A unit rate is a comparison where the second term (the denominator) is 1. When dealing with complex fractions—where either the numerator, the denominator, or both are fractions—calculating the rate requires converting these mixed numbers into improper fractions and then performing division.
How to Calculate Complex Unit Rates
Convert Mixed Numbers: If you have a whole number and a fraction (like 2 1/2), convert it to an improper fraction (5/2).
Set Up the Division: A rate is simply the numerator divided by the denominator. If you travel 1/2 mile in 1/4 hour, your setup is (1/2) / (1/4).
Multiply by the Reciprocal: In fraction division, you "keep" the first fraction, "change" the sign to multiplication, and "flip" the second fraction. Example: (1/2) * (4/1) = 4/2.
Simplify: Reduce the resulting fraction to its simplest form or convert to a decimal. 4/2 simplifies to 2 units per hour.
Real-World Examples
Example 1: Speed
A snail moves 2/3 of a meter in 1/6 of an hour. To find the speed in meters per hour:
(2/3) ÷ (1/6) = (2/3) × (6/1) = 12/3 = 4 meters per hour.
Example 2: Cooking
If a recipe requires 3/4 cup of flour for every 1/2 batch of cookies, the unit rate is:
(3/4) ÷ (1/2) = (3/4) × (2/1) = 6/4 = 1.5 cups per batch.
Common Units for Unit Rates
Miles per Hour (mph)
Grams per Milliliter
Price per Ounce
Kilometers per Liter
Words per Minute
Inches per Foot
function calculateComplexRate() {
// Numerator Inputs
var nW = parseFloat(document.getElementById('num_whole').value) || 0;
var nT = parseFloat(document.getElementById('num_frac_top').value) || 0;
var nB = parseFloat(document.getElementById('num_frac_bottom').value) || 1;
// Denominator Inputs
var dW = parseFloat(document.getElementById('den_whole').value) || 0;
var dT = parseFloat(document.getElementById('den_frac_top').value) || 0;
var dB = parseFloat(document.getElementById('den_frac_bottom').value) || 1;
if (nB === 0 || dB === 0) {
alert("Denominator cannot be zero.");
return;
}
// Convert to Improper Fractions
// Num = (Whole * Denom + Top) / Denom
var improperNumTop = (nW * nB) + nT;
var improperNumBottom = nB;
var improperDenTop = (dW * dB) + dT;
var improperDenBottom = dB;
if (improperDenTop === 0) {
alert("The total denominator value cannot be zero.");
return;
}
// Calculation logic: (A/B) / (C/D) = (A*D) / (B*C)
var finalTop = improperNumTop * improperDenBottom;
var finalBottom = improperNumBottom * improperDenTop;
var decimalRate = finalTop / finalBottom;
// Helper to find Greatest Common Divisor
var gcd = function(a, b) {
return b ? gcd(b, a % b) : a;
};
var commonDivisor = Math.abs(gcd(finalTop, finalBottom));
var simplifiedTop = finalTop / commonDivisor;
var simplifiedBottom = finalBottom / commonDivisor;
// Display
document.getElementById('result_area').style.display = 'block';
document.getElementById('decimal_result').innerHTML = "Unit Rate: " + decimalRate.toLocaleString(undefined, {maximumFractionDigits: 4}) + " units per 1 unit";
var fracText = simplifiedBottom === 1 ? simplifiedTop : simplifiedTop + "/" + simplifiedBottom;
document.getElementById('fraction_result').innerHTML = "Simplified Fraction: " + fracText;
var stepHtml = "Steps:";
stepHtml += "1. Numerator as fraction: " + improperNumTop + "/" + improperNumBottom + "";
stepHtml += "2. Denominator as fraction: " + improperDenTop + "/" + improperDenBottom + "";
stepHtml += "3. Multiply by reciprocal: (" + improperNumTop + "/" + improperNumBottom + ") × (" + improperDenBottom + "/" + improperDenTop + ") = " + finalTop + "/" + finalBottom;
document.getElementById('step_explanation').innerHTML = stepHtml;
}