Converting Decimals to Fractions with JavaScript - Simplify & Improve
highestCommonFactor should take 2 integers as parameter instead of relying on the variable numerator and `denominator'. Also, you could find it using Euclid'a algorithm.
I am wrong is saying that this piece of code :

var rightDecimalPart = decimalArray[1];
// Save decimal part only for later use
var decimalOnly = "0." + rightDecimalPart;
// Find the decimal multiplier
var multiplier = "1";
for (var i = 0; i &lt; rightDecimalPart.length; i++) {
    multiplier += "0";
}
// Create numerator by multiplying the multiplier and decimal part together
var numerator = Number(multiplier) * Number(decimalOnly);



is here to transform a number such as 78924 in 0.78924 and then check that have to multiply it by 100000 to get an integer which is ... 78924. 

Edit
After a first cleanup, I get :

function highestCommonFactor(a,b) {
    if (b==0) return a;
    return highestCommonFactor(b,a%b);
}

var decimal = "1.75";
var decimalArray = decimal.split(".");
var leftDecimalPart = decimalArray[0];
var rightDecimalPart = decimalArray[1];

var denominator = "1";

for (var i = 0; i &lt; rightDecimalPart.length; i++) {
    denominator += "0";
}
document.getElementById("debug").innerText = denominator;
var factor = highestCommonFactor(rightDecimalPart, denominator);

// Simplify the fraction by dividing the numerator and denominator by the factor
var denominator = Number(denominator) / Number(factor);
var numerator = (Number(rightDecimalPart) / Number(factor)) + (leftDecimalPart * denominator);

// Display solution as a proper fraction or improper fraction (depending on input)
document.getElementById("divSolution").innerText = numerator + "/" + denominator;


I'll try to go a step further.

Edit 2
After a rewriting of the calculation, here's what I got :

function highestCommonFactor(a,b) {
    if (b==0) return a;
    return highestCommonFactor(b,a%b);
}

var decimal = "1.75";
var decimalArray = decimal.split(".");
var leftDecimalPart = decimalArray[0]; // 1
var rightDecimalPart = decimalArray[1]; // 75

var numerator = leftDecimalPart + rightDecimalPart // 175
var denominator = Math.pow(10,rightDecimalPart.length); // 100
var factor = highestCommonFactor(numerator, denominator); // 25
denominator /= factor;
numerator /= factor;

document.getElementById("divSolution").innerText = numerator + "/" + denominator;