Solving Number Problems

Number problems are an important research field of mathematics. In mathematical competitions, problems of basic number theory occur frequently. These problems use slight knowledge and have many variations. They are flexible and diverse. The number problems are much related to the human life. Number problems used to calculate the particular value by using the remaining values.

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Solved examples on number problem:

Example 1 on number problem:-

1) Twice the larger of the 2 numbers is 3 more than 5 times the smaller and the sum of four times the larger and three times the smaller is 71. What are the numbers?

The point is in the solving the problem, not in the relative reality of the problem. That said, how do you solve this number problem? The best first step is to start labeling:

The larger number:  x
the smaller number:  y

Twice the larger:  2x
three more than five times the smaller:  5y + 3
relationship between ("is"):  2x = 5y + 3

Four times the larger:  4x
three times the smaller:  3y
relationship between ("sum of"):  4x + 3y = 71

Now I have two equations in two variables:

2x = 5y + 3
4x + 3y = 71

I will solve, say, the first equation for x:

x = (5/2)y + (3/2)

Then I'll plug the right-hand side of this into the 2nd equation in place of the "x":

4[ (5/2)y + (3/2) ] + 3y = 71
10y + 6 + 3y = 71
13y + 6 = 71
13y = 65
y = 65/13 = 5

Now that I have the value for y, I can solve for x:

x = (5/2)y + (3/2)
x = (5/2)(5) + (3/2)
x = (25/2) + (3/2)
x = 28/2 = 14

The answer here is not "x = 14", but is the following sentence:

Answer: The larger number is 14, and the smaller number is 5.

Example 2 on number problem:-

2) Double the bigger of the 2 numbers is 3 more than 5 times the smaller and the sum of four times the bigger and three times the lesser is 71. What are the numbers?

The point is in the solving the number problem, not in the relative reality of the problem. That said, how do you solve this number problem? The best first step is to start labeling:

The larger number:  x
the smaller number:  y

Twice the larger:  2x
three more than five times the smaller:  5y + 3
relationship between ("is"):  2x = 5y + 3

Four times the larger:  4x
three times the smaller:  3y
relationship between ("sum of"):  4x + 3y = 71

Now I have two equations in two variables:

2x = 5y + 3
4x + 3y = 71

I will solving, say, the first equation for x:

x = (5/2)y + (3/2)

Then I'll plug the right-hand side of this into the 2nd equation in place of the "x":

4[ (5/2)y + (3/2) ] + 3y = 71
10y + 6 + 3y = 71
13y + 6 = 71
13y = 65
y = 65/13 = 5

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Now that I have the value for y, I can solving for x:

x = (5/2)y + (3/2)
x = (5/2)(5) + (3/2)
x = (25/2) + (3/2)

x =14

The answer here is not "x = 14", but is the following sentence:

Answer: The larger number is 14, and the smaller number is 5.