A smart way of solving problem -1

Q.)  A sum of Rs. 700 has to be used to give seven cash prizes to the students of a school for their overall academic performance. If each prize is Rs. 20 less than its preceding prize, what is the value of the least and highest value of the prize?

(a) Rs. 30

(b) Rs. 40

(c) Rs. 60

(d) Rs. 80

Solution:

Normal Method:

Let us assume the number to be the highest prize to be  X.

 X+(X-20)+(X - 40)+(X - 60)+(X-80)+(X-100)+(X-120)=700

 7X-(20+40+60+80+100+120)=700

 7X=700+420=1120

X=160

Therefore, required values are:

Highest Value of Prize = Rs. 160; Lowest Value of Prize = 160 – 120 = Rs. 40.

Alternative/Smart Method:

We can write as,

(X-60)+(X-40)+(X-20)+X+(X+20)+(X+40)+(X+60)=700

7x=700

X=100

Therefore,

Highest Value of Prize = X+60 = Rs. 160;  Lowest Value of Prize = 100 – 60 = Rs. 40.

Speed Math: Sum of Consecutive Squares

In this blog post, we will be learning the trick to sum up two consecutive squares. 

Let us assume the numbers to be A, B (i.e A+1).

Rule: A^2 + (A+1)^2 = 2A^2 + 2A +1

 

Now let us look at few examples:

 

Example 1: 25^2 + 26^2 = 2 × 25^2 + 2 × 25 + 1 
           = 2 × 625 + 51= 1301.

Example 2: 58^2 +59^2 = 2 × 58^2 + 2 × 58 + 1  
            = 2 × [33|64] + 117 
            = 6728 +117 = 6845.

 

 

Example 3: 94^2 + 95^2 = 2 × 94^2+ 2 × 94 + 1 
           = 2 × [88|36] + 189 
           = 17672 + 189 = 17861.

 

Example 4: 48^2 + 49^2 =  2 × 48^2 + 2 × 48 + 1 
           = 2 × [23|04] + 97 = 4705.

 

Example 5:  52^2 + 53^2 = 2 ×  52^2+ 2 × 52 + 1 
             = 2 × [27|04] + 105 = 5513.

 

Solve the following by yourself:

1. 12^2 + 13^2 =

2. 97^2 + 98^2 =
   
   

Happy Learning! I hope you read the other published post where the numbers are not consecutive numbers.

 

“The essence of Mathematics lies in its freedom” 

        — Georg Cantor

Multiplying Numbers Close to 100

Hey math folks!

We all have come across multiplication of numbers close to 100 in various competitive exams.  In this blog post, we shall learn the method to multiply two numbers which are close to 100 ( be it more than 100 or less than 100).

Let us dive in!

It would be easy to understand the method clearly with an example.

Q.1) 95 x 94

  Solution: 

        Step 1: Find the difference of each number from base number (here, it is 100)

       How far is 95 from 100? It’s just 05.

        What about 94? It’s just 06. ( 06 and 05  are deficiencies from 100)

      Step 2: Now CROSS subtract the numbers ( deficient number and deficiencies)

       i.e  95 - 06 = 89 or  94 - 05 = 89  
           —→ This forms the left most part of our answer

        Step 3: Multiply VERTICALLY the deficiencies of step 1.

   i.e 05 x 06 = 30 
     —→This forms the right most part of our answer.

        Let us represent the above data in a better way.

                          95  --   05
                          94  --   06

---

        (95 - 06) or (94 -05)  |  ( 05 x 06 )

     Therefore, 95 x 94 =   89 | 30  = 8930

          Note: If the number in right-column is 3 digit number, carry forward the hundreds digit to the left column.

Let us take one more example where the numbers are above 100.

   Q.) 108 x  112 = ?

    We have to follow the same steps. 
 The only difference is we replace the operation of 
 subtraction to addition.

                 108 + 8    ( as 108 is 08 more than 100)
                 112 + 12   ( as 112 is 12 more than 100)

---

---

     (108+12) = (112+8) = 120  |  ( 08 x 12 )  
                           120 | 96       
Therefore, 108 x 112 = 120 | 96 = 1296

It’s now your turn.  Solve the following:

 Q.1) 91 x 89
 Q.2) 112 x 115
 Q.3) 88 x 88
 Q.4) 114 x 106
 Q.5) 85 x AB =7735; Find A and B

Summary: 

 We cross-subtract to get the left-hand part of the answer (deficient and deficiencies). we multiply vertically in the right-hand column (i.e deficiencies) to get the right-hand part of the answer

Thanks for reading. Do share it with your friends. 

