Aptitude: Problems on H.C.F and L.C.M

Aptitude: Problems on H.C.F and L.C.M

problem on H.C.F and L.C.M

   1. Factors and Multiples: –If number a divided another number b exactly, we say that a is a factor of b. In this case, b is called a multiple of a.

2. Highest Common Factor (H.C.F.) or Greatest Common Measure (G.C.M.) or Greatest Common Divisor (G.C.D.): The H.C.F. of two or more than two numbers is the greatest number that divides each of them exactly. There are two methods of finding the H.C.F. of a given set of numbers:

I. Factorization Method: Express the each one of the given numbers as the product of prime factors. The product of least powers of common prime factors gives H.C.F.

II. Division Method: Suppose we have to find the H.C.F. of two given numbers, divide the larger by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor is required H.C.F.

Finding the H.C.F. of more than two numbers: Suppose we have to find the H.C.F. of three numbers, then, H.C.F. of [(H.C.F. of any two) and (the third number)] gives the H.C.F. of three given number. Similarly, the H.C.F. of more than three numbers may be obtained.

3. Least Common Multiple (L.C.M.): The least number which is exactly divisible by each one of the given numbers is called their L.C.M. There are two methods of finding the L.C.M. of a given set of numbers:

I. Factorization Method: Resolve each one of the given numbers into a product of prime factors. Then, L.C.M. is the product of highest powers of all the factors.

II. Division Method (short-cut): Arrange the given numbers in a rwo in any order. Divide by a number which divided exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1. The product of the divisors and the undivided numbers is the required L.C.M. of the given numbers.

4. Product of two numbers = Product of their H.C.F. and L.C.M.

5. Co-primes: Two numbers are said to be co-primes if their H.C.F. is 1.

6. H.C.F. and L.C.M. of Fractions:
1. H.C.F. =H.C.F. of Numerators/L.C.M. of Denominators
2. L.C.M. =L.C.M. of Numerators/H.C.F. of Denominators

8. H.C.F. and L.C.M. of Decimal Fractions: In a given numbers, make the same number of decimal places by annexing zeros in some numbers, if necessary. Considering these numbers without decimal point, find H.C.F. or L.C.M. as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers.

9. Comparison of Fractions: Find the L.C.M. of the denominators of the given fractions. Convert each of the fractions into an equivalent fraction with L.C.M as the denominator, by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest.

QUESTIONS

1.Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.
A.4
B.7
C.9
D.13
2.The H.C.F. of two numbers is 23 and the other two factors of their L.C.M. are 13 and 14. The larger of the two numbers is:
A.276
B.299
C.322
D.345
3.Six bells commence tolling together and toll at intervals of 2, 4, 6, 8 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together ?
A.4
B.10
C.15
D.16
4.Let N be the greatest number that will divide 1305, 4665 and 6905, leaving the same remainder in each case. Then sum of the digits in N is:
A.4
B.5
C.6
D.8
5.The greatest number of four digits which is divisible by 15, 25, 40 and 75 is:
A.9000
B.9400
C.9600
D.9800
6.The product of two numbers is 4107. If the H.C.F. of these numbers is 37, then the greater number is:
A.101
B.107
C.111
D.185
7.Three number are in the ratio of 3 : 4 : 5 and their L.C.M. is 2400. Their H.C.F. is:
A.40
B.80
C.120
D.200
8.The G.C.D. of 1.08, 0.36 and 0.9 is:
A.0.03
B.0.9
C.0.18
D.0.108
9.The product of two numbers is 2028 and their H.C.F. is 13. The number of such pairs is:
A.1
B.2
C.3
D.4
10.The least multiple of 7, which leaves a remainder of 4, when divided by 6, 9, 15 and 18 is:
A.74
B.94
C.184
D.364

Aptitude: Problems on Numbers

Aptitude: Problems on Numbers

FORMULA

  1. a + b)(a – b) = (a2 – b2)
  2.  (a + b)2 = (a2 + b2 + 2ab)
  3.  (a – b)2 = (a2 + b2 – 2ab)
  4.  (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)
  5. (a3 + b3) = (a + b)(a2 – ab + b2)
  6. (a3 – b3) = (a – b)(a2 + ab + b2)
  7. (a3 + b3 + c3 – 3abc) = (a + b + c)(a2 + b2 + c2 – ab – bc – ac)
  8. When a + b + c = 0, then a3 + b3 + c3 = 3abc

