Tag Archives: mathematics

VYP Launches 25 Math Short Cuts Book

In celebration of MSC’s 25th foundation day, Virgilio “Ike”Prudente , President of VYP MSC Institute of Technology, is launching his book, 25 Math Short Cuts.

25 Math Short Cuts is a compilation of the short cuts published in our 25 MSC newsletter. Those shortcuts will show that most calculations can be done mentally.

The foreword was written by former Economic Planning Secretary and Director-General Cielito Habito, PhD.

The book has received testimonials from several academic and public personalities, including:

  • Michael Tan, PhD, Chancellor of the University of the Philippines, Diliman
  • Malou Orijola, Assistant Secretary of Department of Science and Technology and Governor, National Book Development Board
  • Rey Vea, President, Mapua Institute of Technology
  • Gen. Hermogenes Esperon, former Chief of Staf, Armed Forces of the Philippines
  • Admiral Ferdinand Golez, Former Flag Officer In Command, Philippine Navy
  • Kenneth Williams, Author of more than 20 Books in Vedic Math
  • Bobby Castro, Chief Operating Officer, Palawan Pawnshop
  • Isaac Pitcairn Yap, former Dean, College of Engineering, Architecture and Technology, Palawan State University
  • Rex Aurelius Robielos, Dean of the School of Industrial Engineering and Engineering Management, Mapua Institute of Technology
  • Emmanuel Nadela, former Chairman, Mechanical Engineering Department, Cebu Institute of Technology
  • Romeo Fule, Math Supervisor, Department of  Education, San Pablo City
  • Albert Saul, Principal, San Pablo Science High School

Many children nowadays avoid Math. Some even hate Math. With this book, Sir Ike hopes to turn those Math Haters into Math Lovers.

You can purchase the book for you or your children, and it is a perfect gift this Christmas.

Contact Virgilio Prudente or the MATH-Inic/MSC office for your orders.

 

25 MSC book cover

25 MSC book cover

Conversion: Miles to Kilometers

We frequently encounter distances in miles (English System), but in the Philippines, we use the metric system, and need to convert miles to kilometers,

We will use the conversion 1 mile = 1.6 km. And we have several ways to make the computation easier.  Let us consider two ways.

The easiest way to multiply buy 1.6 is by treating 1.6 as 1 + 0.5 + 0.1. Note that 0.5 is half of 1 and 0.1 is one tenth of 1.

  • 30 miles = 30 + ½(30) + 1/10 (30) =  30 + 15 + 3 = 48 km
  • 72 miles = 72 + 36 + 7.2 = 115.2 km
  • 114 miles = 114 + 57 + 11.4 = 182.4 km
  • 205 miles = 205 + 102.5 + 20.5 = 328 km

If we note that 1.6 = 16 / 10, we discover another way to multiply by1.6.  Simply  double the mile measure 4 times (multiply by 16) and move the decimal point of the final product one place to the left ( divide by 10).

  • 30 miles = 30 x 2 → 60 x 2 → 120 x 2 → 240 x 2 → 480 → 48 km
  • 72 miles = 72 x 2 → 144 x 2→ 288 x 2 → 576 x 2 → 1152 →  115.2 km
  • 114 miles = 114 x 2 → 228 x 2→ 456 x 2 → 912 x 2 → 1824 →  182.4 km
  • 205 miles = 205 x 2 → 410 x 2→ 820 x 2 → 1640 x 2 → 3280 → 328 km
1 mile is 1760 yards or 5280 feet which is equivalent to 1609.344 meters. For practical purposes we can use the conversion 1 mile = 1.6 kilometers. This is almost 99.5% accurate

C.)  A third way is to double the Mile measure and save it as the minuend; double it again and move the decimal point one place to the left to get the subtrahend; then perform the subtraction.

30 miles = 30 x 2 = 60;  60 x 2= 120 → 12.0; 60 – 12  → 48 km

72 miles = 72 x 2 = 144; 144 x 2=288  → 28.8; 144 – 28.8 =  115.2 km

114 miles = 114 x 2 = 228; 228 x 2=456  → 45.6; 228 – 45.6 = 182.4 km

205 miles = 205 x 2 = 410; 410 x 2 = 820 → 82.0; 410 – 82  = 328 km

MSC 19: Base Multiplication: One Number Above and One Number Below the Base

When one number is above and the other number is below the base, we either

  • a) add the excess of the number above the base to the other number or
  • b) deduct the deficiency of the number below the base from the other number.

Example 19.1:     Find 104 x 98
Our base is 100, 104 is 4 above the base, and 98 is 2 below the base.
104 x 98
= (100 + 4) *(100 – 2)
= (104 – 2) * 100 + 4 *(-2)

Note that 104 – 2 and 98 + 4 will both yield 102
= 10,200 – 08
= 10,192

Example 19.2:     Find 102 x 93
102 x 93
= (100 + 2)(100 – 7)
= 93 + 2 | 2 * (-7) Easier than 102 – 7
= 95 | -14
= 9500 – 14
= 9,486

Example 19.3:     Find 109 x 97
109 x 92
= 109 – 3 | 9*(-3)
= 106 | -27
= 10,573

Here we just deducted 1 from 106 (109 – 3) and affixed the ten’s complement of 27 = (9 x 3) at the end.

