Enter the values from your word problem below to get the solution.
Distance, Rate, Time
Percentage Increase/Decrease
Ratio & Proportion
Average Speed
Towards Each Other
Away From Each Other
One Catching Up
Increase
Decrease
Understanding Word Problems & Their Solutions
Word problems are a fundamental part of mathematics and physics, requiring us to translate real-world scenarios into numerical equations. This calculator helps solve common types of word problems by applying established mathematical principles. Below, we explore the logic behind each problem type supported.
1. Distance, Rate, and Time Problems
These problems involve the relationship: Distance = Rate × Time. They often describe scenarios with one or more objects moving. We can solve for any of the three variables if the other two are known, or we can set up equations for multiple objects to find when or where they meet, or how far apart they end up.
Key Formula: D = R × T
Towards Each Other: When objects move towards each other, their relative speed is the sum of their individual speeds (Rate1 + Rate2). The time to meet is found by dividing the initial distance between them by their combined rate.
Away From Each Other: When objects move away, their relative speed is also the sum of their individual speeds (Rate1 + Rate2). The distance between them after a certain time is this combined rate multiplied by time.
One Catching Up: When one object chases another, the relative speed is the difference between their speeds (RateFaster – RateSlower). The time to catch up is the initial distance between them divided by this difference in rates.
Example Scenario: Two trains leave stations 300 miles apart, traveling towards each other. Train A travels at 60 mph, and Train B travels at 70 mph. How long until they meet?
Calculation Logic: Combined Rate = 60 mph + 70 mph = 130 mph. Time to Meet = Total Distance / Combined Rate = 300 miles / 130 mph ≈ 2.31 hours.
2. Percentage Increase/Decrease Problems
These problems deal with changes in a value relative to its original amount. They are common in finance, statistics, and everyday scenarios like sales and inflation.
Formula for Increase: New Value = Original Value × (1 + Percentage/100)
Formula for Decrease: New Value = Original Value × (1 – Percentage/100)
Example Scenario: A shirt originally costs $50. It goes on sale with a 20% discount. What is the new price?
Ratios express the relative sizes of two or more quantities. Proportions state that two ratios are equal. These are used for scaling recipes, dividing resources, and understanding proportional relationships.
To find parts: If a total amount is divided according to a ratio A:B, the total number of "parts" is A + B. Each part is worth Total Amount / (A + B). Then, Part A = A × (Each Part Value), and Part B = B × (Each Part Value).
Example Scenario: A sum of $700 is to be divided between two people in the ratio 3:4. How much does each person receive?
Calculation Logic: Total parts = 3 + 4 = 7. Value per part = $700 / 7 = $100. Person 1 receives 3 parts × $100/part = $300. Person 2 receives 4 parts × $100/part = $400.
4. Average Speed Problems
Average speed is calculated as the total distance traveled divided by the total time taken. It's important to note that this is not simply the average of different speeds if the time spent at each speed is different.
Formula: Average Speed = Total Distance / Total Time
Example Scenario: A car travels 150 miles in 3 hours and then travels another 150 miles in 2 hours. What is its average speed for the entire journey?
Calculation Logic: Total Distance = 150 miles + 150 miles = 300 miles. Total Time = 3 hours + 2 hours = 5 hours. Average Speed = 300 miles / 5 hours = 60 mph.
Use this calculator to quickly find solutions and gain a better understanding of how these mathematical concepts are applied.
