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Math Help - Find a fraction between 97/36 and 96/35 with smallest denominator

  1. #1
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    Find a fraction between 97/36 and 96/35 with smallest denominator

    Find a fraction between 97/36 and 96/35 which has the smallest denominator. Since 97/36 > 96/35, I formed the following inequality

    \frac {97}{36} < \frac pq < \frac{96}{35} \, .

    Now, appropriately so, we could write the given fractions with common denominators and thusly yield

    \frac {3395}{1260} < \frac pq < \frac{3456}{1260} \, .

    Now this is where I am stuck. The correct answer is p = 19 and q = 7, which corresponds to the fraction 3420/1260.
    Anyone got any idea?
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: Find a fraction between 97/36 and 96/35 with smallest denominator

    Quote Originally Posted by MathCrusader View Post
    Find a fraction between 97/36 and 96/35 which has the smallest denominator. Since 97/36 > 96/35, I formed the following inequality

    \frac {97}{36} < \frac pq < \frac{96}{35} \, .

    Now, appropriately so, we could write the given fractions with common denominators and thusly yield

    \frac {3395}{1260} < \frac pq < \frac{3456}{1260} \, .

    Now this is where I am stuck. The correct answer is p = 19 and q = 7, which corresponds to the fraction 3420/1260.
    Anyone got any idea?
    Hi MathCrusader!

    To simplify a fraction, you need to know how the numerators and denominators factorize.
    Can you create prime number factorizations of all the numbers you have?
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  3. #3
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    Re: Find a fraction between 97/36 and 96/35 with smallest denominator

    97 is prime.
    96 = 2^5 \cdot 3
    35 = 7 \cdot 5
    36 = 3^2 \cdot 2^2
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  4. #4
    Super Member ILikeSerena's Avatar
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    Re: Find a fraction between 97/36 and 96/35 with smallest denominator

    Quote Originally Posted by MathCrusader View Post
    Find a fraction between 97/36 and 96/35 which has the smallest denominator. Since 97/36 > 96/35, I formed the following inequality

    \frac {97}{36} < \frac pq < \frac{96}{35} \, .

    Now, appropriately so, we could write the given fractions with common denominators and thusly yield

    \frac {3395}{1260} < \frac pq < \frac{3456}{1260} \, .

    Now this is where I am stuck. The correct answer is p = 19 and q = 7, which corresponds to the fraction 3420/1260.
    Anyone got any idea?
    Quote Originally Posted by MathCrusader View Post
    97 is prime.
    96 = 2^5 \cdot 3
    35 = 7 \cdot 5
    36 = 3^2 \cdot 2^2
    Good!

    So you have:

    \frac {97\cdot 35}{35 \cdot 36} < \frac pq < \frac{96 \cdot 36}{35 \cdot 36} \, .

    \frac {3395}{1260} < \frac pq < \frac{3456}{1260} \, .

    \frac {3395}{2^2 \cdot 3^2 \cdot 5 \cdot 7 } < \frac pq < \frac{3456}{2^2 \cdot 3^2 \cdot 5 \cdot 7} \, .

    Now we are looking for a number between 3395 and 3456 that has as much factors in common as possible with 1260.

    I think this is easiest with just trial and error, combining one or more prime factors into a numbers starting with the lowest possible.
    Start with q=2, and see if you can find an acceptable value for p.
    Then q=3, q=4, q=5, q=6, q=7... and there you go!
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