Fraction Calculator

How to Use This Fraction Calculator

How to Use This Fraction Calculator by Student Level

Using This Tool for Fraction Assignments

What Is a Fraction?

What is a fraction and what do the parts mean?

A fraction represents a part of a whole. It is written as one number over another, separated by a dividing line. The top number is the numerator and the bottom number is the denominator. In 3/4, the numerator 3 means you have 3 parts and the denominator 4 means the whole is divided into 4 equal pieces.

What are the different types of fractions?

A proper fraction has a numerator smaller than the denominator, like 2/5. An improper fraction has a numerator equal to or larger than the denominator, like 7/3. A mixed number combines a whole number with a proper fraction, like 2 and 1/3. All three forms represent the same kind of value and can be converted into each other.

What is a unit fraction?

A unit fraction always has a numerator of 1, such as 1/2, 1/5, or 1/12. It tells you the size of one single piece when a whole is divided equally. Every fraction is just a multiple of a unit fraction. For example, 3/8 is three copies of the unit fraction 1/8.

Where do fractions appear in real life?

Fractions show up everywhere. Cooking recipes use 3/4 cup or 1/2 teaspoon. Construction measurements use 5/8 inch. Finance uses fractions when splitting costs. Sports statistics use batting averages as fractions. Understanding how to add, simplify, and compare fractions is a daily skill, not just a classroom exercise.

Adding and Subtracting Fractions

How do you add fractions with the same denominator?

Keep the denominator the same and add only the numerators. For example, 2/7 + 3/7 = 5/7. The size of each piece does not change, you are just counting more of them. Always check if the result can be simplified after adding.

How do you add fractions with different denominators?

Step 1: Find the LCD of both denominators. Step 2: Convert each fraction to that denominator. Step 3: Add the numerators. Step 4: Simplify. For 1/3 + 1/4: LCD is 12. Convert to 4/12 + 3/12 = 7/12. Enter both fractions in the Basic tab, press Add, and every step appears instantly.

How do you subtract fractions with different denominators?

The process mirrors addition. Find the LCD, convert both fractions, then subtract the numerators. For 5/6 minus 1/4: LCD is 12. Convert to 10/12 minus 3/12 = 7/12. If the result is negative, the second fraction was larger. The Basic tab handles this automatically and shows the sign correctly.

Solve: 3/8 + 5/12 step by step

LCD of 8 and 12 is 24. Multiply 3/8 by 3/3 to get 9/24. Multiply 5/12 by 2/2 to get 10/24. Add: 9/24 + 10/24 = 19/24. Since GCD(19, 24) = 1, the fraction is already in lowest terms. Answer: 19/24.

Solve: 7/10 minus 2/15 step by step

LCD of 10 and 15 is 30. Convert 7/10 to 21/30 and 2/15 to 4/30. Subtract: 21/30 minus 4/30 = 17/30. GCD(17, 30) = 1 so the answer stays 17/30. Use the Basic tab with the Subtract button to verify this instantly.

What is the most common mistake when adding fractions?

The biggest mistake is adding both numerators and both denominators separately. For example, writing 1/2 + 1/3 = 2/5. This is wrong. The correct answer is 5/6. You must always find the LCD first. The denominators never get added together.

Multiplying and Dividing Fractions

How do you multiply fractions?

Multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. Then simplify. For 2/3 times 3/4: multiply to get 6/12, then simplify to 1/2. No LCD is needed for multiplication.

What is cross-canceling before multiplying fractions?

Cross-canceling means dividing a numerator and a denominator by a common factor before you multiply, which keeps the numbers smaller. For 4/9 times 3/8: the 4 and 8 share a factor of 4, and the 3 and 9 share a factor of 3. Cancel to get 1/3 times 1/2 = 1/6. The Basic tab shows the simplified result automatically.

How do you divide fractions using the keep-change-flip method?

Keep the first fraction as-is. Change the division sign to multiplication. Flip the second fraction to its reciprocal. Then multiply normally. For 2/3 divided by 4/5: keep 2/3, change to ×, flip 4/5 to 5/4. Result: 2/3 × 5/4 = 10/12 = 5/6.

Solve: (3/4) divided by (9/16)

Keep 3/4. Flip 9/16 to get 16/9. Multiply: 3/4 × 16/9 = 48/36. Simplify: GCD(48, 36) = 12. Divide both by 12: 4/3. Convert to mixed number: 1 and 1/3. Use the Divide button in the Basic tab to check your answer.

