Numbers

5,12,20,29,39,__
Here, the difference between 5 and 12 is 7.
And the difference between 12 and 20 20 is 8.
And the difference between 20 and 29 is 9.
And the difference between 29 and 39 is 10. 
Since the difference is incredible by 1. 
And the next number could be 50.
Because the difference between 39 and 50 is 11 which is the next number of previous difference.
Option d is correct.

The sum of these four integers is 1290, so we can write the equation:
x + (x+1) + (x+2) + (x+3) = 1290

Simplifying this equation, we get:
4x + 6 = 1290

Subtracting 6 from both sides, we get:
4x = 1284

Dividing both sides by 4, we get:
x = 321

So, the four consecutive integers are 321, 322, 323, and 324.
The greatest of these integers is 324.
Therefore, the answer to the question is 324.

Let's say that the number of boys is b and the number of girls is g. We know that the total number of boys and girls is 50, so we can write the equation:
b + g = 50

We also know that the total number of oranges that the boys receive is 5b, and the total number of oranges that the girls receive is 7g. The total number of oranges that are distributed is 280, so we can write the equation:
5b + 7g = 280

We can solve this system of equations using substitution. First, we can solve for b in the first equation by rearranging it as follows:
b = 50 - g

Substituting this expression for b into the second equation, we get:
5(50 - g) + 7g = 280

Expanding and simplifying the left side of the equation, we get:
250 - 5g + 7g = 280
Combining like terms, we get:
2g = 30
Dividing both sides by 2, we get:
g = 15
Therefore, there are 15 girls in the group of 50 people.

Let's say the number we are trying to find is x. We know that the number is multiplied by one-third of itself to get 192, so we can write the equation:

x * (1/3) * x = 192
Simplifying the left side of the equation, we get:

(1/3) * x^2 = 192
Multiplying both sides of the equation by 3, we get:

x^2 = 576
Taking the square root of both sides of the equation, we get:

x = 24
Therefore, the number we are looking for is 24.

Let's say the number that the boy was supposed to multiply by 12 is x. We know that the boy multiplied this number by 21 instead, and got an answer that was 63 more than the correct answer. Therefore, we can write the equation:

12x + 63 = 21x
Subtracting 12x from both sides of the equation, we get:

63 = 9x
Dividing both sides of the equation by 9, we get:

7 = x
Therefore, the number that the boy was supposed to multiply by 12 is 7.

Let's call the first number A, the second number B, and the third number C. We know that the first number is 4 times the second number and 5 times the third number, so we can represent these relationships with the equations:
A = 4B
A = 5C
We also know that the average of the three numbers is 87, so we can represent this with the equation:

(A + B + C)/3 = 87
We can solve this system of equations using substitution or elimination.
Using substitution, we can solve for B in the first equation and substitute it into the second equation:

B = A/4
5C = A
C = A/5
Substituting these values for B and C back into the first equation, we get:

(A + A/4 + A/5)/3 = 87
5A + A + 4A/5 = 261
9A = 1020
A = 180
We can check our solution by substituting this value for A back into the other two equations to make sure they both hold true.

A = 4B => 180 = 4B => B = 45
A = 5C => 180 = 5C => C = 36
Since both equations hold true, we have found a valid solution.
To find the difference between the first and third numbers, we can simply subtract the third number from the first number:

A - C = 180 - 36 = 144
So the difference between the first and third numbers is 144

M - MX = 14
Next, we can combine like terms:

M - MX = 14 MX - MX = 0 M = 14 Finally, we can substitute the value that we found for M, which is 24:
M = 24
So the final equation is:
M = 24

Let H be the number of hens and P be the number of pigs. The first equation is:

H + P = 84 (1)

The second equation is:

2H + 4P = 282 (2)

We can solve for P by dividing equation (2) by 2, which gives us:

H + 2P = 141 (3)

Subtracting equation (1) from equation (3) gives us:

H + 2P - H - P = 141 - 84

P = 57

Substituting this value for P into equation (1) gives us:

H + 57 = 84

H = 84 - 57

H = 27

Therefore, there are 27 hens in the poultry farm.

I hope this rephrased version makes the solution more clear. Let me know if you have any further questions.

Let numbers be A and B

.: (A-B)^2=9

..A^2-2AB+ B^2 = 9

As, A^2+B^2=225

225-2AB = 9

AB = 108 product of the two numbers

 If the sum of the digits of B is 9, then B must be a two-digit number. The only two-digit numbers whose digits add up to 9 are 18 and 27.

