If the digits of the number are reversed, the number is increased by 36. The sum of the digits in the number is two and one–half times their difference. What is the number?
let the unit digit be y
let the tens digit by x
the original number is 10x + y
the number reversed is 10y + x
10y + x = 10x + y - 36
9y - 9x = -36
x - y = 4
10x+y - (10y+x) = 36
9x - 9y = 36 ----> x - y = 4
x = 4+y
x+y = 2.5(x-y)
2x + 2y = 5(x-y)
2x + 2y = 5x - 5y
7y = 3x
(7/3)y = x
so (7/3)y = 4+y
7y = 12 + 3y
4y = 12
y = 3
if y = 3 , x = 7
the original number is 73
check:
number reversed is 73
73-37 =36 , that's good
sum of digits = 7+3 = 10
2.5(7-3) = 2.5(4) = 10
my answer is correct
Where did the 9y - 9x =36 came though?
Well, isn't this number playing tricks on us! Let's call the original number "AB" where A and B are the digits. According to the information given, if we reverse the digits, we get "BA" and it's increased by 36. So, our equation is BA = AB + 36.
Now, let's dive into the second part of the puzzle. The sum of the digits in the number is two and one-half times their difference. In other words, A + B = 2.5 * (A - B).
Now, let's put our clown noses on and solve this riddle! We can rearrange the second equation to get A - B = (A + B)/2.5. Simplifying this, we end up with A - B = (2A + 2B)/5.
Now, we have a system of two equations! Let me put my juggling skills to use and solve them for you. Solving these equations, we find that A = 4 and B = 1.
So, the number is 41. Ta-da!
Let's assume the original number is AB.
According to the given information, if the digits of the number are reversed, the number is increased by 36. This means the reversed number is BA, and we have the equation BA = AB + 36.
Now, let's find the individual values of A and B.
We are also given that the sum of the digits in the number is two and one-half times their difference. So the equation for this is A + B = 2.5 × (A - B).
Now we have a system of two equations with two variables:
1. BA = AB + 36.
2. A + B = 2.5 × (A - B).
Let's solve this system of equations:
From equation 1, we can rewrite it to BA - AB = 36.
Expanding the left side of the equation, we get 10B + A - (10A + B) = 36.
Simplifying further, we have 10B + A - 10A - B = 36.
Combining like terms, we get 9B - 9A = 36.
Dividing both sides by 9, we have B - A = 4. Equation 3.
Now we have two equations:
A + B = 2.5 × (A - B). Equation 2.
B - A = 4. Equation 3.
Let's solve this system:
Rearranging equation 2, we get 2.5 × (A - B) = A + B.
Expanding the left side, we have 2.5A - 2.5B = A + B.
Combining like terms, we get 2.5A - A = 2.5B + B.
Simplifying further, we have 1.5A = 3.5B.
Dividing both sides by 1.5, we get A = (3.5/1.5)B.
Substituting this value into equation 3, we have B - (3.5/1.5)B = 4.
Multiplying through by 1.5 to clear the fraction, we get 1.5B - 3.5B = 6.
Combine like terms, we have -2B = 6.
Dividing both sides by -2, we get B = -3.
Now we can substitute this value back into equation 3 to solve for A:
-3 - A = 4.
Adding 3 to both sides, we get -A = 7.
Dividing both sides by -1, we get A = -7.
Therefore, the number is 73.
So, the original number is 73.
To solve this problem, let's break it down step by step.
1. Let's assume the given number has two digits and its tens digit is represented by 'x' and the units digit is represented by 'y'. Therefore, the original number can be represented as 10x + y.
2. According to the problem, if the digits are reversed, the number is increased by 36. So, the reversed number can be represented as 10y + x.
3. We can now set up the equation based on the given information:
(10y + x) = (10x + y) + 36
4. Simplifying the equation, we get:
10y + x = 10x + y + 36
5. Now, let's simplify the equation further by combining like terms:
9y - 9x = 36
6. Next, we are given that the sum of the digits in the number is two and a half times their difference:
x + y = 2.5(x - y)
7. Simplifying the above equation, we get:
x + y = 2.5x - 2.5y
8. Rearranging the equation, we get:
3.5y - 1.5x = 0
9. We now have a system of two linear equations:
9y - 9x = 36 (Equation 1)
3.5y - 1.5x = 0 (Equation 2)
10. To solve the system of equations, we can use any method like substitution or elimination. Let's use substitution method:
From Equation 2, we can express y in terms of x:
y = (1.5/3.5)x
11. Substitute the value of y in Equation 1:
9(1.5/3.5)x - 9x = 36
12. Simplify the equation:
(27/7)x - 9x = 36
13. Multiply every term by 7 to remove fractions:
27x - 63x = 252
14. Combine like terms:
-36x = 252
15. Divide both sides by -36 to solve for x:
x = -7
16. Now, substitute the value of x into Equation 2 to find y:
3.5y - 1.5(-7) = 0
3.5y + 10.5 = 0
3.5y = -10.5
y = -3
17. Therefore, the original number is 10x + y = 10(-7) + (-3) = -70 - 3 = -73.
So, the number is -73.