function updateInputs() {
var selectedType = document.getElementById('problemType').value;
document.getElementById('distanceInputs').style.display = (selectedType === 'distance') ? 'block' : 'none';
document.getElementById('percentageInputs').style.display = (selectedType === 'percentage') ? 'block' : 'none';
document.getElementById('ratioInputs').style.display = (selectedType === 'ratio') ? 'block' : 'none';
document.getElementById('speedInputs').style.display = (selectedType === 'speed') ? 'block' : 'none';
// Clear previous results
document.getElementById('result').innerHTML = ";
}
function calculate() {
var selectedType = document.getElementById('problemType').value;
var resultHTML = ";
if (selectedType === 'distance') {
var rate1 = parseFloat(document.getElementById('rate1').value);
var time1 = parseFloat(document.getElementById('time1').value);
var rate2 = parseFloat(document.getElementById('rate2').value);
var time2 = parseFloat(document.getElementById('time2').value);
var distanceToCover = parseFloat(document.getElementById('distanceToCover').value);
var meetingPoint = document.getElementById('meetingPoint').value;
if (isNaN(rate1) || isNaN(time1) || isNaN(rate2) || isNaN(time2) || isNaN(distanceToCover)) {
resultHTML = 'Please enter valid numbers for all fields.';
} else {
var combinedRate, timeToMeet, distanceCovered1, distanceCovered2;
if (meetingPoint === 'towards') {
combinedRate = rate1 + rate2;
if (combinedRate === 0) {
resultHTML = 'Rates sum to zero, objects will not meet unless already at the same point.';
} else {
var time = distanceToCover / combinedRate;
distanceCovered1 = rate1 * time;
distanceCovered2 = rate2 * time;
resultHTML = 'Objects will meet in ' + time.toFixed(2) + ' hours.' +
'Object 1 will travel: ' + distanceCovered1.toFixed(2) + ' units.' +
'Object 2 will travel: ' + distanceCovered2.toFixed(2) + ' units.';
}
} else if (meetingPoint === 'away') {
combinedRate = rate1 + rate2;
var totalDistanceApart = combinedRate * time1; // Assuming time1 dictates duration for this scenario
resultHTML = 'After ' + time1 + ' hours, the objects will be ' + totalDistanceApart.toFixed(2) + ' units apart.';
} else if (meetingPoint === 'catchUp') {
// Assuming rate1 is faster, chasing rate2
if (rate1 <= rate2) {
resultHTML = 'Object 1 must be faster than Object 2 to catch up.';
} else {
var relativeSpeed = rate1 – rate2;
var timeToCatchUp = distanceToCover / relativeSpeed;
distanceCovered1 = rate1 * timeToCatchUp;
distanceCovered2 = rate2 * timeToCatchUp;
resultHTML = 'Object 1 will catch up in ' + timeToCatchUp.toFixed(2) + ' hours.' +
'Object 1 will travel: ' + distanceCovered1.toFixed(2) + ' units.' +
'Object 2 will travel: ' + distanceCovered2.toFixed(2) + ' units.';
}
}
}
} else if (selectedType === 'percentage') {
var originalValue = parseFloat(document.getElementById('originalValue').value);
var percentageChange = parseFloat(document.getElementById('percentageChange').value);
var changeType = document.getElementById('changeType').value;
if (isNaN(originalValue) || isNaN(percentageChange)) {
resultHTML = 'Please enter valid numbers for Original Value and Percentage Change.';
} else {
var newValue;
if (changeType === 'increase') {
newValue = originalValue * (1 + percentageChange / 100);
} else { // decrease
newValue = originalValue * (1 – percentageChange / 100);
}
resultHTML = 'The new value is: ' + newValue.toFixed(2) + '';
}
} else if (selectedType === 'ratio') {
var ratioA = parseFloat(document.getElementById('ratioA').value);
var ratioB = parseFloat(document.getElementById('ratioB').value);
var totalValue = parseFloat(document.getElementById('totalValue').value);
if (isNaN(ratioA) || isNaN(ratioB) || isNaN(totalValue) || ratioA <= 0 || ratioB <= 0) {
resultHTML = 'Please enter valid positive numbers for ratio parts and total value.';
} else {
var totalParts = ratioA + ratioB;
var valuePerPart = totalValue / totalParts;
var partAValue = ratioA * valuePerPart;
var partBValue = ratioB * valuePerPart;
resultHTML = 'Value of Part A: ' + partAValue.toFixed(2) + '' +
'Value of Part B: ' + partBValue.toFixed(2) + '';
}
} else if (selectedType === 'speed') {
var totalDistance = parseFloat(document.getElementById('totalDistance').value);
var timeTotal = parseFloat(document.getElementById('timeTotal').value);
if (isNaN(totalDistance) || isNaN(timeTotal) || timeTotal <= 0) {
resultHTML = 'Please enter a valid total distance and a positive total time.';
} else {
var averageSpeed = totalDistance / timeTotal;
resultHTML = 'The average speed is: ' + averageSpeed.toFixed(2) + ' units per time unit';
}
}
document.getElementById('result').innerHTML = resultHTML;
}
// Initialize inputs on page load
document.addEventListener('DOMContentLoaded', updateInputs);