If A is formed by reversing the digits of B, then A is either 81 or 72 if B is 18 or 27, respectively.

If B is 45 less than A, then A must be 72. Therefore, the value of A is 72

To maintain the same expenditure on rice when the price increases from Rs. 6 per kg to Rs. 8 per kg, a person must reduce their consumption of sugar by 25%.

To see this, suppose that a person was originally spending X rupees on rice, and consuming Y kg of sugar. Then, the original cost of the sugar can be expressed as 6X rupees/Y kg.

When the price of rice increases to Rs. 8 per kg, the person must reduce their consumption of sugar in order to maintain the same expenditure on rice. Let's say they reduce their consumption of sugar to Y - Z kg, where Z is the amount of sugar they need to reduce.

The cost of the sugar after the reduction is (8X rupees)/(Y - Z kg). Setting this equal to the original cost of the sugar, we have:

(8X rupees)/(Y - Z kg) = 6X rupees/Y kg

Solving for Z, we find that Z = (3/4)Y. In other words, the person must reduce their consumption of sugar by 25% in order to maintain the same expenditure on rice when the price increases from Rs. 6 per kg to Rs. 8 per kg 

We are given two numbers, A and B, such that A > B. The product of these numbers is 266 and the value of B is equal to 266 divided by A. We are also told that the difference between A and B is 5. Using this information, we can set up the following system of equations:

Equation 1: AB = 266
Equation 2: B = 266/A
Substituting Equation 2 into Equation 1, we get:

(266/A)A = 266

This simplifies to:

A^2 - 5A + 266 = 0

(A – 19)(A – 14) = 0
Therefore , A = 19 or A = 14
Hence, bigger number = A = 19

 In this problem, we are given that 12.5% of a group of books are non-fiction, which can also be written as 1/8th of the books. We are also given that 1/6th of the books are fiction. We are trying to determine the total number of books in the group.

Since 1/6th of the books are fiction and 1/8th are non-fiction, we know that the total number of books must be a multiple of both 6 and 8. This means that the total number of books must be divisible by their least common multiple, which is 24.

We are also told that the total number of books must be between 75 and 100. This means that the possible values for the total number of books are 96, 72, 48, etc.

Of these possible values, only 96 fulfills the criteria of being divisible by 24 and being between 75 and 100. Therefore, we can conclude that the total number of books is 96

Let the incomes of P, Q, and R be 6x, 8x, and 11x, respectively, 
and their expenditures be 5y, 8y, and 12y, respectively. 
Therefore, 
their savings are (6x - 5y), (8x - 8y), and (11x - 12y), respectively.

If we set the ratio of x to y equal to 10/9, 
we can determine the ratio of the savings of P to Q and Q to R. 
The ratio of the savings of P to Q is 15/8, 
and the ratio of the savings of Q to R is 8/2.

The overall ratio of the savings of P, Q, and R is 15:8:2.

Let the numbers be x and (26 - x). 
This means that the numbers are x and 26 - x.

We can express this relationship mathematically as:

x (26 - x) = 168

Expanding this equation gives us:

x^2 - 26x + 168 = 0

This quadratic equation can be factored as:

(x - 14)(x - 12) = 0

This means that x = 14 or x = 12. 
Therefore, the numbers are 14 and 12. 

 The number x is such that 3(2x + 5) = 105.

This equation can be rewritten as 6x + 15 = 105.

 Subtracting 15 from both sides of the equation gives us 6x = 90, 
which means that x = 15.

 Let the integer x be such that x + x^2 = 272.

 This can be rewritten as x^2 + x - 272 = 0, which factors to (x + 17)(x - 16) = 0. 

Therefore, x = 16

 When multiplying numbers, the unit digit of the final product is determined by the product of the unit digits of the individual numbers. 

For example, if we multiply 148 by 163 by 236, the unit digit of the final product will be the product of the unit digits of each of these numbers, which is 8 * 3 * 6 = 24. 

