---
id: 5900f4331000cf542c50ff45
challengeType: 5
title: 'Problem 198: Ambiguous Numbers'
---

## Description
<section id='description'>
A best approximation to a real number x for the denominator bound d is a rational number r/s (in reduced form) with s ≤ d, so that any rational number p/q which is closer to x than r/s has q > d.

Usually the best approximation to a real number is uniquely determined for all denominator bounds. However, there are some exceptions, e.g. 9/40 has the two best approximations 1/4 and 1/5 for the denominator bound 6.
We shall call a real number x ambiguous, if there is at least one denominator bound for which x possesses two best approximations. Clearly, an ambiguous number is necessarily rational.

How many ambiguous numbers x = p/q,
0 < x < 1/100, are there whose denominator q does not exceed 108?
</section>

## Instructions
<section id='instructions'>

</section>

## Tests
<section id='tests'>

```yml
tests:
  - text: <code>euler198()</code> should return 52374425.
    testString: assert.strictEqual(euler198(), 52374425, '<code>euler198()</code> should return 52374425.');

```

</section>

## Challenge Seed
<section id='challengeSeed'>

<div id='js-seed'>

```js
function euler198() {
  // Good luck!
  return true;
}

euler198();
```

</div>



</section>

## Solution
<section id='solution'>

```js
// solution required
```
</section>