Explanation :

Remainder when 51*27*35*62*75 divided by 100

To find remainder when divided by 100:
Take ones place digit of each number: 1, 7, 5, 2, 5
Multiply them: 1 × 7 × 5 × 2 × 5 = 350
Find remainder when 350 ÷ 100 = 50 (remainder)

Therefore, when 51 × 27 × 35 × 62 × 75 is divided by 100, the remainder is 50.

Answer: (a) 50

Explanation :

Let’s solve this problem step by step:

Finding the least four-digit number with remainder 2 when divided by 3, 4, 5, and 6

STEP 1: Find the LCM of 3, 4, 5, and 6
– First, we need the least common multiple (LCM) of 3, 4, 5, and 6
– The LCM of these numbers is 60

STEP 2: Find the smallest four-digit number divisible by LCM
– We need to find the smallest four-digit number divisible by 60
– The smallest four-digit number divisible by 60 is 1020

STEP 3: Add remainder
– Since we want remainder 2 when divided by 3, 4, 5, and 6
– We add 2 to our number: 1020 + 2 = 1022

ANSWER: 1022 is the least four-digit number that leaves remainder 2 when divided by 3, 4, 5, and 6

VERIFICATION:
1022 ÷ 3 = 340 remainder 2
1022 ÷ 4 = 255 remainder 2
1022 ÷ 5 = 204 remainder 2
1022 ÷ 6 = 170 remainder 2

Explanation :

Recurring Decimal 1.272727…

To solve this problem, let x = 1.272727…

Then, 100x = 127.272727…

Subtracting x from 100x, we get:

• 99x = 126
• x = 126/99

We can simplify this fraction by dividing the numerator and denominator by their greatest common factor, which is 3:

x = (126/3)/(99/3) = 42/33

We can simplify this fraction further by dividing the numerator and denominator by their greatest common factor, which is 3:

x = (42/3)/(33/3) = 14/11

Therefore, the recurring decimal representation 1.272727… is equivalent to 14/11, and the correct answer is (b).

Explanation :

For which period was the natural growth rate maximum ?

The natural growth rate is calculated by subtracting the death rate from the birth rate. From the data you provided, we can calculate the natural growth rates for each period:

• 1911-1921: 48.1 – 35.5 = 12.6
• 1921-1931: 46.4 – 36.3 = 10.1
• 1931-1941: 45.2 – 31.2 = 14
• 1941-1951: 39.9 – 27.4 = 12.5
• 1951-1961: 41.7 – 22.8 = 18.9
• 1961-1971: 41.1 – 18.9 = 22.2
• 1971-1981: 37.1 -14 .8 = 22.3

The maximum natural growth rate was 22.3 during the period 1971-1981.

Explanation : 

Largest Number

Let’s solve this problem step by step. A negative exponent means that the base is inverted and the exponent becomes positive. For example, (1/2)^-6 can be written as (2/1)^6.

Using this logic, we can rewrite all the options as follows:
(a) (1/2)^-6 = (2/1)^6 = 64
(b) (1/4)^-3 = (4/1)^3 = 64
(c) (1/3)^-4 = (3/1)^4 = 81
(d) (1/6)^-2 = (6/1)^2 = 36

So among these options, the largest number is 81.

So (c) (1/3)^-4 is indeed the correct answer.

Explanation :

A shop owner offers the following …

Let’s calculate the effective discount for each option, assuming the original price of the article is P.

Option 1:

First discount: 10% -> P * (1 – 0.10) = 0.9P
Second discount: 20% -> 0.9P * (1 – 0.20) = 0.9P * 0.8 = 0.72P
Service tax: 10% -> 0.72P * (1 + 0.10) = 0.72P * 1.1 = 0.792P

Option 2:

First discount: 20% -> P * (1 – 0.20) = 0.8P
Second discount: 10% -> 0.8P * (1 – 0.10) = 0.8P * 0.9 = 0.72P
Service tax: 10% -> 0.72P * (1 + 0.10) = 0.72P * 1.1 = 0.792P

Option 3:

Service tax: 10% -> P * (1 + 0.10) = 1.1P
First discount: 20% -> 1.1P * (1 – 0.20) = 1.1P * 0.8 = 0.88P
Second discount: 10% -> 0.88P * (1 – 0.10) = 0.88P * 0.9 = 0.792P

All three options result in the customer paying the same final price, which is 79.2% of the original price (0.792P). So, all options are equivalent in terms of the final price for the customer.

Explanation : 

A simple mathematical operation

If we subtract 1 from each number in the original sequence, we get the following sequence:

13, 17, 19, 23, 29, 31

Notice that this is simply the sequence of prime numbers starting from 13.

The next prime number after 31 is 37. Therefore, the next number in the original sequence, which corresponds to the prime number 37 after subtracting 1, is:

37 + 1 = 38

Therefore, the correct answer is (c) 38

Explanation :

x+1x+5x+xx+x1=1xx

x + 1x + 5x + xx + x1 = 1xx

=> x + 10+ x + 50 + x +10x + x + 10x + 1 = 100 + 10x + x

=> 24x + 61 = 100 + 11x

=> 13x = 39

=> x = 3

Model Answer:

Let XYZ be a 3 digit number

Let’s write XYZ in expanded form, where X, Y, and Z are the digits in the hundreds, tens, and units places, respectively.

XYZ = 100X + 10Y + Z

Similarly, we can write YZX and ZXY in expanded form:

YZX = 100Y + 10Z + X
ZXY = 100Z + 10X + Y

Adding these three expressions, we get:

XYZ + YZX + ZXY = 100X + 10Y + Z + 100Y + 10Z + X + 100Z + 10X + Y
= 111X + 111Y + 111Z=111(X + Y + Z)

Factors of 111 are 3 and 37. That eliminate option 3 and 37.

Explanation : 

Let x y be the volumes …

Let’s solve this step by step.

1) Let’s say the side length of cube P is ‘a’
Then volume of P (x) = a³
Mass of P is m

2) For cube Q:
Side length is 2a (given: each side of Q is two times that of P)
Volume of Q (y) = (2a)³ = 8a³
Mass of Q (n) = 2m (given: mass of Q is two times that of P)

3) Now, let’s calculate u = m/x
u = m/a³

4) Let’s calculate v = n/y
v = 2m/(8a³)
v = (2m)/(8a³)
v = m/(4a³)
v = (m/a³)/4
v = u/4

5) Therefore:
u = 4v

Looking at the options, (a) u = 4v is correct.

The answer is (a).