                                                                      QUESTION

1.If one-third of one-fourth of a number is 15, then three-tenth of that number is:
A.35
B.36
C.45
D.54
2.Three times the first of three consecutive odd integers is 3 more than twice the third. The third integer is:
A.9
B.11
C.13
D.15
3.The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of that number?
A.3
B.4
C.9

D.Cannot be determined
E.None of these
4.The difference between a two-digit number and the number obtained by interchanging the digits is 36. What is the difference between the sum and the difference of the digits of the number if the ratio between the digits of the number is 1 : 2 ?
A.4
B.8
C.16
D.None of these
5.A two-digit number is such that the product of the digits is 8. When 18 is added to the number, then the digits are reversed. The number is:
A.18
B.24
C.42
D.81
6.The sum of the digits of a two-digit number is 15 and the difference between the digits is 3. What is the two-digit number?
A.69
B.78
C.96
D.Cannot be determined

E.None of these
7.The sum of the squares of three numbers is 138, while the sum of their products taken two at a time is 131. Their sum is:
A.20
B.30
C.40
D.None of these
8.A number consists of two digits. If the digits interchange places and the new number is added to the original number, then the resulting number will be divisible by:
A.3
B.5
C.9
D.11
9.In a two-digit, if it is known that its unit’s digit exceeds its ten’s digit by 2 and that the product of the given number and the sum of its digits is equal to 144, then the number is:
A.24
B.26
C.42
D.46
10. Find a positive number which when increased by 17 is equal to 60 times the reciprocal of the number.

A. 3

B. 10

C. 17

D. 20

Aptitude: Numbers

Aptitude: Numbers

 

Some Basic Formula

i. (a + b)(a – b) = (a2 – b2)

ii. (a + b)2 = (a2 + b2 + 2ab)

iii. (a – b)2 = (a2 + b2 – 2ab)

iv. (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

v. (a3 + b3) = (a + b)(a2 – ab + b2)

vi. (a3 – b3) = (a – b)(a2 + ab + b2)
:-
vii. (a3 + b3 + c3 – 3abc) = (a + b + c)(a2 + b2 + c2 – ab – bc – ac)

viii. When a + b + c = 0, then a3 + b3 + c3 = 3abc

QUESTION

1.Which one of the following is not a prime number?
A.31
B.61
C.71
D.91
2.(112 x 54) = ?
A.67000
B.70000
C.76500
D.77200
3.It is being given that (232 + 1) is completely divisible by a whole number. Which of the following numbers is completely divisible by this number?
A.(216 + 1)
B.(216 – 1)
C.(7 x 223)
D.(296 + 1)
4.What least number must be added to 1056, so that the sum is completely divisible by 23 ?
A.2
B.3
C.18
D.21
E.None of these
5.1397 x 1397 = ?
A.1951609
B.1981709
C.18362619
D.2031719
E.None of these
6.How many of the following numbers are divisible by 132 ? 264, 396, 462, 792, 968, 2178, 5184, 6336
A.4
B.5
C.6
D.7
7.(935421 x 625) = ?
A.575648125
B.584638125
C.584649125
D.85628125
8.The largest 4 digit number exactly divisible by 88 is:
A.9944
B.9768
C 9988
D.8888
E.None of these
9.Which of the following is a prime number ?
A.33
B.81
C.93
D.97
10.What is the unit digit in {(6374)1793 x (625)317 x (341491)}?
A.0
B.2
C.3
D.5

