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Thursday 30 June 2011

Maths - Perimeter and Area

  • Perimeter of a closed plane figure is the length of its boundary.
  • Area of a closed plane figure is the measure of the region (surface) enclosed by its boundary.
  • Area of a triangle: Area of a triangle = (1/2) × base × height
    Also, Area of a triangle = [s (s -a)(s -b)(s-c)] where a, b, c are the lengths of the sides and s = (a +b +c)/2
    Area of an equilateral triangle = 3a²/4, where a is the side.
  • Rectangle: If l = length and b = breadth of a rectangle, then perimeter = 2 (l +b)
    length of diagonal = l²+b², area = l×b.
  • Square: If a is the length of side of a square, then perimeter = 4a
    length of diagonal = 2 a, area = a²
  • Circle: If r is the radius of a circle, then length of a diameter = 2r
    circumference = 2r, area =
    Take = 22/7 (unless given otherwise)
  • Circular ring (track): If R and r are the radii of two concentric circles then
    area of the circular ring (track) = (R²-r²).
                                         Exercise

1. The length of a room is 5.5 m and width is 3.75 m . Find the cost of paving the floor by slabs at the rate of Rs 800 per sq m(in Rs).
          a. 15000 b. 15550 c. 15600 d. 16500 e. None
2. The length of a ractangle plot is 60% more than its breadth. If the diffrence between the length and breadth of that reactangle is 24 cm, what is the area of that ractangle?
          a. 2400 sq cm b. 2480 sq cm c. 2560 sqcm d. Data inadequate e. None
3. The length of a ractangle plot is 20 m more than its breadth. If the cost of fencing the plot @ Rs 26.50 per meter is Rs. 5300, what is the length of the plot in meters.
         a. 40 b. 50 c. 120 d. Data inadequate e. None
4. The ratio between the length and the perimeter of a ractangle is 1 : 3. What is ratio between the length & breadth of the plot?
          a. 1:2 b. 2:1 c. 3:2 d. Data inadequate e. None
5. A ractangular field is to be fenced on three sides leaving a side of 20 ft uncovered. If the area of the field is 680 sq feet, how many feet of fencing will be required?
          a. 34 b. 40 c. 68 d. 88 e. None
6. The ratio between the perimeter and breadth of a ractangle is 5:1. If the area of the ractangle is 216 sq cm. The length is:
          a. 16 cm b. 18 cm c. 24 cm d. Data inadequate e. None
7. A towel when bleached, was found to have lost 20% of its length and 10% of width. The % decrease in area is:
          a. 10 b.15 c. 20 d. 28 e. None
8. What will be the length of the diagonal of that square plot whose area is equal to the area of a ractangular plot of length 45 mt and breadth 40 mt?
          a. 42.5 b. 60 c. 75 d. Data inadequate e. None
9. If the side of a square is increased by 5 cm, the area increases by 165 sq cm. The side of the square is:
        a. 12 b. 13 c. 14 d. 15 e. None
10. The base of a tringle is 15 cm height is 12 cm. The height of the another tringle of the double the area having base 20cm is:
        a. 8 b. 9 c. 12.5 d. 18 e. None
11. The area of a right angle tringle is 40 times its base. What is its height?
        a. 45 b. 60 c. 80 d. Data inadequate e. None
12. The sides of a tringle are in ratio ½:1/3:1/4. if the perimeter is 52cm. Than the length of the smallest side is:
        a. 9 b. 10 c. 11 d. 12 e. None
13. The area of a circle of radius 5 is numerically what % of its circumference?
        a. 200 b. 225 c. 240 d. 250 e. None
14. A circle and a ractangle have the same perimeter. The sides of the ractangle are 18 & 26 cm. The area of circle is:
        a. 88 b. 154 c. 1250 d. Cant find e. None
15. A circular wire of radius of 42 cmis bent in the form of a ractangle whose sides are in ratio 6:5. Smaller side of ractangle is:
        a. 25 b. 30 c. 36 d. 60 e. None
  Ans: 1d, 2c, 3e, 4b, 5d, 6b, 7d, 8b, 9c, 10d, 11c, 12d, 13d, 14e, 15d

Maths - Compound Interest

  • Amount = Principal + Interest
  • Simple Interest = (Principal x Rate x Time)/100
  • A = P Here A = amount, P = principal, r = rate percent yearly (or every fixed period) and n is the number of years (or terms of the fixed period).
  • C.I. = P , where C.I. = compound interest
  • S.I. (simple interest) and C.I. (compound interest) are equal for the first year (or the first term of the fixed period) on the same sum and at the same rate.
  • C.I. of 2nd year (or the second term of the fixed period) is more than the C.I. of 1st year or the first term of the fixed period), and C.I. of 2nd Year -C.I. of 1st year = S.I. on the interest of the first year.
                                                    Exercise

