Number Problem

Word problem solving in Algebra: Number Problem

Most people don't like solving worded problem in Algebra. Except for those who are equipped with the knowledge how to do it. That is why this is made to contribute some a bit of factor to aid you dealing with it.

If you having trouble solving those worded problems, this page is for you. The given problems here was derived from usual worded problems found in some books. So it's like a "training guide" for you to continue the process of dealing it. In "this link" , it will bring you to the basic principles of how to handle it. Click and you'll see.

Now, let's start talking "Number problem". Number problems is simply dealing with the relationship of numbers. Integer, is a whole number. Let say the integer is 5. If we cut 5 into two parts it can be, 3 and 2, 1 and 4, etc. The statement "separate 5 such that one part is larger than the other by 2". What comes in your mind? You can optionally write your answer here.

Example problem 1:

"The sum of two numbers are 45. One number is twice the other. What are the numbers?"

In this case, given are two numbers. We can think of two numbers that when added altogether, the sum would be 45. If we just guess, there are a lot of numbers when added gives us 45, say 20 and 25, 40 and 5, 10 and 35, etc. The struggle starts with the phrase "twice the other". This where we can make a good start we if knew how to translate this phrase mathematically. "Twice the other" simply means doubled the other number. But what number are we going to "double"? Since we do not know yet, we can "assign" temporarily that "number" we are to double. Mostly, the assignment is letter "x", although we are allowed to use any letters. If we go to" this link", we will be guided how to manage the operation.

Given: 15, unknown number
Required: the two numbers
Solution:

Let:
x = first number
2x = second or other number, twice the smaller
x + 2x = first number and other number added together
x + 2x = 45, this is when the first number and other number added together with 45 as their sum. This is now the equation to be solved. Once you have this, your'e all set and with a bit of mathematical manipulation, you'll get what is asked.

Now:


        
          , so this is our first(smaller) number. What is the other number? We can have it by simply substituting the found-value of x.

Hence,

2x = 2(15) = 45, and this is our second (other) number.

Therefore, the two numbers are 15 and 30, 30 is twice 15 and satisfies the statement of the problem

By checking,

  x + 2x = 45
             =15 + 2(15)
             = 15 + 30
             = 45


Example problem 2:

"Three times the number minus 10 is equal to 80. Find the number"

Given this another problem, we should think of a number that is unknown for the meantime. Three times of that unknown number, diminished by 10 gives us 80. We can do it by trial and error of we don't like to deal it Algebraically. But for how long we can go? So. by just ticking this link, guidance was made to translate those vague mathematical statements into programmable one.

Given: unknown number, 10, 80
Required: find the number
Solution:

Let:
         x = the number (unknown)
       3x = three times the unknown number
3x - 10 = 80, justifies the statement "Three times the number minus 10 is equal to 80". This is now our working equation. Algebraic manipulation will bring us to the exact value of x.

So,

3x - 10 = 80
       3x = 80 + 10
       3x = 90
         x = 30, this is the unknown number, 30. We only asked for what is that number and we found it. This is the last thing to do unless you wanna make sure you're right so use the checking method.

     3x - 10 = 80
3(30) - 10 = 80
     90 - 10 = 80
             80 = 80


Example problem 3:

"There are two numbers whose sum is 44. The larger number is 1 less than twice the smaller. What are the numbers?"

Here, there's a combination involved. But let's see how the problem goes. So we have two numbers. Their sum is 44, of course we knew addition is the operator to be used. There is always, always one unknown number (problem with one unknown) to find out. After such, everything will just flow mathematically. "The larger number is twice the smaller" part of the statement gives open-door where to start to. But what is the smaller number? That is your unknown number (say x). Twice of that (x) is the larger number. What is the larger number? of course, subtract the smaller number from the bigger number like, 44 - "smaller number" which is x, and "this link" will clarify why. Let's start working on it.

Given: unknown number, 45, 1
Required: find the numbers
Solution:

Let:
         x = the smaller number, (unknown)
       2x = twice the smaller, see "this link"
  44 - x = the larger number
  2x - 1 = one less than twice the smaller
44 - x = 2x - 1, working equation, and satisfies the statement "The larger number is 1 less than twice the smaller"

Now,

 44 - x = 2x - 1
44 + 1 = 2x+ x
      45 = 3x
      15 = x ,or
        x = 15, this is the smaller number, now let's find the bigger one.

44 - x = the larger number, so substitute the value of our x, 44 - x = 44 - 15 = 29. This concludes finally that the two numbers are, 15 and 29.

Example problem 4:

"The sum of three consecutive integers is 27. What are the integers?"

This is one of the scenario that forced us to think by wild guessing sometimes. But "this link" again reminds you the interpretation. Let's solve this one.

Given: unknown number, 27
Required: find the three numbers
Solution:

Let:
               x = the first number, (unknown)
         x + 2 = second number
(x + 2) + 2 = third number

x + x + 2 + (x + 2) + 2 = 27, working equation. see how the colors were added. This satisfies the statement "The sum of three consecutive integers is 27".

So,
x + x + 2 + (x + 2) + 2 = 27
   x + x + 2 + x + 2 + 2 = 27
                         3x + 6 = 27
                               3x = 27 - 6
                               3x = 21
                                 x = 7, this is the first number.
               x + 2 = 7 + 2 = 9, the second number
(x + 2) + 2 = 7 + 2 + 2 = 11, the third number

Adding altogether, 7 + 9 + 11 = 27.

Pop Quiz:

1. Separate 90 in to two parts such that one part is four times the other part. x = 18
    Can you tell the other part? Write it in here.

2. There are four consecutive integers that has the sum of 64. What integers are they?
    Did you found the smallest integer? Write it here.

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