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Jack has a total of b hardback and paperback books in his...

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Jack has a total of b hardback and paperback books in his...

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Jack has a total of b hardback and paperback books in his library. If the number of hardback books is 1/3 the number of paperback books, and 3/4 of the paperback books are biographies, how many biographies, in terms of b, are in Jack's library?

A) (1/9)b
B) (3/20)b
C) (3/16)b
D) (1/3)b
E) (9/16)b

The OA is E.

I'm really confused by this PS question. Experts, any suggestion about how can I solve it? I don't understand it. Thanks in advance.

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LUANDATO wrote:
Jack has a total of b hardback and paperback books in his library. If the number of hardback books is 1/3 the number of paperback books, and 3/4 of the paperback books are biographies, how many biographies, in terms of b, are in Jack's library?

A) (1/9)b
B) (3/20)b
C) (3/16)b
D) (1/3)b
E) (9/16)b

The OA is E.

I'm really confused by this PS question. Experts, any suggestion about how can I solve it? I don't understand it. Thanks in advance.
Call the number of paperbacks 12. (A multiple of the denominators in the given fractions.)
Hardbacks: 1/3 of 12 = 4
b: 12 + 4 = 16
Paperback biographies: (3/4) * 12 = 9 --> this is our target.

(Presumably, there are no hardback biographies. The prompt could have been a little clearer on this point.)

So now we want plug in 16 in place of b in the answer choices until we find a value equal to 9.

Answer Choice E: (9/16) * 16 = 9.

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LUANDATO wrote:
Jack has a total of b hardback and paperback books in his library. If the number of hardback books is 1/3 the number of paperback books, and 3/4 of the paperback books are biographies, how many biographies, in terms of b, are in Jack's library?

A) (1/9)b
B) (3/20)b
C) (3/16)b
D) (1/3)b
E) (9/16)b

The OA is E.

I'm really confused by this PS question. Experts, any suggestion about how can I solve it? I don't understand it. Thanks in advance.
Algebraically:

If there are three times as many paperbacks as hardbacks, then 3/4 of the books will be paperback and 1/4 of the books will be hardbacks. (3/4 is three times the value of 1/4.)
Paperbacks: (3/4)b
Paperback biographies: (3/4)*(3/4) b = (9/16)b. E is the correct answer.

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LUANDATO wrote:
Jack has a total of b hardback and paperback books in his library. If the number of hardback books is 1/3 the number of paperback books, and 3/4 of the paperback books are biographies, how many biographies, in terms of b, are in Jack's library?

A) (1/9)b
B) (3/20)b
C) (3/16)b
D) (1/3)b
E) (9/16)b
We are given that Jack has a total of b hardback and paperback books in his library. We can let h = the number of hardback books and p = the number of paperback books, and thus h + p = b.

We know that the number of hardback books is 1/3 the number of paperback books, and thus:

h = (1/3)p

We can now substitute (1/3)p for h in the equation h + p = b:

(1/3)p + p = b

Multiplying the entire equation by 3, we have:

p + 3p = 3b

4p = 3b

p = 3b/4

Finally, we are given that 3/4 of the paperback books are biographies. Since p = 3b/4, we have:

(3/4)(3b/4) = biographies

9b/16 = biographies

Answer: E

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