“Without mathematics, there’s nothing you can do.

   Everything around you is mathematics.

    Everything around you is numbers.” 

                                                                      —  Shakuntala Devi.

Multiplying a number by 15

General rule:

Let us assume the number to be ABC, then thumb rule says,

Step 1: Keep on adding the half of left most digit and write the number on the right of the digit till all numbers are covered.

Note: Ignore fractional values.

For instance, A + (A/2) = X (say)

This forms the first digits of the answer.

Step 2: Next digit (B): B + (B/2) = Y

Step 2.a: If the number to left of B is odd, then 5 to the resultant (Y)

This forms the next digits of our answer.

---

Similarly, by taking next digit (C), we have

(C) + (C/2) = Z

Follow the step 2.a as the case may be.

This forms the last part of our answer.

Note: If the resultant is two digit number from step 2 onwards, add the tens digit of resultant to the immediate left number.

Step 3: If the unit digit is

Even: then final answer ends with 0

Odd: then final answer ends with 5

For better understanding, let us take an example:

Example 1: 112 × 15

1 + (1/2) = 1

1 + (1/2) = 1 + 5 = 6 ( left digit is 1, so added 5)

2 + (2/2) = 3 + 5 = 8 ( left digit is 1, so added 5).

Following step 3, we have the unit digit of given number as 2. The final answer will end with 0.

Therefore, the required final answer is 1680.

Example 2: 211 × 15

2 + (2/2) = 3

1 + (1/2) = 1

1 + (1/2) = 1 + 5 = 6

Therefore, the required final answer is 3165

Lastly, let me tell you: it is nothing but simple algebra.

ABC × 15 = (ABC{0/5}) + (ABC/2)

Image credit: http://www.pixabay.com

Number System – 1

Q.1) How many numbers between -11 and 11 are multiples of   2 or 3?

  (a) 11 

  (b) 4

  (c) 9 

  (d) None of these

 

Q.2) What is the remainder when 4^100 is divided by 7?

  (a) 4 

  (b) 2

  (c) 1 

  (d) None of these

Q.3) What is the remainder when 21^(22^(23)^24) is divided   by 20?

  (a) 4 

  (b) -1

  (c) 1 

  (d) None of these

 

Q.4) (7^12 – 4^12) is exactly divisible by which of the following numbers?

  (a) 34 

  (b) 35

  (c) 36 

  (d) 33

 

Q.5) What number should be added to 231228 to make it  exactly divisible by 33?

  (a) 2               

  (b) 4

  (c) 1 

  (d) 3

Number Series Problems – 1

Q.) Find the next term of the following:

 

1.)  5120   1280   320  80   ?

      a.)  16    b.) 20    c.)  30   d.) 40   e.) 27

2.)  8   17   42  91  ?

      a.)  170   b.)   142   c.)  140    d.) 172   e.) 180

3.)   8   7   12   33   128  ?

     a.)   635    b.)  465   c.) 585   d.) 495  e.) 545

4.)  982    977  952   827    822  ?

   a.)  779   b.)  817  c.) 789  d.) 687   e.) 797

5.)   11  23  47  95  191  ?

   a.)   253   b.) 323  c.) 368  d.) 383   e.)   392

 

 

 

 

 

Quantitative Aptitude Series – 1

Here are few questions which have appeared in various government exams. These questions are on various topics. This will help you understand the importance of each and every topic that is part of the aptitude section of the various competitive exams. If this new initiative helps, we will continue with this one.

Please share your score & feedback about this initiative in the comment section.

Q. 1 Find the ratio of Rs. 2.80 and 40 paise.

a.  1: 14

b. 1: 7

c. 7: 1

d. 2: 7

Q.2  If  (x^2+4y^2) = 4xy, then find  y:x.

a. 1: 2

b. 2: 1

c. 4: 1

d. 1: 4

Q. 3  If 40% of a number is equal to 80. Find the number.

a. 160

b. 40

c. 80

d. 100

Q. 4 Find Simple Interest(S.I) if the principal amount is  Rs. 1500 at the end of 5 years at the rate of 6%.

a. Rs. 250

b. Rs. 600

c. Rs. 450

d. Rs. 500

Q.5 If the length of the simple pendulum is increased by 44%, find by what percentage will the time period increase.

