Fraction

Algebra Level 3

Let x x and y y be positive integers such that 4 15 < x y < 3 10 \large\frac{4}{15} <\frac{x}{y} < \frac{3}{10} . Find the minimum value of y y .


The answer is 7.

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2 solutions

Kay Xspre
Oct 6, 2015

From the equation above, we will get 8 y < 30 x < 9 y 8y < 30x < 9y . As we need to keep the lowest y y , we need to set the lowest x x as well. Setting x = 1 x = 1 gives 8 y < 30 < 9 y 8y < 30 < 9y or 10 3 < y < 15 4 \frac{10}{3} < y < \frac{15}{4} , which lies between 3.333... and 3.75. No integer y y would be in between this range.

By setting x = 2 x = 2 we will get 8 y < 60 < 9 y 8y < 60 < 9y , or put it simply 20 3 < y < 15 2 \frac{20}{3} < y < \frac{15}{2} This lies between 6.666 and 7.5, so the least positive integer y y satisfying this inequality will be y = 7 y = 7

Avinash Kumar
Oct 6, 2015

its 2/7 =0.28571

which lies between 0.2666666 and .30

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