When does dividing by a fraction give you a bigger number?

Whenever you divide by a fraction smaller than 1, the result is larger than the original number. For example, 3 divided by 1/2 = 6, because 1/2 fits into 3 exactly 6 times. This is why dividing by 1/4 is the same as multiplying by 4.

Simplifying Fractions to Lowest Terms

What does it mean to simplify a fraction?

A fraction is in its simplest form, also called lowest terms, when the numerator and denominator share no common factor other than 1. For example, 4/6 is not simplified because both 4 and 6 divide by 2. Simplified, it becomes 2/3.

How do you find the GCD to simplify a fraction?

The GCD (greatest common divisor) is the largest number that divides both the numerator and denominator evenly. List the factors of each number and find the biggest one they share. For 36/48: factors of 36 include 1, 2, 3, 4, 6, 9, 12, 18, 36. Factors of 48 include 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The largest shared factor is 12. Divide both by 12: 3/4.

Explain how to simplify 60/84 step by step

Find GCD(60, 84). Both divide by 2 to give 30/42. Both divide by 2 again to give 15/21. Both divide by 3 to give 5/7. Since GCD(5, 7) = 1, the fraction is fully simplified. Answer: 5/7. The Simplify tab in the calculator does this in one step and shows the GCD clearly.

How do you know if a fraction is already in lowest terms?

Calculate the GCD of the numerator and denominator. If the GCD equals 1, the fraction is already fully reduced. For 7/11: GCD(7, 11) = 1 because both 7 and 11 are prime numbers. The Simplify tab confirms this instantly with the message "Already in lowest terms."

Why do teachers require fractions in lowest terms?

Simplified fractions are easier to compare, add, and work with. An answer of 2/3 is clearer than 16/24. Most homework, tests, and standardized exams like SAT and ACT require answers in lowest terms. Leaving a fraction unsimplified is usually marked as incomplete.

Converting Fractions: Decimals, Percentages, and Mixed Numbers

How do you convert a fraction to a decimal?

Divide the numerator by the denominator. For 3/8: 3 ÷ 8 = 0.375. For 1/6: 1 ÷ 6 = 0.1666 (the 6 repeats). Terminating decimals end exactly, like 3/4 = 0.75. Repeating decimals go on forever, like 1/3 = 0.333. The Convert tab shows the full decimal to 8 decimal places.

How do you convert a decimal to a fraction?

Count the decimal places and write the decimal as a fraction over the matching power of 10. For 0.625: three decimal places, so write 625/1000. GCD(625, 1000) = 125. Divide both: 5/8. For 0.2: one decimal place gives 2/10, simplified to 1/5. Use the Decimal to Fraction option in the Convert tab to check any decimal.

How do you convert a fraction to a percentage?

Divide the numerator by the denominator, then multiply by 100 and add a percent sign. For 7/20: 7 ÷ 20 = 0.35, times 100 = 35%. For 3/8: 3 ÷ 8 = 0.375, times 100 = 37.5%. The Convert tab option Fraction to Percentage computes this and shows the full working.

How do you convert a mixed number to an improper fraction?

Multiply the whole number by the denominator, then add the numerator. Keep the same denominator. For 4 and 2/7: multiply 4 times 7 = 28, add 2 = 30. Answer: 30/7. This conversion is required before you can multiply, divide, or compare mixed numbers.

How do you convert an improper fraction to a mixed number?

Divide the numerator by the denominator. The quotient is the whole number. The remainder is the new numerator and the denominator stays the same. For 17/5: 17 ÷ 5 = 3 remainder 2. Answer: 3 and 2/5. The Convert tab option Improper Fraction to Mixed Number shows the division step clearly.

What are some common fraction, decimal, and percentage equivalents students should know?

These are worth memorizing: 1/2 = 0.5 = 50%. 1/4 = 0.25 = 25%. 3/4 = 0.75 = 75%. 1/3 = 0.333 = 33.3%. 2/3 = 0.667 = 66.7%. 1/5 = 0.2 = 20%. 1/8 = 0.125 = 12.5%. Knowing these by heart speeds up any fraction problem significantly.

Comparing Fractions

How do you compare two fractions to find which is larger?

Convert both fractions to the same denominator using the LCD, then compare the numerators. The fraction with the larger numerator is bigger. For 5/6 vs 7/9: LCD is 18. Convert to 15/18 vs 14/18. Since 15 is greater than 14, then 5/6 is greater than 7/9. The Compare tab shows this with a clear greater-than or less-than symbol.