a. 20 %

b. 10%

c. [(44)^(1/2)] %

d. 44%

Q.6 Find the length of the train if it passes a pole in 20 seconds and a platform of 120m in 30 minutes.

a. 220m

b. 240m

c. 200m

d. Insufficient Data

Q.7 The tax on a commodity is diminished by 20% and its consumption increases by 15%. The effect on revenue is

a. Decrease by 12 %

b. Decrease by 9 %

c. Decrease by 8 %

d. None of them

Q.8 If the length and breadth of the rectangular plot are increased by 50% and 20% respectively, the new areas will be how many times than that of an original area?

a. 4

b. 9/6

c. 11/5

d. 9/5

Q.9 The points scored by a basket ball team in a series of matches are as follows
17, 2, 7, 27, 25, 5, 14, 18, 10, 24, 48, 10, 8, 7, 10, 28. The mode of this data is

a. 14

b. 10

c. 7

d. 18

Q. 10 Kim went 20 meters. in the east directions, then turned north and walked 10 meters. And then he turned to the west and covered 25 meters, then he turned south and
covered 10 mtrs. How far Salman is from his original place?

a. 10 meters.

b. 5 meters.

c. 2 meters.

d. 15 meters.

Happy learning!

Sharing is caring – Math Addict (Prep Your Math)

 

MULTIPLICATION (By URDHVA-TIRYAK SUTRA).

The formula itself is very short and terse, consisting of only one compound word and means “vertically & cross-wise”.

Let’s take a simple example:

1. Multiply 12 by 13:

Step-1: We Multiply the left hand most digit 1 of the multiplicand vertically by the left hand most digit 1 of the multiplier, get their product 1 and set it down as the left hand most part of the answer.

Step-2: We then multiply 1 and 3 & 1 and 2 cross-wise, add the two, get 5 as the sum and set it down as the middle part of the answer.

Step-3: We multiply 2 and 3 vertically, get 6 as their product and put it down as the last right hand most part of the answer.

NOTE: Henceforth I’ll be using LHM & RHM suffixes for left hand most and right hand most respectively.

12
13

---

1 : 3+2 : 6 = 156

---

Thus, 12 x 13 = 156.

A few examples:

 21                                        41
x                                         x
 14                                        41

---

2 : 8+1 : 4 = 294                         16 : 4+4 : 1 = 1681

---

 

NOTE: When one of the results contains more than 1 digit, the RHM digit thereof is to be put down there and the preceding, i.e. left-hand side digit or digits should be carried over to the left and placed under the previous digit or digits of the upper row until sufficient practice has been achieved for this operation to be performed mentally. The digits carried over may be shown in the working,

25               37               49
x                x                x
25               33               49

---

405              901              1621
22               32                78

---

625             1221              2401

---

 

The Algebraical principle involved is as follows:

Suppose we have to multiply (ax+b) by (cx+d). The product is acx²+x(ad+bc)+bd. In other words, the first term i.e. the coefficient of x² is got by vertical multiplication of a & c, the middle term, i.e. the coefficient of x is obtained by the cross-wise multiplication of a & d and of b & c and the addition of the two products; and the independent term is arrived at by vertical multiplication of the absolute terms.

Consider,

ax² + bx + c
dx² + ex + f

---

adxˆ4 + x³(ae+bd) + x²(af+be+cd) + x(bf+ce) + cf

---

 

We observe the following facts,

1. that the coefficient of xˆ4 is got by the vertical multiplication of the first digit from the left side.
2. that the coefficient of x³ is got by the cross-wire multiplication of the first two digits and by the addition of the two products.
3. that the coefficient of x² is got by the multiplication of the first digit of the multiplicand by the last digit of the multiplier, of the middle one by the middle one and of the last one by the first one and by the addition of all the three products.
4. that the coefficient of x is got by the cross-wire multiplication of the second digit by the third one and conversely, and by the addition of the two products.
5. that the independent term results from the vertical multiplication of the last digit by the last digit.

We thus follow a process of ascent and of descent going forward with the digits on the upper row and coming rearward with the digits on the lower row. If and when this principle of ordinary algebraic multiplication is properly understood and carefully applied to the arithmetical multiplication on hand, the URDHVA TIRYAK SUTRA may be deemed to have been successfully mastered in actual practice.

