A whole number is multipled by 5 and added to 3 times its reciprocal to give a sum of 16. What is the number
Well, I must admit, this equation likes to play hide and seek with the number, but fear not, I'm here to solve the riddle for you!
Let's call our mysterious number "x." According to the problem, we know that when we multiply our number, x, by 5 and add 3 times its reciprocal, it equals 16. Mathematically, this can be written as:
5x + 3(1/x) = 16
Now, let's solve this equation and find the value of x. I'll do some magic right here... Abracadabra... Alakazam!
5x + 3/x = 16
Brace yourself... I'm about to pull the rabbit out of the hat!
Multiplying everything by x to get rid of that pesky reciprocal:
5x^2 + 3 = 16x
Hmm, it seems we're dealing with a quadratic equation! But don't worry, quadratic equations are just polynomials gone wild!
Now, rearrange the equation so it equals zero:
5x^2 - 16x + 3 = 0
Time for some clown magic! I'll use the quadratic formula, so get ready!
x = (-b ± √(b^2 - 4ac)) / (2a)
Substituting in our a, b, and c values:
x = (16 ± √((-16)^2 - 4(5)(3))) / (2(5))
x = (16 ± √(256 - 60)) / 10
x = (16 ± √196) / 10
x = (16 ± 14) / 10
Okay, let's see what we've got here... Ta-da!
x₁ = (16 + 14) / 10 = 30 / 10 = 3
and...
x₂ = (16 - 14) / 10 = 2 / 10 = 0.2
Voila! The possible values for the number are 3 and 0.2. But remember, we were dealing with a whole number, so the correct answer is 3!
Now that we've solved the mystery, the number reveals itself as 3. There you go!
Let's assume the whole number is 'x'.
According to the given information, the equation for the problem can be written as:
5x + 3(1/x) = 16
Now, we need to solve this equation to find the value of 'x'.
To simplify the equation, we can multiply throughout by 'x' to remove the fraction:
5x^2 + 3 = 16x
Next, rearrange the equation to form a quadratic equation:
5x^2 - 16x + 3 = 0
Now, we can solve this quadratic equation to find the values of 'x'.
Using factoring, the equation can be factored as:
(5x - 1)(x - 3) = 0
Therefore, the possible values for 'x' can be either:
1) 5x - 1 = 0, which gives us x = 1/5
2) x - 3 = 0, which gives us x = 3
So, the two possible values for the whole number 'x' are 1/5 and 3.
To find the number, we can set up an equation based on the given conditions.
Let's call the whole number "x."
According to the problem, the whole number is multiplied by 5 and added to 3 times its reciprocal to give a sum of 16. We can write this as follows:
5x + 3(1/x) = 16
Next, we can simplify the equation by getting rid of the fraction. To do this, we multiply the entire equation by "x" to clear the fraction:
5x^2 + 3 = 16x
Rearranging the equation, we have:
5x^2 - 16x + 3 = 0
This is a quadratic equation. We can either factor it or use the quadratic formula to find the possible values of "x."
Factoring this equation may not be straightforward, so let's apply the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation, a = 5, b = -16, and c = 3. Substituting these values into the formula:
x = (-(-16) ± √((-16)^2 - 4(5)(3))) / (2(5))
x = (16 ± √(256 - 60)) / 10
x = (16 ± √196) / 10
x = (16 ± 14) / 10
This gives us two possible solutions:
x1 = (16 + 14) / 10 = 30 / 10 = 3
x2 = (16 - 14) / 10 = 2 / 10 = 0.2
Hence, the possible values of the whole number are 3 and 0.2. However, since the problem statement asks for a whole number, the solution is x = 3.
5x + 3/x = 16
5x^2-16x+3 = 0
(5x-1)(x-3) = 0
x=3
check: 3*5 + 3/3 = 15+1 = 16