Tag Archive for 'fraction'

Project Euler : Expansion Of Continued Fraction For Square Root Of Two

Published on October 4, 2011 in Problems. 0 Comments

Problem 57 of Project Euler is pretty simple forward. It is very easy to see the following relation:

Numerator = Numerator + 2 * Denominator

Denominator = Numerator + Denominator

Simple code and was done. Here’s that:

    public int getCount(int numberOfExpansions) {
        int count = 0;

        BigInteger numerator = BigInteger.valueOf(3);
        BigInteger denominator = BigInteger.valueOf(2);
        while(--numberOfExpansions > 0) {
            BigInteger a = denominator.multiply(NumberUtil.BIG_TWO).add(numerator);
            BigInteger b = numerator.add(denominator);
            numerator = a;
            denominator = b;
            if(NumberUtil.numberOfDigits(numerator) > NumberUtil.numberOfDigits(denominator)) count++;
        }

        return count;
    }

Popularity: 4% [?]

Project Euler : Fractions With Unorthodox Cancelling Method

Published on August 10, 2010 in Problems. 0 Comments

Problem 33 in Project Euler says there are 4 non-trivial curious fractions which satisfy the following properties:

\frac{ax}{xb} = \frac{a}{b}

where a, b and x are all single digit numbers and a < b

The solution was pretty simple.

\frac{ax}{xb} = \frac{a}{b}

\Rightarrow b [10a + x] = a [10x + b]

\Rightarrow x = \frac{9ab}{10a - b}

Similarly solving for the other form \frac{xa}{bx} = \frac{a}{b} gives the answer x = \frac{9ab}{10b - a}.

Implementing this solution in java:

public class Problem_33 {
	/**
	 * @return All the proper non-trivial curious fractions having numerator, denominator < 100
	 */
	public static List<Fraction> getCuriousFractions() {
		List<Fraction> curiousFractions = new ArrayList<Fraction>();

		for (int a = 1; a < 10; a++)
			for (int b = a + 1; b < 10; b++)
				if (Problem_33.checkCondition(a, b))
					curiousFractions.add(new Fraction(a, b));

		return curiousFractions;
	}

	/**
	 * Checks if the number is of the form xa/bx = a/b or ax/xb = a/b
	 *
	 * @param a The numerator of the fraction
	 * @param b The denominator of the fraction
	 * @return true if it can be formed into a curious fraction, false otherwise
	 */
	private static boolean checkCondition(int a, int b) {
		int numerator = 9 * a * b;
		int denominatorA = 10 * b - a;
		int denominatorB = 10 * a - b;
		return numerator % denominatorA == 0 && numerator / denominatorA < 10 || numerator % denominatorB == 0
		        && numerator / denominatorB < 10 ? true : false;
	}
}

Popularity: 9% [?]