Tag Archives: mathematics

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 103 so 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 4 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 )!

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 )!

 

MSC 9 – Dividing by 5, 50, 0.5, etc.

Five (5) is  ten divided by two ( 10 / 2 ), so to divide a number by 10/2, we

  • multiply it by 2, then
  • divide by 10.

Since dividing by 10 only involves moving the decimal point one place to the left, we can easily divide by 5 by just using doubling or multiplication by 2.

To divide a number by 5 we can either

Method A:

  • double the number first then
  • move the decimal point one place to the left.

Method B

  • move the decimal point first then
  • double the number.

We recommend the method B.

Let us try method A first.

Example 9.1: Find 164 ÷ 5

  • Double 16 is 32 and double 4 is 8. So 164 x 2 is 328.
  • Move the decimal point one place to the left to make it 32.8.

Now let us use the method B.

Example 9.2: Find 832 ÷ 5

  • Shifting the decimal point of the multiplicand one place to the left will make it 83.2. This also fixed the decimal point for the answer.
  • Doubling it gives 166.4.

Example 9.3: Find 1348 ÷ 50

  • Since 50 is half of 100 or 102, we move the decimal point in 1348 two places to the left making it 13.48.
  • We then double it to make it 26.96.

Example 9.4:  Find 24.5 ÷ 0.5

  • 0.5 is half of 1 or 100 so we do not have to adjust the decimal point. Just double the number to produce 49.0.
  • The answer becomes obvious if we double both the dividend and the divisor: 24.5 ÷ 0.5 = 49.0 ÷ 1 = 49

 Example 9.5: How many mint candies costing 50 centavos each can I buy with P 24.50?

 The figures here are the same as in Example 9.4 and the solution here clarifies the technique we used earlier: We can buy 2 candies for one peso; so for 24.50 pesos we can buy 24.5 x 2 or 49 candies.

Example 9.6: Find 376 ÷ 0.05

  • 0.05 is half of 0.1 or 10-1so we have to move the decimal point one place to the right, meaning we have to add a zero making it 3,760.
  • Doubling it would result to 7,520.

Exercise 9: Compute the following:

  1. )     370 ÷ 5 =
  2. )     535 ÷ 5 =
  3. )     2,367 ÷ 5 =
  4. )     9,898 ÷ 5 =
  5. )     4,656 ÷ 50 =
  6. )     24,579 ÷ 50 =
  7. )     5,836 ÷ 500 =
  8. )     34,785 ÷ 500 =
  9. )     4,524 ÷ 0.5 =
  10. )    3,645 ÷ 0.05 =

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!

 

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