Mathematical problems — Problems 61 to 70
The ten problems are presented first, followed by the solutions to the ten problems.
Problem 61:
The sum of a number and 12 is equal to twice the number. What is the number?
Problem 62:
The sum of a number and 3 more than twice the number equals 36. What is the number?
Problem 63:
When 5 is subtracted from two times a given number, the result is two more than the number. What is the number?
Problem 64:
The sum of a number and 8 times its reciprocal is 6. What is the number?
Problem 65:
The sum of two integers is 9. The sum of their squares is 21 greater than the product of the two integers. What are the numbers?
Problem 66:
The sum of two integers is 21. The sum of the squares of these integers is 233. What are the integers?
Problem 67:
The product of two integers is 494. One of the integers is 7 greater than the other. What are the integers?
Problem 68:
If y varies directly with x, and y = 20 when x = 5, then what is the value of y when x = 2 ?
Problem 69:
If the square of y varies directly with the cube of x, and if y is equal to 3 when x is equal to 2, then what is the value of x when y is 24?
Problem 70:
The volume of a sphere varies directly with the cube of its radius. If a sphere with a radius of 3 yards has a volume of 113.1 cubic yards, then what is the volume of a sphere that has a radius of 5 yards?
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Solution to Problem 61:
- x + 12 = 2x
- So, x = 12
Solution to Problem 62:
- x + ( 2x + 3 ) = 36
- 3x + 3 = 36
- 3x = 33
- So, x = 11
Solution to Problem 63:
- 2x – 5 = x + 2
- 2x – x = 2 + 5
- x = 7
Solution to Problem 64:
- x + (8/x) = 6
- Multiplying all components by x yields: x2 + 8 = 6x
- So, x2– 6x + 8 = 0
- Factoring yields: (x – 2)(x – 4) = 0
- So the two solutions are 2 and 4.
Solution to Problem 65:
- x + y = 9 and (x2 + y2) – 21 = x • y
- Substituting (9 – x) for y into the second equation yields: ( x2 + (9 – x)2 ) – 21 = x (9 – x)
- Simplifying yields: x2 + 81 – 18x + x2 – 21 = 9x – x2
- Further simplifying yields: 3x2 – 27x + 60 = 0
- Factoring yields: (3x – 15)(x – 4) = 0
- Hence, x = 4 and the other integer must be 5
Solution to Problem 66:
- x + y = 21 and x2 + y2 = 233
- Substituting (21 – x) for y in the second equation yields: x2 + ( 21 – x )2 = 233
- Simplification yields: x2 + 441 – 42x + x2 = 233
- Further simplification yields: 2x2 – 42x + 208 = 0
- Factoring yields: ( 2x – 26 )( x – 8 ) = 0
- Hence, the two integers are 8 and 13.
Solution to Problem 67:
- x • y = 494 and x + 7 = y
- Substituting ( x + 7 ) for y in the first equation yields: x • ( x + 7 ) = 494
- Simplification yields: x2 + 7x – 494 = 0
- Factoring yields: ( x – 19 )( x + 26 ) = 0
- Hence, the integers are 19 and 26.
Solution to Problem 68:
- 20 / 5 = y / 2
- 5y = 40
- y = 8
Solution to Problem 69:
- 32 / 23 = 242 / x3
- 9 / 8 = 576 / x3
- 9 x3 = 4,608
- x3 = 512
- x = 8
Solution to Problem 70:
- V1 / r13 = V2 / r23
- 113.1 / 33 = V2 / 53
- V2 = (113.1) (125) / 27 = 523.6 cubic yards
Click here to return to the Mathematics index