Natural numbers bounded fractions

CAT 2020 Slot 3 · QA · Hard · Inequalities

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Let mm and nn be natural numbers such that nn is even and 0.2<m20,nm,n11<0.50.2 < \frac{m}{20}, \frac{n}{m}, \frac{n}{11} < 0.5. Then m−2nm - 2n equals

  1. A.

    4

  2. B.

    2

  3. C.

    1

  4. D.

    3

Answer

C

Explanation

From 0.2<n11<0.50.2 < \frac{n}{11} < 0.5: 2.2<n<5.52.2 < n < 5.5. Since nn is an even natural number, n=4n = 4.

Now substitute n=4n = 4 into the inequalities:

  1. 0.2<m20<0.5  ⟹  4<m<100.2 < \frac{m}{20} < 0.5 \implies 4 < m < 10.
  2. 0.2<4m<0.5  ⟹  8<m<200.2 < \frac{4}{m} < 0.5 \implies 8 < m < 20.

Combining 4<m<104 < m < 10 and 8<m<208 < m < 20 gives 8<m<108 < m < 10. Since mm is a natural number, m=9m = 9.

Therefore, m−2n=9−2(4)=1m - 2n = 9 - 2(4) = 1.

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