Aptitude :: Compound Interest

Aptitude :: Compound Interest

  1. Let Principal = P, Rate = R% per annum, Time = n
  2. When interest is compound Annually:
   Amount = P 1 + R n
100
  1. When interest is compounded Half-yearly:
    Amount = P 1 + (R/2) 2n
100
  1. When interest is compounded Quarterly:
    Amount = P 1 + (R/4) 4n
100
  1. When interest is compounded Annually but time is in fraction, say 3
    Amount = P 1 + R 3 x 1 + R
100 100
  1. When Rates are different for different years, say R1%, R2%, R3% for 1st, 2ndand 3rd year respectively.
    Then, Amount = P 1 + R1 1 + R2 1 + R3 .
100 100 100
  1. Present worth of Rs. xdue n years hence is given by:
    Present Worth = x .
1 + R
100
    Question:-
  • A bank offers 5% compound interest calculated on half-yearly basis. A customer deposits Rs. 1600 each on 1st January and 1st July of a year. At the end of the year, the amount he would have gained by way of interest is:
  • Rs. 120
  • Rs. 121
  • Rs. 122
  • Rs. 123
  • The difference between simple and compound interests compounded annually on a certain sum of money for 2 years at 4% per annum is Re. 1. The sum (in Rs.) is:
  • 625
  • 630
  • 640
  • 650
  • There is 60% increase in an amount in 6 years at simple interest. What will be the compound interest of Rs. 12,000 after 3 years at the same rate?
  • Rs. 2160
  • Rs. 3120
  • Rs. 3972
  • Rs. 6240
  • None of these
  • What is the difference between the compound interests on Rs. 5000 for 1 years at 4% per annum compounded yearly and half-yearly?
  • Rs. 2.04
  • Rs. 3.06
  • Rs. 4.80
  • Rs. 8.30
  • The compound interest on Rs. 30,000 at 7% per annum is Rs. 4347. The period (in years) is:
  • 2
  • 212
  • 3
  • 4
  • What will be the compound interest on a sum of Rs. 25,000 after 3 years at the rate of 12 p.c.p.a.?
  • Rs. 9000.30
  • Rs. 9720
  • Rs. 10123.20
  • Rs. 10483.20
  • None of these
  • At what rate of compound interest per annum will a sum of Rs. 1200 become Rs. 1348.32 in 2 years?
  • 6%
  • 6.5%
  • 7%
  • 7.5%
  • The least number of complete years in which a sum of money put out at 20% compound interest will be more than doubled is:
  • 3
  • 4
  • 5
  • 6
  • Albert invested an amount of Rs. 8000 in a fixed deposit scheme for 2 years at compound interest rate 5 p.c.p.a. How much amount will Albert get on maturity of the fixed deposit?
  • Rs. 8600
  • Rs. 8620
  • Rs. 8820
  • None of these
  • The effective annual rate of interest corresponding to a nominal rate of 6% per annum payable half-yearly is:
  • 6.06%
  • 6.07%
  • 6.08%
  • 6.09%
APTITUDE-TIME AND WORKS

APTITUDE-TIME AND WORKS

The money  borrowed or lent out for a certain period is called the principal or the sum.

  1. Interest:

Extra money paid for using other’s money is called interest.

  1. Simple Interest (S.I.):

If the interest on a sum borrowed for certain period is reckoned uniformly, then it is called simple interest.

Let Principal = P, Rate = R% per annum (p.a.) and Time = T years. Then

(i). Simple Intereest = P x R x T
100
(ii). P = 100 x S.I. ; R = 100 x S.I. and T = 100 x S.I. .
R x T P x T P x R

QUESTION

1.A sum of money at simple interest amounts to Rs. 815 in 3 years and to Rs. 854 in 4 years. The sum is:

  1. Rs. 650
  2. Rs. 690
  3. Rs. 698
  4. Rs. 700

2.Mr. Thomas invested an amount of Rs. 13,900 divided in two different schemes A and B at the simple interest rate of 14% p.a. and 11% p.a. respectively. If the total amount of simple interest earned in 2 years be Rs. 3508, what was the amount invested in Scheme B?

  1. Rs. 6400
  2. Rs. 6500
  3. Rs. 7200
  4. Rs. 7500
  5. None of these

3.A sum fetched a total simple interest of Rs. 4016.25 at the rate of 9 p.c.p.a. in 5 years. What is the sum?

  1. Rs. 4462.50
  2. Rs. 8032.50
  3. Rs. 8900
  4. Rs. 8925
  5. None of these

4.How much time will it take for an amount of Rs. 450 to yield Rs. 81 as interest at 4.5% per annum of simple interest?