Since 4 is the unit digit of 24, the unit digit of the final product will be 4.

x = (2/5)x + 45
x = (3/5)x + 45
x = 75 

Letting the positive number be x and then expressing the first condition in terms of x, we get:

2/3 x = 16/216 * 1/x

We can then multiply both sides by 216 and simplify to get:

x^2 = 16/144

Taking the square root of both sides gives:

x = sqrt(16/144)

Simplifying this expression gives:

x = sqrt(1/9)

Finally, since the number must be positive, we take the positive square root, which gives us:

x = 1/3 or 4/12

Therefore, the positive number that satisfies the given conditions is 1/3. 

 To find the unit digit in the product of the numbers 624, 708, 913, and 463, 
we can use the fact that the unit digit of a product is the same as the unit digit of each of the factors.

The unit digit of 624 is 4, the unit digit of 708 is 8, the unit digit of 913 is 3, and the unit digit of
 463 is 3. Therefore, the unit digit of the product of these numbers is the same as
 the unit digit of 4 * 8 * 3 * 3, which is equal to 8.

Therefore, the unit digit in the product (624 * 708 * 913 * 463) is 8. 

 To solve this problem, we can use the fact that the total number of 
marks a student receives is equal to the number of correct answers multiplied by 3, minus the
 number of wrong answers. We can represent this relationship with the equation:

3x - y = 38

Where x is the number of correct answers and y is the number of wrong answers.

We are also given that the total number of questions is 70, so we can set up
 a second equation to represent this relationship:

x + y = 70

We can solve this system of equations by first solving for y in the first equation:

y = 3x - 38

Substituting this expression for y into the second equation gives:

x + (3x - 38) = 70

This simplifies to:

4x - 38 = 70

Adding 38 to both sides gives:

4x = 108

Dividing both sides by 4 gives:

x = 27

Therefore, the student answered 27 questions correctly.   

Let p be the total number of pages in the book and d be the number of digits used per page.
We can represent the given information as follows: Number of digits used in 1-digit numbers = 1 * 9 = 9
Number of digits used in 2-digit numbers = 2 * 90 = 180
Number of digits used in 3-digit numbers = 3 * 900 = 2700
Number of digits used in 4-digit numbers = 3209 - (9 + 180 + 2700) = 320
Number of pages with 4-digit numbers = 320 / 4 = 80
Total number of pages = p = 1079
This can be written as a system of equations as follows:
1 * 9 + 2 * 90 + 3 * 900 + (p - 9 - 180 - 2700) * 4 = 3209
(p - 9 - 180 - 2700) / 4 = 80
p = 1079
Solving this system of equations will confirm that the given information is consistent.

We can represent the given information as follows:
52 % 6 = 1
53 % 6 = 5
54 % 6 = 1
55 % 6 = 5

For any positive integer x, we have:
5(2x) % 6 = 1
5(2x - 1) % 6 = 5
Therefore, for x = 285, we have:
5(2 * 285) % 6 = 570 % 6 = 1
This can be written as an equation as follows:
5(2 * 285) % 6 = 1
570 % 6 = 1
Solving this equation will confirm that the given information is consistent

To find the number of numbers between 100 and 600 that are divisible by 2, 3, and 7 simultaneously, we can first find the least common multiple (LCM) of 2, 3, and 7. The LCM of 2, 3, and 7 is 42, which means that any number that is divisible by 2, 3, and 7 must also be divisible by 42.
We can then count the number of multiples of 42 between 100 and 600. The first multiple of 42 greater than or equal to 100 is 105, and the last multiple of 42 less than or equal to 600 is 588. The number of multiples of 42 between these two numbers is:
(588 - 105) / 42 + 1 = 12
Therefore, there are 12 numbers between 100 and 600 that are divisible by 2, 3, and 7 simultaneously.

In this case, let x and y be the two given numbers. We are given that the sum of x and y is 73 and that their difference is 28. We can express these two pieces of information as equations:
(x + y) = 73
(x - y) = 28
Using the difference of squares formula, we can now find the difference between the squares of x and y:
(x^2 - y^2) = (x + y)(x - y)
= 73 * 28
= 2044

Therefore, the difference between the squares of x and y is 2044

To find the quarter of the number, we need to first find the total number. We can set up an equation to represent the given information:
6/7 * x = 96


To solve for x, we can multiply both sides of the equation by 7/6:
x = (7/6) * 96
= 112


Now that we know the total number is 112, we can find one quarter of it by dividing it by 4:
1/4 * 112 = 28

Therefore, one quarter of the number is 28

We can set up an equation to represent the given information:
8x + 3y = 203

where x represents the first part and y represents the second part. We know that the total of the two parts is 36, so we can add another equation to represent this:
x + y = 36

To solve for x, we can use elimination method by multiplying the second equation by 8 and subtracting it from the first equation:
8x + 3y - 8x - 8y = 203 - 288
-5y = -85
y = 17


We can substitute this value of y back into the second equation to solve for x:
x + 17 = 36
x = 36 - 17
= 19

Therefore, the first part is 19.