A few examples will serve to explain this URDHVA TIRYAK SUTRA process of vertical and cross-wise multiplications,

108            116             582              321
x              x               x                x
108            114             231              52

---

10 60 4        12 1 0 4        12 32 6          0 5 692
 1  6           1 1 2           1 13            1 1

---

11 66 4        13 2 2 4        13 45 6          1 6 692

---

 

 

6471                            87265
x                               x
6212                            32117

---

36 6 6 6 752                    2 4 7 8 7 2 7 5 7 5
 3 5 3 1 1                        3 2 3 9 6 2 4 3

---

40 1 9 7 852                    2 8 0 2 6 9 0 0 0 5

---

 

Will be posting many more useful methods soon. THANK YOU.

Daily Questions – 27/10/2017

Hello guys!!

Hope you guys are doing well.

How is your preparation going on for various competitive exams?

Do you have some initiatives that would help others to learn?If yes please mail to us. So that you can be part of our team.

Your work will be appreciated 🙂

 

Due to lack of time. I was/ will not be able to update the blog post regularly. Instead of daily questions. I will be writing down the most important 50 questions and posting it every Sunday. You guys can send the solutions through some means. It would benefit others too.

Daily Questions – 27/10/2017
Q.1) If the difference between compound interest & simple interest for 2 years at the rate of 5% p. a is Rs.5. Find the sum.
Q.2) If the ratio of circumferences of two circles is 2:3. What is the ratio of their areas?
Q.3) A single 6 sided die is rolled. Find the probability of getting a 2 or a 5.
Q.4) At what time between 9 o’clock & 10 o’clock, will the hands of the clock be in the same straight line but not together?
Q.5) What was the day of the week on 27th Oct 2017?
Q.6) There are 5 numbers. HCF of each possible pair is 4. & LCM of all the 5 numbers is 27720. What will be the product of all the 5 numbers?
Q.7) Find the LCM & HCF of 0.6, 9.6 and 0.12.
Q.8) If a+b+c =13, then find the maximum value of (x-2)(y+1)(z-3).
Q.9) Five years ago, the average of P and Q is 15 years. Now, average age of P, Q & R is 20 years. What would be the age of R after 10 years?
Q.10) The marked price of a painting (Sketch) is Rs. 500. If artist thinks to give two successive discounts of 20% and 10%. Help the artist to calculate the selling price of painting (As he is weak in Mathematics).

 
© Prep Your Math

 

Happy Learning 🙂

 

 

 

 

 

 

Daily​ ​Questions​ ​-​ ​19/10/2017

Q.1) If the minute hand of a clock has moved 300°, how many degrees has the hour
hand moved?
                       a) 25°                       b) 27°                         c) 30°             d) 36°

 

Q.2) Find the odd one out 212, 331, 824, 632, 422, 121
                      a) 212                       b) 824                          c) 422                d) 121

 
Q.3) The least number which when divided by 5, 6, 7 and 8 leaves a remainder 3, but
when divided by 9 leaves no remainder, is:

                       a) 1677                     b) 1683                         c) 2523             d) 3363

 
Q.4) The ratio between a two-digit number and the sum of the digits of that number is 4: 1.If the digit in the unit’s place is 3 more than the digit in the ten’s place, what is the number?
a) 30                        b) 35                           c) 40                    d) 36

 
Q.5) Ravi’s salary was reduced by 25%.Percentage increase to be effected to bring the salary to the original level is
a) 20%                    b) 25%                       c) 33.33%            d) 30%

 
Q.6) Suresh’s salary is 60% more than Ramesh’s salary. Suresh got a 50% increment in his salary while Ramesh got 20% increment. By what percent is Suresh’s salary more than Ramesh?
a) 90%                    b) 96%                       c) 92%                 d) 100%

 
Q.7) The odds against an event are 11:9. Find the probability that the event will occur.
a) 9/11
b) 9/20
c) 11/20
d) NoT

 
Q.8) If the ratio of present ages of Rekha and Ram is 5:7 and after 6 years the ratio will be 3:4, what is the present age of Ram?
                      a) 42                     b) 30                        c) 31                            d) 25

 
Q.9) A alone can finish a work in 10 days and B alone can do it in 15 days. If they work together and finish, then out of total wages of Rs.75. A will get 
                      a) Rs. 35              b) Rs. 40                  c) Rs. 45                      d) None of these

 
Q.10) In a 100m race, A beats B by 10 m and C by 20 m. In a 180m race, by how many
meters will B beat C?

                     a) 10 m              b) 18 m                     c) 20 m                           d) 25 m