How do you compare fractions using cross multiplication?

Multiply the numerator of the first fraction by the denominator of the second. Then multiply the numerator of the second by the denominator of the first. Compare those two products. For 3/5 vs 4/7: 3 × 7 = 21 and 4 × 5 = 20. Since 21 is greater, 3/5 is greater than 4/7. This shortcut avoids finding the LCD.

How do you order fractions from least to greatest?

Convert all fractions to decimals or to the same denominator, then sort by value. For 1/2, 3/8, and 2/5: convert to decimals 0.5, 0.375, and 0.4. Sort to get 3/8 less than 2/5 less than 1/2. Use the Compare tab repeatedly to check each pair during sorting.

What is a benchmark fraction and how does it help with comparing?

Benchmark fractions are easy reference points: 0, 1/4, 1/2, 3/4, and 1. When comparing two fractions, check whether each is above or below 1/2. If one fraction is above 1/2 and the other is below, no calculation is needed. For example, 5/9 is above 1/2 and 3/7 is below 1/2, so 5/9 must be larger without doing any math.

Least Common Denominator

What is the least common denominator?

The least common denominator (LCD) is the smallest number that all denominators in a set of fractions can divide into evenly. It equals the least common multiple (LCM) of those denominators. You need the LCD every time you add, subtract, or compare fractions with different denominators.

How do you find the LCD of two fractions?

Use the formula: LCD = (denominator 1 × denominator 2) ÷ GCD(denominator 1, denominator 2). For denominators 4 and 6: GCD(4, 6) = 2. LCD = (4 × 6) ÷ 2 = 12. Enter both denominators in the LCD tab to confirm instantly and see the conversion for each fraction.

How do you find the LCD of three or more fractions?

Find the LCM of the first two denominators, then find the LCM of that result with the third. Repeat for each additional denominator. For 3, 4, and 6: LCM(3, 4) = 12, then LCM(12, 6) = 12. For 2, 3, 4, and 5: LCM(2, 3) = 6, LCM(6, 4) = 12, LCM(12, 5) = 60. The LCD tab handles up to 4 denominators at once.

What equivalent fractions does the LCD give you?

Once you have the LCD, multiply each fraction's numerator and denominator by the factor that brings the denominator up to the LCD. For fractions with denominators 4 and 6 and LCD of 12: multiply 1/4 by 3/3 to get 3/12, and multiply 1/6 by 2/2 to get 2/12. Now both fractions share the same denominator and can be added or compared directly. The LCD tab shows these conversions automatically.

What is the difference between LCD and LCM?

They are the same calculation applied to different contexts. LCM (least common multiple) is the general math term for the smallest shared multiple of two or more integers. LCD (least common denominator) is the LCM of the denominators of two or more fractions. The LCD tab on this calculator finds this value for any set of denominators you enter.

Adding, Subtracting, Multiplying, and Dividing Mixed Numbers (Mixed # Tab)

What is a mixed number?

A mixed number has a whole number part and a fraction part written side by side, like 3 and 1/2. It represents a value greater than 1. Mixed numbers appear in real life constantly: recipes call for 2 and 1/4 cups of flour, lumber is sold in 1 and 1/2 inch widths, and track events are timed to the nearest 1/100 of a second.

How do you add mixed numbers step by step?

Convert each mixed number to an improper fraction. Then find the LCD, convert to equal denominators, add the numerators, simplify, and convert back to a mixed number. For 1 and 1/2 plus 2 and 1/3: convert to 3/2 and 7/3. LCD is 6. That gives 9/6 plus 14/6 = 23/6 = 3 and 5/6. The Mixed # tab does all five steps and shows each one.

How do you subtract mixed numbers when the fraction part of the first is smaller?

This is called borrowing. Rewrite the first mixed number by taking 1 from the whole number and adding it as a fraction. For 5 and 1/4 minus 2 and 3/4: rewrite 5 and 1/4 as 4 and 5/4. Now subtract: 4 and 5/4 minus 2 and 3/4 = 2 and 2/4 = 2 and 1/2. Converting to improper fractions first avoids borrowing entirely.

How do you multiply two mixed numbers?

Convert both to improper fractions, then multiply. For 2 and 1/3 times 1 and 1/2: convert to 7/3 and 3/2. Multiply: 7/3 × 3/2 = 21/6. Simplify: GCD(21, 6) = 3, so 21/6 = 7/2. Convert to mixed number: 3 and 1/2. Select Multiply in the Mixed # tab to see this computed instantly.

