the larger of two numbers is three times the smaller number, and their difference id 34. find the two numbers.
Ah, numbers playing mind games, I see! Let me help you unravel this mathematical riddle.
Let's call the smaller number "x" and the larger number "y."
According to the information given, we know that y = 3x (the larger number is three times the smaller number) and y - x = 34 (their difference is 34).
Now, we can solve for x by substituting y in terms of x in the second equation:
3x - x = 34
2x = 34
x = 34/2
x = 17
Now that we have the value of x (the smaller number), we can substitute it back into the first equation to find y:
y = 3x
y = 3 * 17
y = 51
Therefore, the smaller number is 17 and the larger number is 51. Voila!
Let's assume the smaller number is "x".
According to the given information, the larger number is three times the smaller number. So, the larger number is 3x.
The difference between the two numbers is 34. This can be expressed as:
3x - x = 34.
Simplifying the equation:
2x = 34.
Dividing both sides by 2:
x = 34 / 2,
x = 17.
So, the smaller number is 17.
Now, we can find the larger number by substituting the value of x:
Larger number = 3x = 3 * 17 = 51.
Therefore, the two numbers are 17 and 51.
To solve this problem, let's assign variables to the two numbers. Let's say the larger number is represented by 'x' and the smaller number is represented by 'y'.
According to the problem statement, the larger number is three times the smaller number. So we can express this relationship as:
x = 3y -- Equation 1
The problem also states that their difference is 34. Mathematically, the difference between two numbers can be calculated by subtracting the smaller number from the larger number:
x - y = 34 -- Equation 2
We now have a system of two equations (Equation 1 and Equation 2) with two unknowns (x and y). We can use these equations to solve for x and y.
Method 1: Substitution Method
Using Equation 1, we can substitute the expression for x in terms of y into Equation 2:
(3y) - y = 34
2y = 34
y = 17
Substituting this value of y back into Equation 1, we can find x:
x = 3(17)
x = 51
So, the two numbers are 51 and 17.
Method 2: Elimination Method
We can also solve the system of equations by eliminating one variable. Multiply Equation 1 by -1, which gives us:
-x = -3y
Then we can add this new equation to Equation 2:
(-x) + (x - y) = (-3y) + 34
-x + x - y = -3y + 34
Simplifying,
-y = -3y + 34
Rearranging the equation,
2y = 34
Solving for y,
y = 17
Substituting the value of y back into Equation 1, we can find x:
x = 3y
x = 3(17)
x = 51
So, the two numbers are 51 and 17.
Both methods yield the same solution: the larger number is 51 and the smaller number is 17.