Rectangular Parallelepiped Problem, 3

Category: Solid Geometry, Physics

"Published in Vacaville, California, USA"

Counting 38 cu. ft. of coal to a ton, how many tons will a coal bin 19 ft. long, 6 ft. wide, and 9 ft. deep contain, when level full?

Solution:

To illustrate the problem, it is better to draw the figure as follows 

Photo by Math Principles in Everyday Life

The volume of a rectangular parallelepiped is given by the formula

Substitute the values of length, width, and height of a bin, we have

The density of a substance is given by the formula

where ρ is the density, W is the weight, and V is the volume of a substance respectively. 

Therefore, the weight of a coal in a bin is

Finding Missing Digit - Divisibility Rule, 14

Category: Arithmetic

"Published in Vacaville, California, USA"

Find the missing digit so that it becomes divisible by 15 for

a. 13?2
b. 337?

Solution:

a. Consider the given number

A number is divisible by 15 if it is both divisible by 3 and 5. The given number is not divisible by 5 because the last digit is not 5 or 0. Because of this, the given number is not divisible by 15. The multiples of 15 ends with 5 or 0 always.There's nothing that we can do in order to become divisible by 15 since the last digit of a given number is not 5 or 0. You can assign any number to the missing digit but still, the given number will never become divisible by 15.

b. Consider the given number

A number is divisible by 15 if it is both divisible by 3 and 5. Since the last digit of a given number is missing, then we can assign 5 or 0. A number is divisible by 5 if the last digit ends with 5 or 0 and a number is divisible by 3 if the sum of the digits is a multiple of 3. Let's do the test for 5 and 0 as their last digit.

If the last digit is 0, then the sum of the digits is

Since 4 is not a multiple of 3, then the given number is not divisible by 3. 0 is not the last digit of a given number. Let's do the test again for the other last digit which is 5.

If the last digit is 5, then the sum of the digits is

Since 9 is a multiple of 3, then the given number is divisible by 3. 5 is the last digit of a given number. Because of this, the given number is divisible by 15. Therefore, the possible number is only 3375.  

Finding Missing Digit - Divisibility Rule, 13

Category: Arithmetic

"Published in Newark, California, USA"

Find the missing digit so that it becomes divisible by 14 for

a. 5?3
b. 137?

Solution:

In finding the missing digit, this method is completely different from the previous divisibility by other numbers because we will use the principles of Algebra in solving for the unknown digit.

a. Consider the given number

Since the last digit of a given number is an odd number, then it is not divisible by 14. A number is divisible by 14 if it is both divisible by 2 and 7. All even numbers are divisible by 2. There's nothing that we can do in order to become divisible by 14 since the last digit of a given number is not an even number. You can assign any number to the missing digit but still, the given number will never become divisible by 14. 

b. Consider the given number

A number is divisible by 14 if it is both divisible by 2 and 7. In short, an even number that is divisible by 7. We can use 0, 2, 4, 6, and 8 as the last digit of a given number. To test the divisibility of a number by 7, double the last digit and then subtract it to the remaining digits. If the result is a multiple of 7, then the given number is divisible by 7. Let's do this test to all the last digits.

If the last digit is 0, then the given number becomes

 

 

 

Since the result is not a multiple of 7, then the given number is not divisible by 7. 0 is not the last digit of a given number. Let's do the test again for the other even number digit.

If the last digit is 2, then the given number becomes

Since the result is a multiple of 7, then the given number is divisible by 7. 2 is the last digit of a given number. Let's do the test again for the other even number digit.

If the last digit is 4, then the given number becomes

Since the result is not a multiple of 7, then the given number is not divisible by 7. 4 is not the last digit of a given number. Let's do the test again for the other even number digit.

If the last digit is 6, then the given number becomes 

Since the result is not a multiple of 7, then the given number is not divisible by 7. 6 is not the last digit of a given number. Let's do the test again for the other even number digit.

If the last digit is 8, then the given number becomes 

Since the result is not a multiple of 7, then the given number is not divisible by 7. 8 is not the last digit of a given number. In this case, only 2 is the last digit of a given number. Therefore, the possible number is only 1372.