How do you divide two mixed numbers?

Convert both to improper fractions. Keep the first, change the sign to multiplication, and flip the second. For 3 and 1/2 divided by 1 and 1/4: convert to 7/2 and 5/4. Flip 5/4 to get 4/5. Multiply: 7/2 × 4/5 = 28/10 = 14/5 = 2 and 4/5. Select Divide in the Mixed # tab to verify each step.

Simplifying Ratios

What is a ratio?

A ratio compares two quantities and shows how much of one there is relative to the other. It is written as a : b or as a fraction a/b. For example, if a class has 12 boys and 18 girls, the ratio of boys to girls is 12:18. Ratios appear in recipes, maps, finance, and science.

How do you simplify a ratio?

Find the GCD of both numbers and divide each by it. For 12:18: GCD(12, 18) = 6. Divide both by 6 to get 2:3. For 15:25: GCD(15, 25) = 5. Divide both by 5 to get 3:5. A simplified ratio has no common factor between its two parts. Enter any ratio into the Ratio tab to simplify it with full working shown.

What is the difference between a ratio and a fraction?

A fraction always represents a part of one whole. A ratio can compare any two quantities, even from different wholes. The ratio 2:3 could mean 2 red balls to 3 blue balls, which is not a fraction of anything in particular. However, any ratio can be written as a fraction, so the math of simplification is identical.

How do you use ratios to scale a recipe?

Write the original ratio and decide on your multiplier. If a cookie recipe uses flour to sugar in a 3:1 ratio and you want to triple the batch, multiply both parts by 3 to get 9:3, which simplifies back to 3:1. The proportions stay the same. For non-integer scaling, the Proportion tab is more useful.

Solve: simplify the ratio 48:36

GCD(48, 36) = 12. Divide both: 48 ÷ 12 = 4, 36 ÷ 12 = 3. Simplified ratio: 4:3. As a fraction: 4/3 = 1.333. Enter 48 and 36 in the Ratio tab and press Calculate to confirm.

Unit Rate Calculator

What is a unit rate?

A unit rate compares a quantity to exactly one unit of another quantity. The denominator is always 1. Examples: 60 miles per 1 hour, $2.50 per 1 pound, 120 words per 1 minute. Any rate can be turned into a unit rate by dividing both numbers so the second becomes 1.

How do you calculate a unit rate?

Divide the first quantity by the second. For 240 miles in 4 hours: 240 ÷ 4 = 60 miles per hour. For $7.50 for 3 cans: $7.50 ÷ 3 = $2.50 per can. Enter the two numbers in the Unit Rate tab and press Calculate. The result is the rate per 1 unit.

How do unit rates help with comparison shopping?

Compare the unit price of two products to find the better deal. A 12-ounce box at $3.60 gives a unit price of $0.30 per ounce. A 20-ounce box at $5.40 gives $0.27 per ounce. The larger box is cheaper per ounce. Use the Unit Rate tab to calculate each price before deciding.

What is the unit rate in a word problem?

Look for phrases like "per," "for each," "every," or "in one." A car travels 375 miles on 15 gallons: unit rate = 375 ÷ 15 = 25 miles per gallon. A typist types 1,200 words in 8 minutes: unit rate = 1,200 ÷ 8 = 150 words per minute. The Unit Rate tab solves any of these with a single calculation.

How is unit rate connected to fractions?

A unit rate is simply a fraction with a denominator of 1 after simplification. Expressing 60/1 miles per hour is the same as saying the fraction 60/1. Unit rates are also the bridge between ratios and proportions. Once you know the unit rate, you can scale any proportion by multiplying.

Equivalent Fractions

What are equivalent fractions?

Equivalent fractions represent the same value but are written with different numbers. Multiply or divide both the numerator and denominator by the same non-zero number and you always get an equivalent fraction. For 1/2: multiply both by 2 to get 2/4, by 3 to get 3/6, by 10 to get 10/20. All equal 0.5.

How do you find equivalent fractions for any fraction?

Pick any multiplier and apply it to both parts of the fraction. For 3/5: multiplier 2 gives 6/10, multiplier 3 gives 9/15, multiplier 4 gives 12/20, multiplier 5 gives 15/25. The Equiv. tab generates 8 equivalent fractions for any fraction you enter, starting from the simplified form.

How do you check if two fractions are equivalent?