Example 19.4:     Find 114 x 88
114 x 88
= (100 + 14) (100 – 12)
= 114 – 12 | 14(-12)
= 102 | -168
= 100 | 200 – 168
= 10,032

Here the right hand part is -168, so we deducted 2 from the left hand part.

Proof:

Let x = base; a, b = excess / deficiency from the base

(x + a)(x – b)
= x2 + (a – b)x + a(-b)
= (x + a – b)x – ab
= [(x + a) – b]x – ab
or
= [(x – b) + a]x – ab

Exercise 19: Find the following products using the base method

  1. ) 12 x 9 =
  2. ) 103 x 98 =
  3. ) 102 x 97 =
  4. ) 102 x 98 =
  5. ) 103 x 97 =
  6. ) 105 x 93 =
  7. ) 75 x 103 =
  8. ) 112 x 89 =
  9. ) 1012 x 991 =
  10. ) 1125 x 995 =

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!

Conversion: Meters to Feet and Inches

Those who are accustomed to expressing their heights in feet and inches find it hard to estimate the heights of NBA players when expressed in meters.  It can be easily done if we follow these simple steps:

  • express the measure in centimeters.
  • divide the number of centimeters by 30 to get the number of feet.
  • divide the remainder by 2.5 to get the number of inches (remember, 10 ÷ 2.5 = 4)

Note these conversions

  • 1.60 m = 160 cm ÷ 30 =  5 r 10 = 5 feet 4 inches
  • 1.95 m = 195 ÷ 30 =  6 r 15 = 6 feet 6 inches
  • 2.15 m = 215 ÷ 30 =  7 r 5 = 7 feet 2 inches
Again, we used 1 inch = 2.5 cm instead of the exact 1 inch = 2.54 cm. To make it more accurate we could deduct 0.5 cm per foot from the remainder of step B to make the conversion more accurate.
  • 1.60 m = 160 cm ÷ 30 =  5 r 10 -2.5  = 5 feet 3 inches
  • 1.95 m = 195 ÷ 30 =  6 r 15 – 3 = 6 feet 4.8 inches
  • 2.15 m = 215 ÷ 30 =  7 r 5 – 3.5  = 7 feet 0.6 inches

MSC 18: Base Multiplication – Numbers Below the Base

When multiplying numbers below a power of 10, we subtract one number’s deficiency from the base from the other number and then add the product of the deficiencies.

 Example 18.1:    Find 98 x 97
 = (100 – 2)(100 – 3)
= (100 – 2 – 3)(100) + (-2 x -3)
= (98 – 3) x 100 + 2 * 3
= 95 | 06
 = 9,506

Remember that the right hand side has the same number of digits as the number of zeroes in the base.

Example 18.2:    Find 89 x 87

89 x 87
= (100 – 11)(100 – 13)
= 87 – 11 | 13 * 11 
= 76 | 143
= 77 | 43

It is good practice to use the smaller deficiency as the subtrahend.

Example 18.3:    Find 6,879 x 9,998

6,879 x 9,998
 = 6,879 – 2 | 3,121 * 2
 = 6,877 | 6242
 = 68,776,242

It is definitely easier to subtract 2 from 6,879 than to deduct 3,121 from 9,998.

Here is the algebraic proof of the method used:

Let x = base; a, b = deficiency from the base

(x – a)(x – b)
 = x2 – (a + b)x + a*b
 = (x – a – b)x + a*b
 =[(x – a) – b]x + a*b

 Exercise 18:      Find the following products using the base method.

  1. 6 x 9 =
  2. 99 x 98 =
  3. 98 x 93 =
  4. 88 x 98 =
  5. 75 x 97 =
  6. 87 x 88 =
  7. 97 x 67 =
  8. 94 x 91 =
  9. 995 x 975 =
  10. 997 x 778 =

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!

Conversion: Feet to Meters

To convert feet to meters, we could approximate by multiplying the measurement in feet by 0.3 to convert it into meters. This is the exact reverse of the method we used to convert meters to feet.

  • 10 feet = 10 x 0.3 = 3 meters
  • 20 feet = 20 x 0.3 = 6 meters
  • 35 feet = 35 x 0.3 = 10.5 meters
  • 67.5 feet = 67.5 x 0.3 = 20.25 meters
I foot is exactly 0.3048 meters. If we use the conversion factor 1 foot = 0.3 meters, we would be 98.4% accurate as in the previous section.

 

MSC 17 – Base Multiplication: Multiplying “Teen” Numbers and Others

Most of us have learned by heart the multiplication table up to 10 x 10. A simple technique will enable us to extend our multiplication power up to 20 x 20.

