previous section we learned that 1 yard = 36 inches = 36 x 2.54 = 91.44 cm. Therefore 1 yard is exactly is 0.9144 meters . Using 1 yard = 0.9 meter is about 98.4 % accurate

 

MSC 15: Squaring Numbers Ending in 5

Squaring is multiplying a number by itself.

It is very easy to square two-digit numbers ending in five by using the Vedic Math sutra or word formula By one more than the one before.

The square of a two-digit number ending in 5 is composed of 2 parts.

  • The first or left hand part is obtained by multiplying the ten’s digit “By one more than the one before” or by the next number.
  • The second part is always 5 x 5 or 25.

Example 15.1:     Find 652

  • The first part is the ten’s digit 6 multiplied by the next number 7, (By one more than the one before) so  6 x 7 =  42.
  • The second part is always 25 ( 5 x 5) .
  • 652= 42|25 or 4,225

Even larger numbers can be squared.

Example 15.2:     Find 9952

          9952  =  99 x 100|25

          9952  = 990,025

Proof P15: Let t = the ten’s digit; 10t + 5 = two digit number ending in 5(10t + 5)2 = (10t)2 + 2(10t)(5) + 52= 100t2  + 100t + 25= 100(t2 + t) + 25= 100(t)(t + 1)+ 25

We can extend the technique to decimals and fractions.

Example 15.3:     3.52 = 3 x 4 | .5 x .5  = 12.25

Example 15.4:     (7 ½)2 = 7 x 8 | ½  x ½  =  56 1/4

Exercise 15:     Find the square of the following numbers

  1. )   45
  2. )   95
  3. )   195
  4. )   10,005
  5. )   10.5
  6. )   7.5
  7. )   6 ½
  8. )   9 ½
  9. )   100.5
  10. )   0.025

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!

Conversion: Meters to Feet

Since 1 yard  =  3 feet, we can use the conversion factor 1 meter  =  3.3 feet.  We can simplify this computation using  one of our favorite shortcuts, multiplying by 11. Remember that multiplying by 1.1 is simply multiplying by 11, and dividing by 10.

We can convert meters into feet  in two ways, both involving only mental calculations:

  1. Multiply the meter measure by 3 then by 1.1.
  • 7 meters = 7 x 3 x 1.1 = 21 x 1.1 = 23.1 feet
  • 23 meters = 23 x 3 x 1.1 = 69 x 1.1 = 75.9 feet
  • 30 meters = 30 x 3 x 1.1 = 90 x 1.1 = 99.0 feet

2.  Multiply the meter measure by 1.1 then by 3

  • 7 meters = 7 x 1.1 x 3 = 7.7 x 3 = 23.1 feet
  • 23 meters = 23 x 1.1 x 3 = 25.3 x 3 = 75.9 feet
  • 30 meters = 30 x 1.1 x 3 = 33 c 3 = 99 feet

Many will find the first method, easier to use.

For rough calculations, we can just multiply the meter measure by 10 and then divide by 3

  • 7 meters = 70 ÷ 3 = 23 1/3 feet
  • 23 meters = 230 ÷ 3 = 76 2/3 feet
  • 30 meters = 300 ÷ 3 = 100 feet

As seen in last example, the result is only 1% different from the previous method.

In this last example we used the conversion rate of 1 foot = .30 m whereas the exact conversion is 1 foot = 0.3048 meters. This is 98.4% accurate which is good enough for estimates.

This is another example how we can creative apply the 25 Math Short Cuts (and combinations thereof) in our everyday computing problems

MSC 14 – Dividing by 25, 250, 125

Remember that 25 is 100 divided by 4, so dividing by 25 is the same as dividing by 100 and multiplying by 4:

  1. move the decimal point two places to the left then
  2. multiply by 4 ( or double twice)

Of course, the multiplication can be done first, but using the recommended procedure has the advantage of providing us the magnitude of the answer first.

Example 14.1:    Find 700 ÷ 25 = (7 x 100) ÷ 25 = 7 x (100 ÷ 25) = 7 x 4 = 28

This example shows the basis of our shortcut. There are 4  25s  in every 100, so there are 7 x 4 or 28  25s in 7 hundreds.