Simplify both fractions to lowest terms. If the simplified forms are identical, the fractions are equivalent. For 6/10 and 9/15: simplify both. GCD(6, 10) = 2 so 6/10 = 3/5. GCD(9, 15) = 3 so 9/15 = 3/5. They are equal. You can also cross-multiply: 6 × 15 = 90 and 10 × 9 = 90. Equal products confirm equivalence.

Why do equivalent fractions matter in homework and tests?

You need equivalent fractions to add and subtract fractions with different denominators. You also need them to verify that two answers that look different are actually the same value. For example, a student answers 6/9 and the answer key shows 2/3. Both are correct because 6/9 simplifies to 2/3.

Explain how to use equivalent fractions to add 1/4 + 1/6

LCD of 4 and 6 is 12. Create equivalent fractions: 1/4 = 3/12 (multiply by 3) and 1/6 = 2/12 (multiply by 2). Now add: 3/12 + 2/12 = 5/12. Every addition problem with different denominators secretly requires creating equivalent fractions with a shared denominator first.

Fraction Exponents

How do you raise a fraction to a whole number exponent?

Raise both the numerator and denominator separately to that power. For (2/3)^4: numerator = 2^4 = 16, denominator = 3^4 = 81. Answer: 16/81. No simplification is needed here because 16 and 81 share no common factors. Enter the fraction and exponent into the Exponent tab to see this worked out clearly.

Solve: (3/4)^3 step by step

Raise numerator: 3^3 = 27. Raise denominator: 4^3 = 64. Result: 27/64. GCD(27, 64) = 1 because 27 = 3^3 and 64 = 2^6, which share no factors. Answer: 27/64. Decimal: 27 ÷ 64 = 0.421875.

Solve: (2/5)^3 and simplify

2^3 = 8, 5^3 = 125. Result: 8/125. GCD(8, 125) = 1 so the fraction is already simplified. Decimal: 0.064. The Exponent tab also gives the decimal so you can quickly check whether the result is reasonable.

What happens when a fraction less than 1 is raised to a higher power?

The result gets smaller. For 1/2: (1/2)^2 = 1/4, (1/2)^3 = 1/8, (1/2)^4 = 1/16. Each time you raise by one more power, the fraction is halved. This is why compound interest shrinks debt over time when the rate is expressed as a fraction, and why probabilities of multiple events multiplied together get very small quickly.

What does a fractional exponent like x^(1/2) mean?

A fractional exponent means a root. x^(1/2) means the square root of x. x^(1/3) means the cube root. x^(2/3) means the square of the cube root. This comes up in algebra, pre-calculus, and physics. The Exponent tab on this calculator works with whole number exponents applied to fraction bases.

Proportion Solver

What is a proportion?

A proportion says two ratios or fractions are equal: a/b = c/d. If you know three of the four values, you can always find the fourth. Proportions show up in cooking, map reading, scale drawings, speed problems, and percent calculations. They are one of the most practical math tools in everyday life.

How do you solve a proportion using cross multiplication?

Multiply the top-left by the bottom-right, and the top-right by the bottom-left. Set them equal and solve. For 3/4 = x/12: cross multiply to get 3 × 12 = 4 × x. So 36 = 4x. Divide by 4: x = 9. Enter 3, 4, blank, and 12 in the Proportion tab. The tool solves for the blank automatically.

Solve: 5/8 = x/40

Cross multiply: 5 × 40 = 8 × x. So 200 = 8x. Divide both sides by 8: x = 25. Check: 5/8 = 25/40 = 5/8. Correct. Use the Proportion tab: enter 5, 8, leave c blank, enter 40. The answer x = 25 appears instantly.

How do you set up a proportion from a word problem?

Identify the two quantities being compared and keep them consistent across both ratios. If a car travels 150 miles in 3 hours, how far does it travel in 7 hours? Set up: 150/3 = x/7. Cross multiply: 150 × 7 = 3x. 1050 = 3x. x = 350 miles. The key is making sure both ratios compare the same units in the same order.

What is a direct proportion versus an inverse proportion?

In a direct proportion, when one quantity increases the other increases at the same rate. More hours worked means more pay. In an inverse proportion, when one quantity increases the other decreases. More workers on a job means fewer hours to finish. This calculator's Proportion tab handles direct proportions of the form a/b = c/d.

How is proportion used in percentage problems?

Any percentage problem can be written as a proportion. "What is 30% of 80?" becomes 30/100 = x/80. Cross multiply: 30 × 80 = 100x. 2400 = 100x. x = 24. This is one of the most common uses of the proportion solver in middle school and high school math assignments.