1. Albert invested an amount of Rs. 8000 in afixed deposit scheme for 2 years at compound interest rate 5% pa. How much amount will Albert get on maturity of the fixed deposit?
      a. 8600 b. 8620 c. 8800 d. 8840 e. None
2. What will be the compound interest on a sum of Rs. 25000 after 3 years at the rate of 12 % pa?
     a. 9000.30 b. 9720 c. 10123.20 d. 10483.20 e. None
3. Sam invested Rs. 15000 at the rate of 10% pa for one year. If the interest is compouned half yearly, than the amonut received by Sam at the end of one year will be:
     a. 16500 b. 16525.50 c. 16537.50 d. 18150 e. None
4. Find the compoune interest on Rs. 15625 for 9 months at 16% per annum compounded quarterly.
     a. 1851 b. 1941 c. 1951 d. 1961
5. If the simple interest on asum of money for 2 years at 5% pa is Rs. 50, what is the Compound interest on the same sum at the same rate and for the same period?
     a. 51.25 b. 52 c. 54.25 d. 60
6. The difference between simple interest and compound interest on Rs 1200 for one year at the rate of 10% per annum reckoned half yearly is Rs. Is:
     a. 2.50 b. 3 c. 3.75 d. 4 e. None
7. The present worth of Rs. 169 due in 2 years at 4% pa componud interest is:
     a. 150.50 b. 154.75 c. 156.25 d. 158
8. The compound interest on a certain sum for 2 years at 10% pa is Rs. 525. The simple interest on the same sum for double the time at half the rate percent per annum is:
    a. 400 b. 500 c. 600 d.800
9. The difference between compound interest and simple interest on an amount of Rs. 15000 for 2 years is Rs. 96. What is the rate of interest per annum?
    a. 8 b. 10 c. 12 d. Can't find e. None
10. The effective annual rate of interest corrosponding to a nominal rate of 6% pa payable half yealy is;
    a. 6.06 % b. 6.07% c. 6.08% d. 6.09%
Ans: 1e, 2c, 3c, 4c, 5a, 6b, 7c, 8b, 9a, 10d, 11c

Maths - Ratio and Proportion

  • Compounded Ratio of two ratios a/b and c/d is ac/bd, i.e., ac : bd.
    Duplicate ratio
    of a : b is a² : b²,         Triplicate ratio of a : b is a³ : b³
    Reciprocal ratio
    of a : b is b : a
  • Proportion. If four numbers a, b, c, d are in proportion, written as a : b :: c : d, ie a/b = c/d
  • Invertendo. If a : b :: c : d then b : a :: d : c,   Alternendo. If a : b :: c : d then a : c :: b : d
    Componendo.
    If a : b :: c : d then (a +b) : b :: (c +d) : d
    Dividendo.
    If a : b :: c : d then (a -b) : b :: (c -d) : d
    Componendo and dividendo:
    If a : b :: c : d then (a +b) : (a -b) :: (c +d) : (c -d)
        i.e., a/b = c/d => (a +b)/(a - b) = (c +d)/(c +d)
  • If a/b = c/d = e/f = ..., then each ratio = (a +c +e +...)/(b +d +f +...)

1. If A : B = 8 : 15, B : C = 5 : 8 and C : D = 4 : 5 than A : D is .
          a. 2 : 7 b. 4 : 15 C. 8 : 15 d. 15 : 4 e. None
2. If 2A = 3B and 4B = 5C than A : C is:
          a. 4 : 3 b. 8 : 15 c. 15 : 8 d. 3 : 4 e. None
  3. Two numbers are in the ratio of 1 : 2. If 7 is added to both the ratio is to 3 : 5. The greatest no:
          a. 24 b. 26 c. 28 d. 32 e. None
4. In a bag, there are coins of 25 paisa, 10 paisa & 5 paisa in 1 : 2 : 3 ratio. Their sum is Rs. 30 how many 5 paisa coins are there?
          a. 50 b. 100 c. 150 d. 200 e. None
5. Salaries of Ravi & Sumit are in 2 : 3 ratio. If salary of each is increased by Rs 4000, new ratio becoms 40 : 57. Sumits present salary?
          a. 17000 b. 20000 c. 25500 d. 30000 e. None
6. Two no are respectively 20% & 50% more than a third numbers. The ratio of two no are:
          a. 2 : 5 b. 3 : 5 c. 4 : 5 d. 6 : 7 e. None
7. In a mixture of 60 litres, water and milk is in 2 : 1 ratio. If this ratio is to be 1 : 2, than the quantity of water to be added is:
         a. 20 litres b. 30 litres c. 40 litres d. 60 litres e. None
8. The fourth proportional to 5, 8, 15 is:
         a. 18 b. 24 c. 19 d. 20 e. 21 
9. The ages of A & B are in 3 : 1. Fifteen years hence the ratio will be 2 : 1. Their present age is:
a. 30 & 10 b. 45 & 15 c. 21 & 7 d 60 & 20 e. None
10. If 10% of x = 20% of y, then x : y is:
       a. 1 : 2 b. 2 : 1 c. 5 : 1 d. 10 : 1 e. None
11. The compount ratio of (2 : 3), (6 : 11) & (11 : 2) is:
        a. 1 : 2 b. 2 : 1 c. 11 : 24 d. 36 : 121 e. None
Ans: 1b, 2c, 3c, 4c, 5d, 6c, 7d, 8b, 9b, 10b, 11b.