To find the number that is as much greater than 36 as it is less than 86, we can use the following steps:
Let the number be represented by the variable x.
Set up the equation x - 36 = 86 - x.
Solve the equation to obtain 2x = 122, which simplifies to x = 61.
The required number is x, which is equal to 61.

To find the number such that when 15 is subtracted from 7 times the number, the result is 10 more than twice the number, we can use the following steps:
Let the number be represented by the variable x.
Set up the equation 7x - 15 = 2x + 10.
Solve the equation to obtain 5x = 25, which simplifies to x = 5.
The required number is x, which is equal to 5.

To find the rational number such that the sum of the number and its reciprocal is 13/6, we can use the following steps:
Let the number be represented by the variable x.
Set up the equation x + (1/x) = 13/6.
Multiply both sides of the equation by x to obtain x^2 + 1 = 13/6 * x.
Simplify the right side of the equation to obtain (x^2 + 1)/x = 13/6.
Set up a quadratic equation in x by multiplying both sides of the equation by 6: 6x^2 - 13x + 6 = 0.
Rewrite the equation as (3x - 2)(2x - 3) = 0.
Solve the equation to obtain x = 2/3 or x = 3/2.
The required number is x, which is equal to 2/3 or 3/2.

To find the smaller of two numbers whose sum is 184, where one-third of the one exceeds one-seventh of the other by 8, we can use the following steps:
Let the two numbers be represented by the variables x and y, where y is equal to 184 - x.
Set up the equation (x/3) - (y/7) = 8.
Substitute y with 184 - x to obtain (x/3) - ((184 - x)/7) = 8.
Simplify the equation to obtain 7x - 3(184 - x) = 168.
Simplify the equation to obtain 10x = 720.
Solve the equation to obtain x = 72.
The two numbers are x and 184 - x, which are equal to 72 and 112, respectively.
The smaller number is x, which is equal to 72.

Given:

x - y = 11
(x + y)/5 = 9
Add the two equations:

(6x + 4y)/5 = 20
6x + 4y = 100
Solve for x:

x = (100 - 4y)/6
We could have solved for x by substituting the value of y from the first equation into the second equation.
This would give us the equation:

x = 28
Substituting this value for x into the first equation, we would get:

y = 17
Therefore, the two numbers are x = 28 and y = 17.

Given:

x + y = 42
xy = 437
Find:

|x - y|
Substitute the first equation into the expression for |x - y| to get:

|x - y| = √((x + y)^2 - 4xy)
Substitute the given values for x + y and xy into this expression to get:

|x - y| = √((42^2) - 4(437)) = √(1764 - 1748) = √(16) = 4
Therefore, the absolute difference between the numbers is 4.

Given:

x + (15 - x) = 15
x^2 + (15 - x)^2 = 113
Find:

x and (15 - x)
Substitute the first equation into the expression for (15 - x) to get:

x^2 + (15 - x)^2 = 113
x^2 + 225 + x^2 - 30x + x^2 = 113
3x^2 - 30x + 112 = 0
Factor the quadratic equation:

(x - 7)(x - 8) = 0
Solve for x:

x = 7 or x = 8
Therefore, the two numbers are x = 7 and (15 - x) = 8.

Given:

The average of four consecutive even numbers is 27
Find:

The largest of these numbers
Let the four consecutive even numbers be x, x + 2, x + 4, and x + 6. Then the sum of these numbers is 4*27 = 108.

Solve for x:

x + (x + 2) + (x + 4) + (x + 6) = 108
4x = 96
x = 24
Therefore, the largest number is x + 6 = 24 + 6 = 30.

Given:

The sum of the squares of three consecutive odd numbers is 2531
Find:

The values of these numbers
Let the three consecutive odd numbers be x, x + 2, and x + 4. Then the sum of their squares is given by:

x^2 + (x + 2)^2 + (x + 4)^2 = 2531
Expand and rearrange the terms:

3x^2 + 12x - 2511 = 0
Factor the quadratic equation:

(x - 27)(x + 31) = 0
Solve for x:

x = 27
Therefore, the three numbers are x = 27, x + 2 = 27 + 2 = 29, and x + 4 = 27 + 4 = 31.

