**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?

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**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: x
^{2}+ 8 = 6x - So, x
^{2}– 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 (x
^{2}+ y^{2}) – 21 = x • y - Substituting (9 – x) for y into the second equation yields: ( x
^{2}+ (9 – x)^{2}) – 21 = x (9 – x) - Simplifying yields: x
^{2}+ 81 – 18x + x^{2}– 21 = 9x – x^{2} - Further simplifying yields: 3x
^{2}– 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 x
^{2}+ y^{2}= 233 - Substituting (21 – x) for y in the second equation yields: x
^{2}+ ( 21 – x )^{2}= 233 - Simplification yields: x
^{2}+ 441 – 42x + x^{2}= 233 - Further simplification yields: 2x
^{2}– 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:*

- 3
^{2}/ 2^{3}= 24^{2}/ x^{3} - 9 / 8 = 576 / x
^{3} - 9 x
^{3 }= 4,608 - x
^{3 }= 512 - x = 8

*Solution to Problem 70:*

- V
_{1}/ r_{1}^{3}= V_{2}/ r_{2}^{3} - 113.1 / 3
^{3}= V_{2}/ 5^{3} - V
_{2}= (113.1) (125) / 27 = 523.6 cubic yards

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