Wednesday 29 June 2011

Maths - LCM and HCF

Prime factorisation:If a natural number is expressed as the product of prime numbers, then the factorisation of the number is called its prime factorisation.

For example: (i) 24 = 2×2×2×3 = 23×3, (ii) 60 = 2×2×3×5 = 2² ×3×5.
Least Common Multiple (L.C.M.) of two natural numbers is the smallest natural number which is a multiple of both the numbers.
Highest Common Factor (H.C.F.) of two natural numbers is the largest common factor (or divisor) of the given natural numbers. In other words, H.C.F. is the greatest factor of the given numbers. H.C.F. is also called Greatest Common Divisor (G.C.D.)
Relation between L.C.M. and H.C.F. of two natural number: No 1 x No 2 = Their L.C.M. x H.C.F
Example: Find the L.C.M. of 72, 240, 196.
Using Prime factorisation method
   72 = 2×2×2×3×3 = 2³×3²,      240 = 2×2×2×2×3×5 = 24×3×5
   196 = 2×2×7×7 = 2²×7²
L.C.M. of the given numbers = product of all the prime factors of each of the given number with greatest index of common prime factors
     = 24×3²×5×7² = 16×9×5×49 = 35280.
Using Division method
  2 | 72, 240, 196
  2 | 36, 120, 98
   2 | 18, 60 , 49
  3 | 9 , 30 , 49
      | 3  , 10 , 49
 L.C.M. of the given numbers
    = product of divisors and the remaining numbers
    = 2×2×2×3×3×10×49
    = 72×10×49 = 35280.

Example: Find the H.C.F. of 72, 126 and 270.

Using Prime factorisation method
    72 = 2×2×2×3×3 = 2³×3²,      126 = 2×3×3×7 = 21×3²×71
  270 = 2×3×3×3×5 = 21×3³×51
H.C.F. of the given numbers = the product of common factors with least index
  = 21×3² = 2×3×3 = 18
Using Division method
First find H.C.F. of 72 and 126
72|126|1
      72       
       54| 72|1
              54
              18| 54| 3
                     54
                     
 H.C.F. of 72 and 126 = 18
Similarly calculate H.C.F. of 18 and 270 as 18
Hence H.C.F. of the given three numbers = 18

Exercise


1. 1095/1168 when expressed in simplest form is :
      a. 13/16 b. 15/16 C. 17/26 d. 25/26 e. None
2. HCF of 4 x 27 x 3125, 8 x 9 25 x 7 and 16 x 81 x 5 x 11 x 49 :
      a. 180 b. 360 c. 540 d. 1260 e. None
3. Which of the following is co-prime:
      a. 16,62 b. 18,25 c. 21,35 d. 23,92 e. None
4. Find the lowest common multiple of 24, 36 and 40 :
       a. 120 b. 240 c. 360 d. 480 e. None
5. Three numbers are in the ratio of 1:2:3 and their HCF is 12. numbers are:
       a. 4, 8, 12 b. 5, 10, 15 c. 10, 20, 30 d. 12, 24, 36 e. None
6. The LCM of two no is 48, and are in ratio of 2:3. The sum of no is:
      a. 28 b. 32 c. 40 d. 64 e. None
7. LCM of two no is 495 and HCF is 5. If their sum is 100, then diffrence is :
      a. 10 b. 46 c. 70 d. 90 e. None
8. The HCF of two numbers is 8, which one of following can never be their LCM:
      a. 24 b. 48 c. 56 d. 60 e. None
9. The greatest number that exactly divides 105, 1001 and 2436 is:
      a. 3 b. 7 c. 11 d. 21 e. None
10. The greatest possible length which can be used to measure exactly 7m, 3m 85cm and 12m 95cm:
      a. 15 cm b. 25 cm c. 35 cm d. 42 cm e. None
11. Find the greatest number that will divide 43, 91 & 183 so as to leave same remainder:
      a. 4 b. 7 c. 9 d. 13 e. None
12. The greatest number of four digit which is divisible by 15, 25, 40 and 75:
      a. 9000 b. 9400 c. 9600 d 9800 e. None
13. The least number which should be added to 2497 so that the sum is exactly divisible by 5, 6, 4 and 3:
      a. 3 b. 13 c. 23 d. 33 e. None
14. The smallest number whcih when diminished by 7, is divisible by 12, 16, 18, 21 and 28:
      a. 1008 b. 1015 c. 1022 d. 1032 e. None
15. The least number, which when divided by 12, 15, 20 and 54 leaves in each case a reminder of 8 is:
      a. 504 b. 536 c. 544 d. 548 e. None
16. The HCF of 0.54, 1.8 and 7.2 is:
      a. 1.8 b. 0.18 c. 0.018 d. 18 e. None 
Answer: 1b, 2a, 3b, 4c, 5d, 6c, 7a, 8d, 9b, 10c, 11a, 12c, 13c, 14b, 15d, 16b

