These are the solutions and answers to the problems in Week 8 Review on Number Problems.

Problem 1

One number is 3 more than the other. Their sum is 27. What are the numbers?

Let x – smaller number
x + 3 – larger number

Their sum is 27, so
x + (x + 3) = 27
2x + 3 = 27
2x = 27 – 3
2x = 24
x = 24/2
x = 12 (smaller number)
x + 3 = 15 (larger number).

Problem 2
One number is 5 less than the other. Their sum is 51. What are the numbers?

Let x – larger number
x – 5 –  smaller number

And their sum is 51. So,

x + (x – 5) = 51
2x – 5 = 51
2x = 51 + 5
2x = 56
x = 56/2
x = 28 (larger number)
x – 5 = 28 – 5 = 23 (smaller number).

Answer: 23 and 28

Problem 3

One number is 3 times the other number. Their sum is 48. What are the numbers?

Let x – smaller number
3x – larger number

And their sum is 48. So,

x + 3x = 48
4x = 48
x = 48/4
x = 12(1st number)

2nd number = 3x
3(12) = 36

Answer: 12 and 36

Problem 4
One number is 5 times the other number. Their difference is 52. What are the numbers?

Let x – smaller number
5x – larger number

And their difference is 52. So,

5x – x = 52
4x = 52
x = 52/4
x = 13 (smaller number)
5x = 5(13) = 65.

Checking: -13 – (-65)
-13 + (65) = 52

Answer: 13 and 65

Problem 5
The sum of three numbers is 36. The second number is 5 more than the first number and the third number is 8 less than the first number. What are the three numbers?

Let x – 1st number
x + 5 – 2nd number
x – 8 – 3rd number

Their sum is 36. So,
x + (x + 5) + (x – 8) = 36
3x – 3 = 36
3x = 36 + 3
3x = 39
x = 13 (1st number)
2nd number = x + 5 => (13) + 5 => 18
3rd number = x – 8 => (13) – 8 => 5

Checking: 13 + 18 + 5 = 36

Problem 6

The sum of three numbers is 98. The second number is twice the first number and the third number twice the second number. What are the three numbers?

14, 28 & 56

Let x = 1st number
2x = 2nd number (twice the first)
2(2x)=3rd number (twice the second)

And their sum is 98. So,

x + (2x) + 2(2x) = 98
x + 2x + 4x =98
7x = 98
x = 98/7
x = 14 (1st number)

2nd number = 2x => 2(14) => 28

3rd number = 2(2x) => 2(2(14)) => 2(28) => 56

Problem 7

One number is two more than thrice the other. Their sum is 26. What are the two numbers?

Let x – 1st number
3x + 2 = 2nd number (two more than thrice the other)

And their sum is 26.

x + (3x + 2) = 26
4x + 2 = 26
4x = 26 – 2
4x = 24
x = 24/4
x = 6 (1st number)

2nd number = (3x + 2) => 3(6) + 2 => 18 + 2 => 20

Answer: 6 and 20

Problem 8

One number is thrice the other. When 3 is added to the larger and 7 is subtracted from the smaller, their sum becomes 32. What are the two numbers?

Let x – smaller number
3x – larger number (thrice the other)

When 3 is added to larger number… = 3x + 3

…and 7 is subtracted to smaller = x – 7

Their sum becomes 32. So,

(3x + 3) + (x – 7) = 32
4x – 4 = 32
4x = 32 + 4
4x = 36
x = 36/4
x = 9(smaller number)

Larger number = 3x = 3(9) = 27

Checking:
When 3 is added to larger number = 27 + 3 = 30
And 7 is subtracted to smaller number = 9-7 = 2

Their sum is 32 = 30 + 2 = 32

Answer: 9 and 27

Problem 9

The sum of two consecutive numbers is 91. What are the two numbers?

Let x – first number
x + 1 – 2nd number

x + (x + 1) = 91
2x + 1 = 91
2x = 91 – 1
2x = 90
x = 90/2
x = 45 (1st number)

2nd number => x + 1 => 45 + 1 => 46

Answer: 45 and 46

Problem 10
The sum of two positive consecutive EVEN integers is 66. What are the two numbers?

