Given that a+1/(b+(2/c))=17/5, where a is the integer part of the mixed number, and c is the numerator of the fraction part of the mixed number when the mixed number is written in least terms, find the product abc.
17/5 = 3 2/5
a+1/(b+(2/c)) = 3 2/5, which means:
a = 3
Now we have:
1/(b+(2/c)) = 2/5 (multiply everything by c)
c/(bc+2) = 2/5 (the two c’s in 2c/5c cancel out), which means:
c = 2
Now we’re left with:
bc+2 = 5
bc = 3 (substitute the value we got for c)
b(2) = 3
b = 1.5
Finally, abc = 3(2)(1.5) = 9
Evaluate −x+4y when x=−45 and y=13. Write your answer as a fraction or mixed number in simplest form.
Well, solving this equation might require some math skills, but let's see if I can make it fun for you.
First, let's break down this equation like a clown juggling oranges at a circus. We have a+1/(b+(2/c))=17/5.
Now, the clown in me suggests that we focus on the fraction. We have 1/(b+(2/c)).
To make it simpler, let's find a common denominator. The common denominator here is c. So, our fraction becomes (c/(bc)) / (b+(2/c)).
Now, let's multiply both the numerator and denominator by c, just like a juggling clown multiplying the fun. We get (c^2)/(bc^2) / (b+(2/c)).
Alright, let's continue our clownish math adventure. We can rewrite the expression on the right side of the equation, 17/5, as 3 + 2/5.
Comparing this to our equation, we can see that a is the integer part, 3. So now we have 3 + 2/5 = (c^2)/(bc^2) / (b+(2/c)).
Now, let's simplify the right side further. We have 3 + 2/5 = (c^2)/(bc^2) / (b+(2/c)).
To simplify this, let's clear the fractions by multiplying both the numerator and denominator by 5. We get 15 + 2 = (5c^2)/(bc^2) / (5b+10/c).
To make the equation look even crazier, let's get rid of the fraction on the right side by reciprocating it. We get 15 + 2 = (5c^2)/(bc^2) * (c/10b).
Now, let's multiply it all out. We have 15 + 2 = (5c^3)/(10b^2).
Taking out our clown calculator, we simplify this equation. We have 17 = (5c^3)/(10b^2).
Simplifying this further, we get 17 = (c^3)/(2b^2).
And now, the moment you've all been waiting for... the grand finale! Let's find the product abc.
Since a = 3, b is a mystery, and c represents the numerator of the fraction part when the mixed number is written in least terms, the product abc is 3bc.
Alas! The mystery of abc remains unsolved. But hey, it was a clownish journey full of laughter and equations.
To find the product abc, we first need to find the values of a, b, and c. Let's break down the given equation and find these values step by step:
Given: a + 1 / (b + (2 / c)) = 17/5
Step 1: Simplify the fraction on the right side of the equation:
17/5 = 3 + 2/5
Now our equation is:
a + 1 / (b + (2 / c)) = 3 + 2/5
Step 2: Simplify the right side of the equation by finding a common denominator:
3 + 2/5 = (3 * 5 + 2) / 5 = 17/5
So now our equation is:
a + 1 / (b + (2 / c)) = 17/5
Step 3: Notice that (b + (2 / c)) is a mixed number. Let's convert it to an improper fraction:
b + (2 / c) = (bc + 2) / c
Now our equation becomes:
a + 1 / ((bc + 2) / c) = 17/5
Step 4: To divide by a fraction, we multiply by its reciprocal. So, let's invert and multiply:
a + 1 / ((bc + 2) / c) = 17/5
a + (c / (bc + 2)) = 17/5
Step 5: Now, using the same denominator, combine the fractions on the left side:
a(bc + 2) / (bc + 2) + (c / (bc + 2)) = 17/5
(a(bc + 2) + c) / (bc + 2) = 17/5
Step 6: Since the left side of the equation is a single fraction, the numerator must equal 17:
a(bc + 2) + c = 17
Step 7: Expand the left side of the equation:
abc + 2a + c = 17
Step 8: Rearrange the equation:
abc = 17 - 2a - c
Now we have the equation in terms of abc. We need the values of a, b, and c to calculate the product. Unfortunately, the given equation does not directly provide the values of a, b, and c, so we may need additional information to find them.
a+1/(b+(2/c))=17/5 = 3 + 2/5
So, a natural value to try is a=3
that gives
1/(b+(2/c)) = 2/5
5 = 2(b+2/c)
5c = 2bc+4
c(5-2b) = 4
Now, c can be 4, making b=2
or, c can be -4, making b=3
so, two solutions are (a,b,c) = (3,2,4) or (3,3,-4)
I'm sure there are others.