Adding Fractions with Different Denominators Calculator

Adding Fractions with Different Denominators Calculator

Fraction 1

Fraction 2

Step-by-Step Explanation:

// Function to calculate Greatest Common Divisor (GCD) using Euclidean algorithm 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 to calculate Least Common Multiple (LCM) function lcm(a, b) { if (a === 0 || b === 0) return 0; return Math.abs(a * b) / gcd(a, b); } function calculateFractions() { var num1 = parseInt(document.getElementById('numerator1').value); var den1 = parseInt(document.getElementById('denominator1').value); var num2 = parseInt(document.getElementById('numerator2').value); var den2 = parseInt(document.getElementById('denominator2').value); var resultDiv = document.getElementById('result'); var explanationDiv = document.getElementById('explanation'); var explanationContent = '

Step-by-Step Explanation:

'; // Input validation if (isNaN(num1) || isNaN(den1) || isNaN(num2) || isNaN(den2)) { resultDiv.innerHTML = 'Please enter valid numbers for all fields.'; resultDiv.style.borderColor = '#dc3545'; resultDiv.style.backgroundColor = '#f8d7da'; resultDiv.style.color = '#dc3545'; explanationDiv.innerHTML = explanationContent + 'Invalid input detected. Please ensure all fields contain numbers.'; return; } if (den1 === 0 || den2 === 0) { resultDiv.innerHTML = 'Denominator cannot be zero.'; resultDiv.style.borderColor = '#dc3545'; resultDiv.style.backgroundColor = '#f8d7da'; resultDiv.style.color = '#dc3545'; explanationDiv.innerHTML = explanationContent + 'Division by zero is undefined. Please enter non-zero denominators.'; return; } explanationContent += 'You want to add the fractions: ' + num1 + '/' + den1 + ' and ' + num2 + '/' + den2 + '.'; // Step 1: Find the Least Common Multiple (LCM) of the denominators var commonDen = lcm(den1, den2); explanationContent += 'Step 1: Find the Least Common Multiple (LCM) of the denominators.'; explanationContent += 'The denominators are ' + den1 + ' and ' + den2 + '.'; explanationContent += 'The LCM of ' + den1 + ' and ' + den2 + ' is ' + commonDen + '.'; // Step 2: Convert each fraction to an equivalent fraction with the common denominator var factor1 = commonDen / den1; var newNum1 = num1 * factor1; var factor2 = commonDen / den2; var newNum2 = num2 * factor2; explanationContent += 'Step 2: Convert each fraction to an equivalent fraction with the common denominator.'; explanationContent += 'For ' + num1 + '/' + den1 + ':'; explanationContent += 'Multiply numerator and denominator by ' + factor1 + ' (since ' + den1 + ' * ' + factor1 + ' = ' + commonDen + ').'; explanationContent += 'New fraction 1: ' + num1 + ' * ' + factor1 + ' / ' + den1 + ' * ' + factor1 + ' = ' + newNum1 + '/' + commonDen + '.'; explanationContent += 'For ' + num2 + '/' + den2 + ':'; explanationContent += 'Multiply numerator and denominator by ' + factor2 + ' (since ' + den2 + ' * ' + factor2 + ' = ' + commonDen + ').'; explanationContent += 'New fraction 2: ' + num2 + ' * ' + factor2 + ' / ' + den2 + ' * ' + factor2 + ' = ' + newNum2 + '/' + commonDen + '.'; // Step 3: Add the numerators var sumNum = newNum1 + newNum2; explanationContent += 'Step 3: Add the numerators of the new fractions.'; explanationContent += " + newNum1 + ' + ' + newNum2 + ' = ' + sumNum + '.'; explanationContent += 'The sum is now ' + sumNum + '/' + commonDen + '.'; // Step 4: Simplify the resulting fraction var commonDivisor = gcd(sumNum, commonDen); var simplifiedNum = sumNum / commonDivisor; var simplifiedDen = commonDen / commonDivisor; explanationContent += 'Step 4: Simplify the resulting fraction (if possible).'; if (commonDivisor > 1) { explanationContent += 'The Greatest Common Divisor (GCD) of ' + sumNum + ' and ' + commonDen + ' is ' + commonDivisor + '.'; explanationContent += 'Divide both numerator and denominator by ' + commonDivisor + ':'; explanationContent += simplifiedNum + ' / ' + simplifiedDen + '.'; } else { explanationContent += 'The fraction ' + sumNum + '/' + commonDen + ' cannot be simplified further as the GCD of ' + sumNum + ' and ' + commonDen + ' is 1.'; } var finalResult = simplifiedNum + '/' + simplifiedDen; if (simplifiedDen === 1) { finalResult = simplifiedNum; // If denominator is 1, it's a whole number } resultDiv.innerHTML = 'The sum is: ' + finalResult + ''; resultDiv.style.borderColor = '#28a745'; resultDiv.style.backgroundColor = '#e2f0e5'; resultDiv.style.color = '#28a745'; explanationDiv.innerHTML = explanationContent; } // Initial calculation on load document.addEventListener('DOMContentLoaded', calculateFractions);