Let x – 1st number
x + 2 = 2nd number

x + (x + 2) = 66
2x + 2 = 66
2x = 66 – 2
2x = 64
x = 64/2
x = 32 (1st number)

2nd number => x + 2 => 32 + 2 => 34

Answer: 32 and 34

PCSR Problem 11

The sum of two positive consecutive ODD integers is 36. What are the two numbers?

Let x – 1st odd number
x + 2 – 2nd odd number

And their sum is 36.

x + (x + 2) = 36
2x + 2 = 36
2x = 36-2
2x = 34
x = 34/2
x = 17(1st number)
x + 2 = 17 + 2 = 19 (2nd number)

Checking:

17 + 19 = 36
And 17 and 19 are both odd numbers

Answer: 17 and 19

Problem 12

The sum of three positive consecutive ODD integers is 81. What are the three integers?

Let x – 1st odd integer
x + 2 – 2nd odd integer
x + 4 – 3rd odd integer

Their sum is 81.

x + (x + 2) + (x + 4) = 81
3x + 6 = 81
3x = 81 – 6
3x = 75
x = 75/3
x = 25 (1st int)

2nd int = (x + 2) => 25 + 2 => 27
3rd int = (x + 4) => 25 + 4 => 29

Checking:

25 + 27 + 29 = 81
They are consecutive ODD integers.

Answer: 25, 27 & 29

Problem 13

The sum of the smallest and the largest of five positive consecutive integers is 108. What is the third integer?

Let x – 1st integer
x + 1 = 2nd integer
x + 2 = 3rd integer
x + 3 = 4th integer
x + 4 = 5th integer

Since the sum of the first and the fifth is 108,

x + (x + 4) = 108
2x + 4 = 108
2x = 108 – 4
2x = 104
x = 104/2
x = 52 (smallest number).

2nd int. => (x + 1) => 52 + 1 => 53
3rd int => ( x + 2) => 52 + 2 => 54
4th int. => (x + 3) => 52 + 3 => 55
5th int. => (x + 4) => 52 + 4 => 56

Since we are looking for the third integer, the answer is 54.

Problem 14
The average of four positive consecutive EVEN integers is 19. What is the largest integer?

Let x – 1st even integer
x + 2 = 2nd even integer
x + 4 = 3rd even integer
x + 6 = 4th even integer

Their average is 19.

(x + (x + 2) + (x + 4) + (x + 6))/4 = 19
(4x + 12)/4 = 19

Multiplying both sides of the equation by 4,

4x + 12 = 19(4)
4x + 12 = 76
4x = 76 – 12
4x = 64
x = 64/4
x = 16(1st even int).

2nd even int. = x + 2 => 16 + 2 => 18
3rd even int. = x + 4 => 16 + 4 => 20
4th even int. = x + 6 => 16 + 22 => 22

Checking:

(16 + 18 + 20 + 22)/4 = 19
(76)/4 = 19
19 = 19

Answer: 22(largest number)

PCSR Problem 15
The average of seven positive consecutive integers is 31. What is the smallest integer?

Let x – 1st integer
x + 1 = 2nd integer
x + 2 = 3rd integer
x + 3 = 4th integer
x + 4 = 5th integer
x + 5 = 6th integer
x + 6 = 7th integer

Their average is 31.

(x + (x + 1) + (x + 2) + (x +3) + (x + 4) + (x + 5) + (x + 6))/7 = 31
(7x + 21)/7 = 31
7x + 21 = 31(7)
7x + 21 = 217
7x = 217 – 21
7x = 196
x = 196/7
x = 28(1st integer)

2nd int. = x + 1 => 28 + 1 => 29
3rd int. = x + 2 => 28 + 2 => 30
4th int. = x + 3 => 28 + +3 => 31
5th int. = x + 4 => 28 + 4 => 32
6th int. = x + 5 => 28 + 5 => 33
7th int. = x + 6 => 29 + 6 => 34

Checking:
(28 + 29 + 30 + 31 + 32 + 33 + 34)/7 = 31
217/7 = 31
31 = 31

Since we are looking for the smallest integer, the answer is 28.