 Example 14.2:    Find 3,400÷ 25

1.   Move the decimal point two places to the left to get 34.00.
2.   Double it to get 68.
3.   Double it again to get 136.

Example 14.3:    Find 3562÷ 250

1.   250 is 1/4 of 1000 or 10so we move the decimal point three places to the left giving us 3.562.
2.   Multiplying it by 4 gives us 14.248.

To divide a number by 125,

  1. Move the decimal point three places to the left then
  2. Multiply it by 8 (or double thrice)

 Example 14.4:    Find 20,400÷ 125

Moving the decimal point three places to the left gives 20.4

20.4 x 8 = 40.8 x 4 = 81.6 x 2 = 163.2

Example 14.5:    Find 758 ÷ 12.5

1.   This time 12.5 is 1/8 of 100 so we have to shift the decimal point two places to the left making it 7.58.
2.   Multiplying it by 8 gives us 60.64

Exercise 14:

  1. ) 1050 ÷ 25 =
  2. ) 6025÷ 25 =
  3. ) 8375 ÷ 25 =
  4. ) 12,400 ÷ 250 =
  5. ) 71000÷ 125 =
  6. ) 5,125÷ 125 =
  7. ) 3,475÷ 250 =
  8. ) 830÷ 12.5 =
  9. ) 2475÷ 2.5 =
  10. ) 5264÷ 0.25 =

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!

MSC 13 – Multiplying by 25, 250, 125

Remember that 25 is 100/4. This means we can multiply the number by 100, and then divide by 4.

To multiply by 25, move the decimal point of the multiplicand two places to the right and divide the resulting number by 4.

Example 13.1:    Find 64 x 25

  1. Since the multiplicand is a whole number, add two zeroes at the end to make it 6400.
  2. Divide 6400 by 2 giving 3200
  3. Divide 3200 by 2 to get the final answer, 1600

Example 13.2:    Find 9.3 x 25

  1. Move the decimal point two places to the right to make it 930
  2. Divide by using the one line method to get 5.

 Example 13.3:    How much is $250 in pesos if the exchange rate is P45.15 = $1

  1. Since 250 is one-fourth of 1000, move the decimal point three places to the right to get 45,150.
  2. Divide 45,150 by 4 to get P11,287.50

 Example 13.4:    How much is the selling price of a pair of running shoes originally priced at P4,795 if it is offered at 75% discount?

A 75% discount means the selling price is just 25% or 1/4 of the original price, so the selling price is just 4,795 divided by 4 or P 1,198.75.

How about multiplying by 125?

Remember 125 is simply 1000/8.

To multiply by 125, move the decimal point three places to the left and divide the result by 8.

Example 13.5:

2 x 4 x 5 x 8 x 25 x 125 =

(2 x 5) x (4 x 25) x (8 x 125)

= 10 x 100 x 1000 = 1,000,000

Example 13.6:    384 x 125 = 384,000/8 = 48,000

Exercise 13:

  1.    36 x 25 =
  2.    78 x 25 =
  3.    67 x 25 =
  4.    256 x 125 =
  5.    55 x 125 =
  6.    123 x 250 =
  7.    466 x 2.5 =
  8.    3599 x 25% =
  9.    2468 x 2.5% =
  10.    2552 x 1.25% =

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!

MSC 12 – Dividing by 4 and 8

We can easily divide a number by 4 by halving the number twice and by 8, by halving thrice.  We shall, however try dividing by 4 and 8 using the one line solution introduced in MSC 6 – Division by Two.

Example 12.1:    586 ÷ 4 =

When dividing a whole number by 4, there are only 4 possible remainders: 0, which means that the number is evenly divisible by 4; and 1, 2 and 3 which are equivalent to 0.25, 0.50 and 0.75 respectively. In this example, the dividend is an even number so the remainder can only be 0 or 2.

             1                         1   4                     1   4   6 
          4) 5 18  6           4) 5 18 26             4) 5 18 26  ( r 2

a) Dividing 5 by 4 gives a quotient of and a remainder of 1 which we place in front of the next digit of the dividend,

b) Dividing the next dividend 18 by 4 results in a quotient of 4 and a remainder of 2, which we then place in front of the next digit

c) Finally, we divide 26 by 4 to give 6 and a remainder of Thus the final quotient is 146 2/4 or 146.5.

