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help with word problems

hello:  
 
here is the word problem i am having trouble with:
 
a second number is five times the first number.  a third number is a hundred times more than the first number.  if the sum of the three numbers is 415, find the numbers.
 
thanks!
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5 Answers

Vickie:
 
I'm a little confused by the wording of your problem.  Assuming that it should read "A second number is five times the first number.  A third number is a hundred more than the first number.  if the sum of the three numbers is 415, find the numbers.",    solution is, as follows:
 
Let n = the first number;  The 2nd number is therefore 5n (five times the number);  The 3rd number is n+100 (a hundred more than the 1st number)
 
Since the sum of the three numbers is 415: n+5n+(n+100)=415
 
We can now solve for n:  7n+100=415
                                     7n=415-100
                                     7n=315
                                       n=45 (this is the 1st number)
 
The 2nd number is 5n or 5*45 which equals 225
The 3rd number is n+100 which equals 145
 
We can then check our work, by adding the 3 numbers:  45+225+145=415 (check)
 
Hi Vickie,
 
Are you sure you wrote the problem correctly?
 
There is a whole number solution if the 3rd number is "a hundred more than the first number." If the third number is, in fact, a hundred times more than the third number, then your solutions will be decimals.
 
I will help you solve the problem assuming the 3rd number is a hundred more than the first.
 
This problem has three unknown values: the 1st, 2nd, and 3rd number. To avoid using three different variables, we can write each value in terms of the 1st number.
 
The 1st number is the value for which we are given no information. So let's use x to represent that 1st number.
 
If we use to represent the 1st number, we can write the 2nd number as 5x (or five times x).
Then we can write the 3rd number as x+100 (or 100 more than the 1st number.)
 
We have:
1st number = x
2nd number = 5x
3rd number = x +100
 
The problem also tell us that the sum of the three number is 415.
To find the sum, we need to add all the values : x + 5x + x + 100 = 415
When we combine like terms our equation reads 7x + 100 = 415
 
To solve for x, we need to isolate the x. That can be done by using inverse operations to "undo" what has been done to the x. 7x+100 means that x is being multiplied by 7, and is being added to 100. To undo, subtract 100 from both sides. We are subtracting from both sides so that this remains an equality.
 
7x + 100 - 100 = 415 - 100
7x = 315
 
To get x alone on the left side of the equation, we must divide both sides by 7.
 
7x/7 = 315/7
x = 45
 
Since x = 45, the 1st number is 45. That would make the 2nd number 225, since 5 times 45 is 225. Finally, the third number is 145, since 100 more than 45 is 145.
 
To check our work, we add all three values together to achieve a sum of 415
 
45 + 225 + 145 = 315
 
I hope this helped, Vickie!
When solving problems such as this, always choose a variable x to represent the answer and perform the operations as described in the problem.  Let x be the first number.
 
The second number is five times the first number, so the second number is 5x
 
The third number is 100 times more than the first number.  If this is meant to mean that the third number is 100 times the first number and since we know the first number is x, the third number is 100x.
 
We now have the three numbers in terms of x as x, 5x and 100x.
 
The term Sum in word problems means the result of adding numbers.  Since the sum of the three numbers is given as 415 and the sum from our numbers in terms of x is x + 5x + 100x = 106x
 
The given value should therefore be equal to the sum in terms of x, or 
 
106x=415
 
x ~ 3.915
The first number we will call X
That makes the second number 5X
The third number would be X + 100
 
so the equation would be X + 5X + X +100 = 415
Simplifying the equation 7X + 100 = 415
Subtract 100                      - 100 = -100
 
                                            7X = 315
Divide by 7                             7       7
 
therefore X = 45
 
Dan
 
Ok so this is a step by step, trying to convert words into numbers. First start out identifying how you will represent your numbers. Lets call them X, Y, and Z.
So lets write equations of what we know. The second number is five times the first number. So if Y is our second number then 
 
Y = 5X
 
Then a third number is a hundred times more than the first number. So that means 
Z = 100X
 
and we know when u total the 3 numbers
 
X + Y + Z = 415
 
This is what is called a system of equations. To solve it, we need to get everything in terms of 1 variable. So lets use X since it will be the biggest help to know...and well plug everything we know so far into our last equation. SO...
 
X + (5X) + 100X = 415
Simplify
106X = 415
 
Solve for X
X = 3.9 (rounded)
 
So that means that 
 Y= 5*3.9 = 19.5  AND Z = 100* 3.9 = 390
 
Tada!