  1. 3.5 years
  2. 4 years
  3. 4.5 years
  4. 5 years

 

5.Reena took a loan of Rs. 1200 with simple interest for as many years as the rate of interest. If she paid Rs. 432 as interest at the end of the loan period, what was the rate of interest?

  1. 3.6
  2. 6
  3. 18
  4. Cannot be determined
  5. None of these

6.A sum of Rs. 12,500 amounts to Rs. 15,500 in 4 years at the rate of simple interest. What is the rate of interest?

  1. 3%
  2. 4%
  3. 5%
  4. 6%
  5. None of these

7.utomobile financier claims to be lending money at simple interest, but he includes the interest every six months for calculating the principal. If he is charging an interest of 10%, the effective rate of interest becomes:

  1. 10%
  2. 10.25%
  3. 10.5%
  4. None of these

8.lent Rs. 5000 to B for 2 years and Rs. 3000 to C for 4 years on simple interest at the same rate of interest and received Rs. 2200 in all from both of them as interest. The rate of interest per annum is:

  1. 5%
  2. 7%
  3. 71%
  4. 10%

9.Rs. 725 is lent in the beginning of a year at a certain rate of interest. After 8 months, a sum of Rs. 362.50 more is lent but at the rate twice the former. At the end of the year, Rs. 33.50 is earned as interest from both the loans. What was the original rate of interest?

  1. 3.6%
  2. 4.5%
  3. 5%
  4. 6%
  5. None of these

10 an took loan from a bank at the rate of 12% p.a. simple interest. After 3 years he had to pay Rs. 5400 interest only for the period. The principal amount borrowed by him was:

  1. Rs. 2000
  2. Rs. 10,000
  3. Rs. 15,000
  4. Rs. 20,000
aptitude :: height and distance

aptitude :: height and distance

Trigonometry:
In a right angled OAB, where BOA = ,
i. sin = Perpendicular = AB ;
Hypotenuse OB
ii. cos = Base = OA ;
Hypotenuse OB
iii. tan = Perpendicular = AB ;
Base OA
iv. cosec = 1 = OB ;
sin AB
v. sec = 1 = OB ;
cos OA
vi. cot = 1 = OA ;
tan AB
Trigonometrical Identities:
sin2 + cos2 = 1.
1 + tan2 = sec2 .
1 + cot2 = cosec2 .
Values of T-ratios:
0° (/6)
30° (/4)
45° (/3)
60° (/2)
90°
sin 0
1
2
3
2
1
cos 1
3
2
1
2
0
tan 0
1
3
1 3 not defined
Angle of Elevation:
Suppose a man from a point O looks up at an object P, placed above the level of his eye. Then, the angle which the line of sight makes with the horizontal through O, is called the angle of elevation of P as seen from O.
Angle of elevation of P from O = AOP.
Angle of Depression:
Suppose a man from a point O looks down at an object P, placed below the level of his eye, then the angle which the line of sight makes with the horizontal through O, is called the angle of depression of P as seen from O.
QUESTION
1.
Two ships are sailing in the sea on the two sides of a lighthouse. The angle of elevation of the top of the lighthouse is observed from the ships are 30° and 45° respectively. If the lighthouse is 100 m high, the distance between the two ships is:
A. 173 m
B. 200 m
C. 273 m
D. 300 m
2.
A man standing at a point P is watching the top of a tower, which makes an angle of elevation of 30º with the man’s eye. The man walks some distance towards the tower to watch its top and the angle of the elevation becomes 60º. What is the distance between the base of the tower and the point P?
A. 43 units
B. 8 units
C. 12 units
D. Data inadequate
E. None of these
3.
The angle of elevation of a ladder leaning against a wall is 60º and the foot of the ladder is 4.6 m away from the wall. The length of the ladder is:
A. 2.3 m
B. 4.6 m
C. 7.8 m
D. 9.2 m
4.
An observer 1.6 m tall is 203 away from a tower. The angle of elevation from his eye to the top of the tower is 30º. The heights of the tower is:
A. 21.6 m
B. 23.2 m
C. 24.72 m
D. None of these
5.
From a point P on a level ground, the angle of elevation of the top tower is 30º. If the tower is 100 m high, the distance of point P from the foot of the tower is:
A. 149 m
B. 156 m
C. 173 m
D. 200 m

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