 Example 12.2:     743  ÷ 4 =

               1   8   5                                                
           4) 7 323 (r 3

Here the dividend is an odd number, so we expect a remainder of 1 or 3.

  1. Dividing 7 by 4 gives 1 remainder
  2. 34 divided by 4 gives 8 remainder23 divided by 4 gives 5 remainder
  3. The quotient is 185 3/4 or 185.75. 

 The following are the values of the remainders when dividing by 8.

Remainder                         decimal equivalent
        1                                              0.125
        2                                              0.25
        3                                              0.375
        4                                              0.5
        5                                              0.625
        6                                              0.75
        7                                              0.875

Example 12.3:          983÷ 8 =

                    1   2  2                                  
                8) 9 123 (7

Using the one line solution, we have

  1. 9 divided by 8 is 1 remainder 1
  2. 18 divided 8 is 2 remainder 2
  3. 23 divided by 2 is 2 remainder 7

So we arrive at an answer of 122 remainder 7. This is equivalent to 122 7/8 or 122.875

Example 12.4:         6352 ÷ 8 =

                 7   9  4                                               
           8) 63 75 32 (0

Using the one line method, we got the quotient 794. For the careful observer, the dividend, 6,352 is very near 6,400 or exactly 48 less.  6,400÷ 8 = 800 and 48÷ 8 = 6 so 6352 ÷ 8 is simply 800 – 6 = 794.

 Exercise 12:

  1. )  456 ÷ 4 =
  2. )  527 ÷ 4 =
  3. )  983 ÷ 4 =
  4. )  2,538 ÷ 4 =
  5. )  6,789 ÷ 4 =
  6. )  745 ÷ 8 =
  7. )  1,278 ÷ 8 =
  8. )  3,684 ÷ 8 =
  9. )  6,454 ÷ 8 =
  10. )  10,522 ÷ 8 =

 

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!

Divisibility Test: 2, 4, and 8

The divisibility rules for 2, 4 and 8 are simple and similar.

A number is divisible by 2 if its last digit is divisible by 2. 

Since 10 is divisible by 2, the divisibility of a number by 2 depends only on the unit’s digit. So all even numbers, i.e. those numbers ending in 0, 2, 4, 6 and 8 are divisible by 2.

A number is divisible by 4 if its last 2 digits are divisible by 4.

100 is evenly divisible by 4 so only the last two digits determines the number’s divisibility by 4. First the number must be even to be divisible by 2.  All odd numbers are not.

A number is divisible by 8 if its last 3 digits are divisible by 8.

1000 is evenly divisible by 8, so we only have to determine if the last 3 digits are divisible by 8.

There are several handy methods to simplify the division of the last digits by 4 or 8:

a) Divide the last 2 digits by 4 or the last 3 digits by 8 using the method in MSC 12.

b) Divide the last two digits by 2, if the quotient is an even number, the number is divisible by 4. Divide the last three digits by 2 twice. If the quotient is an even number, the number is divisible by 8.

c) Add the ultimate (last) digit to twice the penultimate (second to the last) digit and if the sum is divisible by 4, the number is divisible by 4. For 8, add twice the hundreds digit and four times the tens digit to the unit’s digit. If the total is divisible by 8, the number is divisible by 8. This method is not recommended but is instructive in understanding divisibility of 3, 9, and 11.

d) Add or subtract from the digits to arrive at numbers obviously divisible by the number tested.

d.1) Add or subtract 4 or 8 from the last digit to make it zero and if the resulting ten’s digit is an even number, the number is divisible by 4.

Examples:   52 + 8 = 60, 52 is divisible by 4; but 74 – 4 = 70, so 70 is not divisible by 4.

d.2) To find out if a number is divisible by 8, add or subtract 8 or 16 from the last three digits to make the last digit zero and see if the number resulting is divisible by four.  Add or subtract 40 or 80 to make the ten’s digit also zero if needed.  If the hundreds digit becomes even, the number is divisible by 8.