Understanding How to Add Fractions with Different Denominators

Adding fractions can seem daunting, especially when their denominators (the bottom numbers) are not the same. However, with a clear step-by-step approach, it becomes a straightforward process. This calculator is designed to help you understand and perform this operation with ease.

What is a Fraction?

A fraction represents a part of a whole. It consists of two numbers: the numerator (the top number), which tells you how many parts you have, and the denominator (the bottom number), which tells you how many equal parts make up the whole.

For example, in the fraction 1/2, the numerator is 1 (you have one part), and the denominator is 2 (the whole is divided into two equal parts).

Why Do Denominators Need to Be the Same for Addition?

Imagine you have 1/2 of a pizza and 1/3 of a different-sized pizza. You can't simply add the numerators (1+1=2) to get 2/something, because the "parts" (halves and thirds) are not the same size. To add them, you need to express both quantities in terms of equally sized pieces. This is where a common denominator comes in.

The Steps to Add Fractions with Different Denominators:

1. Find the Least Common Multiple (LCM) of the Denominators:

   The LCM is the smallest positive integer that is a multiple of both denominators. This will be your new common denominator. For example, if you're adding 1/2 and 1/3, the multiples of 2 are 2, 4, 6, 8… and the multiples of 3 are 3, 6, 9… The LCM is 6.

2. Convert Each Fraction to an Equivalent Fraction with the Common Denominator:

   For each fraction, determine what you need to multiply its original denominator by to get the LCM. Then, multiply both the numerator and the denominator by that same number. This creates an equivalent fraction that has the common denominator without changing its value.

   Using our example (1/2 and 1/3, LCM is 6):

   • For 1/2: To get a denominator of 6, you multiply 2 by 3. So, multiply the numerator (1) by 3 as well: (1 * 3) / (2 * 3) = 3/6.
   • For 1/3: To get a denominator of 6, you multiply 3 by 2. So, multiply the numerator (1) by 2 as well: (1 * 2) / (3 * 2) = 2/6.
3. Add the Numerators:

   Once both fractions have the same denominator, you can simply add their numerators. The denominator remains the common denominator you found.

   Continuing the example: 3/6 + 2/6 = (3 + 2) / 6 = 5/6.

4. Simplify the Resulting Fraction (if possible):

   After adding, the resulting fraction might be able to be simplified. To do this, find the Greatest Common Divisor (GCD) of the new numerator and denominator. Divide both by their GCD to get the fraction in its simplest form.

   In our example, 5/6 cannot be simplified further because the GCD of 5 and 6 is 1.

Example Using the Calculator:

Let's say you want to add 3/4 and 5/6.

1. Input:
   • Fraction 1: Numerator = 3, Denominator = 4
   • Fraction 2: Numerator = 5, Denominator = 6
2. Calculate LCM: The LCM of 4 and 6 is 12.
3. Convert Fractions:
   • 3/4 becomes (3*3)/(4*3) = 9/12
   • 5/6 becomes (5*2)/(6*2) = 10/12
4. Add Numerators: 9/12 + 10/12 = (9+10)/12 = 19/12.
5. Simplify: 19/12 is an improper fraction (numerator is greater than denominator) but cannot be simplified further as the GCD of 19 and 12 is 1.

The calculator above automates these steps, providing you with both the final answer and a detailed explanation of how it arrived at the result.