Monday 27 June 2011

Maths - Operations on Numbers

Face Value: The face value of a digit in a numeral is its own value, at  whatever place it may be. eg. in 6872 the face value of 7 is 7.
Place Value: The Place value of a digit in a numeral is its value as per position, at  whatever place it may be. eg. in 6872 the Place value of 7 is 70.
Natural Numbers:Counting no. are called natural numbers. eg. 1, 2, 3, 4, .......
Whole Numbers: All counting no. including zero are called Whole numbers. eg. 0, 1, 2, 3, 4, .......
Integers: All counting numbers, zero and negative numbers are called integers. eg. ..., -3, -2, -1, 0, 1, 2, 3, .....
Even Numbers: All numbers divisible by 2 are called even no. eg. 2, 4, 6, .....
Odd Numbers: All numbers not divisible by 2 are called odd no. eg. 1, 3, 5, .....
Prime Number: A no which is divisible by 1 or itself is called Prime number. eg. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53....
Co-Prime: Two natural no are said to be co-prime if their HCF is 1. eg. (2, 3), (7, 9), (23,24), etc
Divisibility:
Divisibility by 2: A no is divisibile by 2 if its unit digit is 0, 2, 4, 6, 8. eg. 58888
Divisibility by 3: A no is divisible by 3 if the sum of all its numeral is divisible by 3. eg. 111111 total is 6 which is multiple of 3.
Divisibility by 4: A no is divisible by 4 if its last two numerals are divisible by 4. eg. 111116, here 16 is divisible by4.
Divisibility by 5: A no is divisible by 5 if its unit digit is 0 or 5. eg. 22225 or 11110.
Divisibility by 6: A no is divisible by 6 if it is divisible by 2 and 3.
Divisibility by 8: A no is divisible by 8 if its last three no are divisible by 8. 123123352 .
Divisiblity by 9: A no is divisible by 9 if the sum of its numerals is divisible by 9. eg 111111111 here total of numerals is 9 so it is divisible by 9.
Divisibility by 10: A no is divisible by 10 if its unit digit is zero. eg 1230.
Divisibility by 11: A no is divisible by 11 if the difference between the sum of its digits at odd places and the sum of digits at even places is either 0 or a no divisible by 11. eg. 29435417. now (7+4+3+9)-(1+5+4+2)=11 which is divisible by 11.
                                                  EXERCISE
1.  The diffrence between the place value and the face value of 6 in the numeral 856973 is : 
     a. 973 b. 6973 c. 5994 d. None 
2.   The unit digit in (784 x 618 x 917 x 463 ) is: 
      a. 2 b. 3 c. 4 d. 5,   
3.    If the no 481*673 is fully divisible by 9, than smallest whole no in place of * is: 
.      a. 2 b. 5 c. 6 d. 7 e. None 
4.    If the no 97215*6 is fully divisible by 11, than smallest whole no in place of * is: 
       a. 3 b. 2 c. 1 d. 5. e. None 
5      Which one of the following no is completely divisible by 45:  
        a. 181560 b. 331145 c. 202860 d. 1033550 e. None. 
6.     The smallest prime no is:  
       a. 0 b. 1 c. 2 d. 3 e. None 
7.     Which one of the following cannot be square of a natural no.: 
      a. 32761 b. 81225 c. 42437 d. 20164 e. None 
 8.     Which one of the following is divisible by 11 : 
      a. 235641 b. 245642 c. 315624 d. 415624 e. None
ANSWER: 1  C, 2 A, 3 D, 4 A, 5 C, 6 C, 7 C, 8 D