SOLUTION: 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

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: 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      Log On


   

Question 1198341: Given that +a+%2B++1%2F%28+b%2B2%2Fc+%29+=+17%2F5++, 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 the least terms, find the product abc.
A) 9
B) 18
C) 6
D) 30
E) 3
Found 5 solutions by josgarithmetic, MarkSingh-, math_tutor2020, MathTherapy, greenestamps:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
The way the equation appears and the description of it doe not match.
THE DESCRIPTION DOES NOT CORRESPOND TO THE EQUATION.

Answer by MarkSingh-(4) About Me  (Show Source):
You can put this solution on YOUR website!
josgarithmetic(37819), if you can't solve it, just say that

Answer by math_tutor2020(3817) About Me  (Show Source):
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Rewrite the improper fraction 17/5 into its mixed number format.
17/5 = (15+2)/5
17/5 = (15/5)+(2/5)
17/5 = 3+(2/5)
17/5 = 3 & 2/5

'a' is the integer part, or whole part, of the mixed number.
Therefore a = 3.

c is the numerator of the fractional part when the fraction is fully reduced.
We have c = 2.

Let's simplify the left hand side of the given equation.
+a+%2B++1%2F%28+b%2B2%2Fc+%29+=+17%2F5++

+a+%2B++1%2F%28+%28bc%29%2Fc%2B2%2Fc+%29+=+17%2F5++

+a+%2B++1%2F%28+%28bc%2B2%29%2Fc+%29+=+17%2F5++

+a+%2B++c%2F%28bc%2B2%29+=+17%2F5++

Then plug in a = 3 and c = 2.
Also, let's replace the 17/5 with 3 + 2/5 we found earlier.
+a+%2B++c%2F%28bc%2B2%29+=+17%2F5++

+3+%2B++2%2F%28b%2A2%2B2%29+=+17%2F5++

+3+%2B++2%2F%282b%2B2%29+=+17%2F5++

+3+%2B++2%2F%282b%2B2%29+=+3+%2B+2%2F5++
It's clear that the fractional part 2%2F%282b%2B2%29 must be equal to 2/5.
Furthermore, the denominator 2b%2B2 must be 5.

We'll use this fact to find b.
2%2F%282b%2B2%29+=+2%2F5

2b%2B2+=+5

2b+=+5-2

2b+=+3

b+=+3%2F2

b+=+1.5

------------------------

Summary:
a = 3
b = 3/2 = 1.5
c = 2

a%2Ab%2Ac+=+3%2A1.5%2A2+=+9

Answer: Choice A) 9

Answer by MathTherapy(10552) About Me  (Show Source):
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Given that +a+%2B++1%2F%28+b%2B2%2Fc+%29+=+17%2F5++, 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 the least terms, find the product abc.
A) 9
B) 18
C) 6
D) 30
E) 3
             highlight%28a%29+%2B+1%2F%28b+%2B+2%2Fc%29
matrix%281%2C5%2C+17%2F5%2C+%22=%22%2C+3%262%2F5%2C+or%2C+highlight%283%29+%2B+2%2F5%29
            
We see clearly that a = 3 

In addition, since c is the numerator of the reduced fraction in the mixed number, then c = 2

Now, we need to change the 2 in the numerator in the mixed number on the right (highlight%282%29%2F5) to MATCH
the 1 in the numerator in the mixed number on the left highlight%281%29%2F%28b+%2B+2%2Fc%29.
How do we achieve that? 
We simply divide the 2 in the numerator in the mixed number on the right by 2 to get 1. We then do the same to the 
denominator (divide it by 2), as follows: 
                                       We now see that: matrix%281%2C3%2C+5%2F2%2C+%22=%22%2C+b+%2B+2%2Fc%29 
                                                        matrix%281%2C3%2C+5%2F2%2C+%22=%22%2C+b+%2B+2%2F2%29 ---- Substituting 2 for c
                                                        matrix%281%2C3%2C+5%2F2%2C+%22=%22%2C+%282b+%2B+2%29%2F2%29
                                                         5 = 2b + 2 --- Denominators are equal and so are the numerators
                                                        matrix%282%2C3%2C+3%2C+%22=%22%2C+2b%2C+3%2F2%2C+%22=%22%2C+b%29

                                      With matrix%283%2C3%2C+a%2C+being%2C+3%2C+b%2C+being%2C+3%2F2%2C+c%2C+being%2C+2%29, we finally get: 

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CHECK
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Answer by greenestamps(13200) About Me  (Show Source):
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17/5 as a mixed number is 3 2/5. The whole number part is 3; the mixed number part is 2/5.

We are told that a is the whole number part, so a=3.
We are also told that c is the numerator of the fraction part of the mixed number, so c=2.

Then we have, for the fraction part of the mixed number,

1%2F%28b%2B2%2F2%29=2%2F5
1%2F%28b%2B1%29=2%2F5
2%28b%2B1%29=5
2b%2B2=5
2b=3
b=3%2F2

a=3; b=3/2; c=2. abc = (3)(3/2)(2) = 9.

ANSWER: A) 9