Given:

4 times the smaller of two numbers is less than 3 times the larger number by 5
The sum of the two numbers is larger than 6 times their difference by 6
Let the two numbers be x and y, such that x > y. 
Then the given conditions can be expressed as the following equations:

3x - 4y = 5
(x + y) - 6 (x - y) = 6
Solve these equations simultaneously to find the values of x and y:

(x + y) - 6 (x - y) = 6
-5x + 7y = 6
Solve for x:

x = 59
Substitute this value for x into the first equation to solve for y:

3(59) - 4y = 5
177 - 4y = 5
4y = 172
y = 43
Therefore, the two numbers are x = 59 and y = 43.

Given:

The ratio between a two-digit number and the sum of the digits of that number is 4:1
The digit in the unit's place is 3 more than the digit in the ten's place
Let the digit in the ten's place be x.
Then the digit in the unit's place is x + 3 and the sum of the digits is 2x + 3. 
The two-digit number can be expressed as:

10x + (x + 3) = 11x + 3
The ratio of the number to the sum of its digits is given by:

(11x + 3) / (2x + 3) = 4/1
Solve for x:

3x = 9
x = 3
Therefore, the two-digit number is 11x + 3 = 11(3) + 3 = 36.

To find the values of x and y, we can use the information provided to create a system of equations and solve for x and y.

The first equation is given: x + y = 42
The second equation is given: xy = 437

To find the difference between x and y, we can use the equation x - y = sqrt[(x + y)^2 - 4xy], which is also given.

Substituting the values from the first two equations into the third equation, we get: x - y = sqrt[(42)^2 - 4 x 437] = sqrt[1764 - 1748] = sqrt[16] = 4.

From this equation, we can see that the difference between x and y is 4.
To find the values of x and y, we can set up the following system of equations:

x + y = 42
x - y = 4

Solving this system of equations using substitution, we get:

x = 23 and y = 19

Therefore, the values of x and y are 23 and 19, respectively.

0.008 = 8/100 = 1/1250.

To evaluate the expression "28% of 450 + 45% of 280", you can first find the value of 28% of 450 and 45% of 280.

To find 28% of 450, you can multiply 450 by 0.28, which is equal to 126.

To find 45% of 280, you can multiply 280 by 0.45, which is equal to 126.

Then, you can add these two values together to get the final result:
 126 + 126 = 252.

So, the expression "28% of 450 + 45% of 280" is equal to 252.

In this approach, you first convert the units to the same type of mass measurement (1 metric tonne = 10 quintals), 

And then divide the number of quintals by the number of metric tons and multiply by 100 to express the result as a percentage.

The expression "40/(2 * 10)" simplifies to "4", and when this value is multiplied by 100%, the result is 200%. 

This shows that 40 quintals is equal to 200% of 2 metric tons.

Required difference = [3 ½ % of Rs.8400] – [3 1/3 % of Rs.8400]
= [(7/20-(10/3)]% of Rs.8400 =1/6 % of Rs.8400
= Rs. [(1/6)8(1/100)*8400] = Rs. 14

Let x and y be two numbers such that 7.5% of x is equal to 12.5% of y. 

This can be expressed as the equation:

0.075x = 0.125y

We can solve for x in terms of y by rearranging this equation to solve for x:

x = (0.125y) / (0.075)
= (125y) / (75)
= 5y / 3

Now, we are given that x - y = 1660. Substituting our expression for x in terms of y into this equation, we get:

(5y/3) - y = 1660
2y/3 = 1660
y = (1660*3)/2 = 2490

Now that we have solved for y, we can use our expression for x in terms of y to find x:

x = (5y/3) = (5 * 2490) / 3 = 4150

Therefore, the two numbers are x = 4150 and y = 2490.

To find the number of meters that need to be examined, you can set up an equation using the given information. 
The number of meters to be examined, represented by x, is equal to 0.08% of x, which is equal to 2.

You can then solve for x by multiplying both sides of the equation by the reciprocal of 0.08%. 
The reciprocal of 0.08% is 100*100/8, so multiplying both sides of the equation by this value gives you x = 2500.

Therefore, the number of meters that need to be examined is 2500.