Example 17.1:     Compute 14 x 12

  • Cover one of the ten’s digit and add what remains to the other number: 14 + 2 or 4 + 12 will both give 16. This will be the first or left hand part of the product.
  • Cover both ten’s digit and multiply what remains: 2 x 4 = 8. This will be the second or right hand part of the answer.
  • Thus 14 x 12 = 16 | 8 or 168
 What we actually did in the first step is to add the excess over the base (10) of a number to the other.In the second step we multiplied the excesses. The algebraic proof of this method, known as base multiplication, is shown below:
(x + a) * (x + b)
= x2 + (a + b) * x + a * b
= (x + a + b) * x + a * b
= [ (x + a) + b] * x + a * b

In our example above, x = 10, a = 4 and b = 2. So we have
14 x 12
= [  (10 + 4) + 2 ] * 10 + 4 * 2
= 160 + 8
= 168   

In this case both multiplicands are above the base.

Example 17.2:     Compute 16 x 13

(10 + 6) x (10 + 3)

  • Add 3 (excess of 13 over 10) to 16 to get 19
    = ( 10 + 6) + 3 |
    = 19 |
  • Multiply the excesses 6 x 3 = 18.
    = 19 | 6 x 3
    = 19 | 18
  • Since there is only one zero in the base, only one place is allotted for the right hand part. Therefore the 1 of 18 must be carried or added to 19 to get a final answer of 208
    = 20 | 8 = 208

Example 17.3:     Find 107 x 104

  • The left part is 107 + 4 = 111, and the right part is 7 * 4 = 28
    (100 + 7) * (100 + 4)
    = 107 + 4 | 7 x 4
    = 111 | 28

= 11,128Note that in this example we did not write the two zeroes after 111 but two places are reserved for the 28.

Example 17.4:     Find 1,025 x 1,012

  • The left part is (1025 + 12 ) = 1037 and the right part is 25 * 12 = 300
    1,025 x 1,012
    = (1000 + 25) * (1000 + 12)
    = 1025 + 12 | 25 * 12
     = 1037 | 300
     = 1,037,300

Example 17.5:     Find 115 x 111

  • The left part is (115 + 11) and the right part 165
    115 x 111
    = (100 + 15) (100 + 11)
    = 115 + 11 | 15 * 11
    = 126 | 165
  • Since we have only two zeroes in our base, we can allot only two spaces for the right hand part of the answer, so we must “carry” the 1 of 165 into the left side.
    = 12,765

Example 17.6:     Find 102 x 104

  • This is pretty straightforward with the left 102 + 4 = 106 and the right is 2 * 4 = 8
    102 * 104
    = (100 + 2) (100 + 4)
    = 102 + 4 | 2(4)
    = 106 | 08
  • In this case, the product of the excesses is 8 but since the base is 100, two spaces are allotted to it. We write it as 08.
    = 10,608

Example 17.7:     103 x 119

  • The left side is 103 + 19 = 3 + 119 = 122 and the right side is 3 * 19 = 57
    (100 + 3) * (100 + 19)
    = 119 + 3 | 3 * 19
    = 12,257
  • Note that in this example we chose to add the smaller excess to the other number. This always leads to a simpler calculation.

Exercise 17: Find the following products using base multiplication

  1. 12 x 13 =
  2. 14 x 17 =
  3. 15 x 18 =
  4. 108 x 101=
  5. 116 x 102 =
  6. 108 x 112 =
  7. 112 x 114=
  8. 123 x 106 =
  9. 1021 x 1006 =
  10. 1432 x 1002 =

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!

 

 

MSC 16 – Multiplying Complementary Numbers

Complementary numbers are numbers with the same initial digits and the sum of the last digits equal to 10.

The technique in multiplying complementary numbers is similar to that of squaring numbers ending in 5 (By one more than the one before) except that the right hand part is not 25 but the product of the last digits

Example 16.1:    Find 46 x 44

             =  4 x 5 | 6 x 4

             = 20 | 24

              = 2,024

P16: Let t = the common ten’s digit; a, b = the units digit such that a + b = 10

(10t + a)(10t + b)

= (10t)(10t) + 10ta + 10tb + ab

= 100t2 + 10t(a + b) + ab

= 100t2 + 10t(10) + ab

= 100(t2 + t)+ ab

= 100 t(t + 1)+ ab

 

Example 16.2:    Find 39 x 31

= 3 x 4 | 9 x 1

= 12|09 Note that the right part has two digits, 09 not 9.

= 1,209

Again, this can be applied to decimals and fractions as long as they have the same whole number part and their decimal and fractional parts total to 1.

Example 16.3:    8.3 x 8.7 = 8 x 9 | .3 x .7 = 72.21

Example 16.4:    4  2/7 x 4  5/7 = 4 x 5 | 2/7 x  5/7 = 20 10/49

Example 16.5:    699 x 601

Since 99 + 01 = 100, we can also apply complimentary rule

699 x 601 = 6 x 7 | 99 x 01 = 42| 0099 = 420,099

 Exercise 16:

  1. ) 24 x 26 =
  2. ) 73 x77 =
  3. ) 81 x 89 =
  4. ) 998 x 992 =
  5. ) 4.3 x 4.7 =
  6. ) 7.4 x 7.6 =
  7. ) 3 3/7 x 3 4/7 =
  8. ) 4 1/6 x 4 5/6 =
  9. ) 2 4/9 x 2 5/9 =
  10. ) 297 x 203 =

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!

 

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