392 + 8 = 400; 392 is divisible by 8;             296 – 16 = 280;  280 – 80 = 200;  296 is divisible by 8

324 + 16= 340;  340-40 = 300; 324 is not divisible by 8.

MSC 11 – Division by 9

Most of us want to avoid the number 9 in almost all calculations. But we can make calculations easier by thinking of 9 as (10 – 1). This fact is particularly useful in division by 9.

Dividing-by-9

Every 10 contains a 9 and a remainder of 1. So every multiple of ten that is less than 90 will have a quotient and remainder equal to its tens digit.

So                           20/9 = 2 r 2

                                40/9 = 4 r 4

and                        70/9 = 7 r 7.

Extending this observation, we can readily obtain the quotient when small numbers are divided by 9.

Take the case of 34. When divided by 9, the quotient is equal to the tens digit, 3 and the remainder is equal to the sum of the tens and units digits, 3 + 4 or 7.

Similarly,

                42/9 =  4 r  (4+2) = 4 r 6

                71/9 = 7 r (7+1)  = 7 r 8

                26/9 = 2 r (2+6) = 2 r 8

                69/9 = 6 r (6+9) = 6 r 15

But wait!  Since the remainder 15 is greater than 9, we can divide 15 by 9 to get 1 r 6.

So 69/9 = 6 r 15  = (6+1) r 6 = 7 r 6.

At this point, we would like to stress that the following results are equivalent:

                69/9 = 6 r 15 = 7 r 6 = 8 r -3 but 7 r 6 is the best form.

Example 11.1:                    1321/9

We can write the procedure as:     1 3 2 1 / 9

Step 1. Bring down the first digit (1) to the answer row.

      1 3 2 1 / 9
      1

Step 2. Add the next digit of the dividend to this number to get the next digit of the quotient: (1+3=4)
      1 3 2 1 / 9 
      1 4

Step 3. Repeat the preceding procedure to get the next digit of the quotient:    (4+2=6)
     1 3 2 1 / 9
      1 4 6

Step 4. The last sum is the remainder: (6+1=7)                                                   1 3 2 1 / 9
      1 4 6 r 7

Example 11.2:             2023/9

      2 0 2 3 / 9
      2 2 4 r 7

To check: the sum of the digits of the dividend should be equal to the remainder.

      2 + 0 + 2 + 3 = 7

Example 11.3:            4352/9

      4   3    5    2 / 9
      4   7   12  r 14

Here, we see that we have a 12 and a 14 in the quotient. The 1 in the 12 must be carried over to the 7 to yield 482.  There is also one 9 in the remainder 14.

So the final answer is 483 r 5.

We can modify our procedure to avoid double digits in the quotient.

      4   3   5    2 / 9
      4  

Before writing down the 7 (4 + 3), we see that the next addition 7 + 5 will give a two digit result, 12. So we anticipate the carry operation and write down 8 instead of 7.

      4   3   5    2 / 9
      4   8

We then proceed as before

4   3   5    2 / 9
    4   8

8 + 5 = 13. But since we have performed the carry operation in the previous step, we will write down only the last digit 3.

4   3   5    2 / 9
    4   8   3

Finally we have the remainder: 3 + 2 = 5

4   3   5    2 / 9
     4   8   3 r 5      

check:         4 + 3 + 5 + 2 = 14; 1 + 4 = 5

The following are the decimal values of the remainder when dividing by 9.

1 – 1/9 = .1111… = 0.1

2 – 2/9 = .2222… = 0.2

3 – 3/9 = .3333… = 0.3

4 – 4/9 = .4444… = 0.4

5 – 5/9 = .5555… = 0.6

6 – 6/9 = .6666… = 0.7

7 – 7/9 = .7777… = 0.8

8 – 8/9 = .8888… = 0.9

Exercise 11: Divide the following numbers by 9

  1. )      134 / 9 =
  2. )      215 / 9 =
  3. )      2231 / 9 =
  4. )      4202 / 9 =
  5. )      625 / 9 =
  6. )      3030 / 9 =
  7. )      7135 / 9 =
  8. )      5672 / 9 =
  9. )      3692 / 9 =
  10. )      46893 / 9 =

 

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!