function gcd(a, b) {
a = Math.abs(a);
b = Math.abs(b);
while (b) {
var temp = b;
b = a % b;
a = temp;
}
return a;
}
function calculateFractionDivision() {
var num1 = parseFloat(document.getElementById('numerator1').value);
var den1 = parseFloat(document.getElementById('denominator1').value);
var num2 = parseFloat(document.getElementById('numerator2').value);
var den2 = parseFloat(document.getElementById('denominator2').value);
var resultFractionElem = document.getElementById('resultFraction');
var resultDecimalElem = document.getElementById('resultDecimal');
var calculationStepsElem = document.getElementById('calculationSteps');
resultFractionElem.innerHTML = ";
resultDecimalElem.innerHTML = ";
calculationStepsElem.innerHTML = ";
if (isNaN(num1) || isNaN(den1) || isNaN(num2) || isNaN(den2)) {
calculationStepsElem.innerHTML = 'Please enter valid numbers for all fields.';
return;
}
if (den1 === 0 || den2 === 0) {
calculationStepsElem.innerHTML = 'Denominator cannot be zero.';
return;
}
if (num2 === 0) {
calculationStepsElem.innerHTML = 'The numerator of the second fraction (divisor) cannot be zero. Division by zero is undefined.';
return;
}
// Keep, Change, Flip method
// (num1 / den1) / (num2 / den2) = (num1 / den1) * (den2 / num2)
var newNumerator = num1 * den2;
var newDenominator = den1 * num2;
// Simplify the resulting fraction
var commonDivisor = gcd(newNumerator, newDenominator);
var simplifiedNumerator = newNumerator / commonDivisor;
var simplifiedDenominator = newDenominator / commonDivisor;
// Determine sign for simplified fraction
var sign = 1;
if ((newNumerator 0) || (newNumerator > 0 && newDenominator < 0)) {
sign = -1;
}
simplifiedNumerator = Math.abs(simplifiedNumerator) * sign;
simplifiedDenominator = Math.abs(simplifiedDenominator);
// Handle cases where simplifiedDenominator is 1 or -1
if (simplifiedDenominator === 1) {
resultFractionElem.innerHTML = simplifiedNumerator;
} else if (simplifiedDenominator === -1) { // Should not happen if we handle sign correctly
resultFractionElem.innerHTML = -simplifiedNumerator;
}
else {
resultFractionElem.innerHTML = simplifiedNumerator + ' / ' + simplifiedDenominator;
}
var decimalResult = newNumerator / newDenominator;
resultDecimalElem.innerHTML = decimalResult.toFixed(6); // Display with 6 decimal places
calculationStepsElem.innerHTML =
'Step 1: Keep, Change, Flip' +
'Original: (' + num1 + ' / ' + den1 + ') ÷ (' + num2 + ' / ' + den2 + ')' +
'Flip the second fraction and multiply: (' + num1 + ' / ' + den1 + ') × (' + den2 + ' / ' + num2 + ')' +
'Step 2: Multiply Numerators and Denominators' +
'New Numerator: ' + num1 + ' × ' + den2 + ' = ' + newNumerator + " +
'New Denominator: ' + den1 + ' × ' + num2 + ' = ' + newDenominator + " +
'Resulting fraction: ' + newNumerator + ' / ' + newDenominator + " +
'Step 3: Simplify the Fraction' +
'Greatest Common Divisor (GCD) of ' + newNumerator + ' and ' + newDenominator + ' is ' + commonDivisor + " +
'Simplified Numerator: ' + newNumerator + ' ÷ ' + commonDivisor + ' = ' + simplifiedNumerator + " +
'Simplified Denominator: ' + newDenominator + ' ÷ ' + commonDivisor + ' = ' + simplifiedDenominator + ";
}
// Initial calculation on page load
window.onload = calculateFractionDivision;
Understanding Fraction Division
Dividing fractions might seem intimidating at first, but it's a fundamental operation in mathematics with a straightforward method. Essentially, when you divide by a fraction, you're asking how many times one fraction fits into another. The key to solving these problems lies in a simple technique known as "Keep, Change, Flip."
What is Fraction Division?
Fraction division is the process of dividing one fraction by another. For example, if you have 3/4 of a pizza and you want to divide it among friends, giving each friend 1/8 of a pizza, you'd use fraction division to find out how many friends can get a slice. The operation is represented as (a/b) ÷ (c/d).
The "Keep, Change, Flip" Method
This method simplifies fraction division into a multiplication problem, which is generally easier to solve. Here's how it works:
Keep: Keep the first fraction exactly as it is.
Change: Change the division sign (÷) to a multiplication sign (×).
Flip: Flip the second fraction (the divisor). This means you swap its numerator and its denominator. The flipped fraction is called the reciprocal.
Once you've applied these three steps, you simply multiply the two fractions as you normally would: multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. Finally, simplify the resulting fraction if possible.
Formula for Fraction Division
Given two fractions, a/b and c/d, the division is calculated as follows:
(a / b) ÷ (c / d) = (a / b) × (d / c) = (a × d) / (b × c)
Example Calculation
Let's divide 3/4 by 1/2 using the "Keep, Change, Flip" method:
Keep the first fraction: 3/4
Change the division sign to multiplication: 3/4 × ...
Flip the second fraction (1/2 becomes 2/1): 3/4 × 2/1
Now, multiply the fractions:
Multiply the numerators: 3 × 2 = 6
Multiply the denominators: 4 × 1 = 4
The result is 6/4. This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
6 ÷ 2 = 3
4 ÷ 2 = 2
So, 3/4 ÷ 1/2 = 3/2, or 1 and 1/2 as a mixed number, or 1.5 as a decimal.
Why is it Important?
Fraction division is crucial in many real-world scenarios, from cooking and baking (scaling recipes up or down) to engineering and finance. It helps us understand proportions, ratios, and how quantities relate to each other when they are expressed as parts of a whole. Mastering this concept builds a strong foundation for more advanced mathematical topics.