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id	title	challengeType	forumTopicId	dashedName
5900f3b31000cf542c50fec6	Problem 71: Ordered fractions	5	302184	problem-71-ordered-fractions

--description--

Consider the fraction, \frac{n}{d}, where n and d are positive integers. If n < d and highest common factor, {{HCF}(n, d)} = 1, it is called a reduced proper fraction.

If we list the set of reduced proper fractions for d ≤ 8 in ascending order of size, we get:

\frac{1}{8}, \frac{1}{7}, \frac{1}{6}, \frac{1}{5}, \frac{1}{4}, \frac{2}{7}, \frac{1}{3}, \frac{3}{8}, \frac{\textbf2}{\textbf5}, \frac{3}{7}, \frac{1}{2}, \frac{4}{7}, \frac{3}{5}, \frac{5}{8}, \frac{2}{3}, \frac{5}{7}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \frac{7}{8}

It can be seen that \frac{2}{5} is the fraction immediately to the left of \frac{3}{7}.

By listing the set of reduced proper fractions for d ≤ limit in ascending order of size, find the numerator of the fraction immediately to the left of \frac{3}{7}.

--hints--

orderedFractions(8) should return a number.

assert(typeof orderedFractions(8) === 'number');

orderedFractions(8) should return 2.

assert.strictEqual(orderedFractions(8), 2);

orderedFractions(10) should return 2.

assert.strictEqual(orderedFractions(10), 2);

orderedFractions(9994) should return 4283.

assert.strictEqual(orderedFractions(9994), 4283);

orderedFractions(500000) should return 214283.

assert.strictEqual(orderedFractions(500000), 214283);

orderedFractions(1000000) should return 428570.

assert.strictEqual(orderedFractions(1000000), 428570);

--seed--

--seed-contents--

function orderedFractions(limit) {

  return true;
}

orderedFractions(8);

--solutions--

function orderedFractions(limit) {
  const fractions = [];
  const fractionValues = {};
  const highBoundary = 3 / 7;
  let lowBoundary = 2 / 7;

  for (let denominator = limit; denominator > 2; denominator--) {
    let numerator = Math.floor((3 * denominator - 1) / 7);
    let value = numerator / denominator;
    if (value > highBoundary || value < lowBoundary) {
      continue;
    }
    fractionValues[value] = [numerator, denominator];
    fractions.push(value);
    lowBoundary = value;
  }

  fractions.sort();
  return fractionValues[fractions[fractions.length - 1]][0];
}

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