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Fraction Word Problems with Step-by-Step Solutions

Grade: 6-7 | Topic: Arithmetic

What You Will Learn​

After working through this page, you will be able to translate real-world situations into fraction operations, choose the correct operation (addition, subtraction, multiplication, or division), solve the problem step by step, and verify that your answer makes sense. These skills are essential for tests and for using fractions in everyday life -- cooking, building, budgeting, and more.

Theory​

A strategy for solving word problems​

Word problems can feel overwhelming, but a consistent approach makes them manageable. Follow these four steps every time:

Step 1: Read and identify. What information is given? What is the question asking for?

Step 2: Choose the operation. Use clue words and the context to decide:

Clue wordsOperation
total, combined, altogether, in allAddition
remaining, left over, difference, less thanSubtraction
of, times, each (with groups)Multiplication
shared equally, divided into, per, how many groupsDivision

Step 3: Set up and solve. Write the fraction expression, convert mixed numbers to improper fractions if needed, then compute step by step.

Step 4: Check. Does the answer make sense in context? Is it simplified? Does it have the correct unit?

"Of" means multiply​

One of the most important patterns in fraction word problems is the word "of," which signals multiplication:

23 of 24=23×24=483=16\frac{2}{3} \text{ of } 24 = \frac{2}{3} \times 24 = \frac{48}{3} = 16

This appears in problems like "Maria used 23\frac{2}{3} of her flour" or "A store sold 34\frac{3}{4} of its stock."

"How many groups" means divide​

When a problem asks how many equal groups or portions fit into a quantity, you divide:

6÷34=6×43=243=86 \div \frac{3}{4} = 6 \times \frac{4}{3} = \frac{24}{3} = 8

This appears in problems like "How many 34\frac{3}{4}-cup servings are in 6 cups?"

Working with mixed numbers in word problems​

Real-world quantities are often mixed numbers (like 2122\frac{1}{2} cups or 3343\frac{3}{4} hours). Always convert to improper fractions before operating, then convert back to a mixed number for your final answer, since that is more meaningful in context.

Worked Examples​

Example 1: Addition -- combining amounts (easy)​

Problem: Jake jogged 34\dfrac{3}{4} of a mile in the morning and 58\dfrac{5}{8} of a mile in the afternoon. How far did he jog in total?

Step 1: We need to add the two distances. 34+58\frac{3}{4} + \frac{5}{8}

Step 2: Find the LCD of 4 and 8, which is 8. 34=68\frac{3}{4} = \frac{6}{8}

Step 3: Add. 68+58=118\frac{6}{8} + \frac{5}{8} = \frac{11}{8}

Step 4: Convert to a mixed number. 118=138\frac{11}{8} = 1\frac{3}{8}

Check: He jogged less than 1 mile each time, so a total between 1 and 2 miles makes sense.

Answer: Jake jogged 1381\dfrac{3}{8} miles in total.

Example 2: Subtraction -- finding what remains (easy)​

Problem: A water tank holds 5125\dfrac{1}{2} gallons. After watering the garden, 2232\dfrac{2}{3} gallons were used. How much water remains?

Step 1: Subtract the amount used from the total. 512−2235\frac{1}{2} - 2\frac{2}{3}

Step 2: Convert to improper fractions. 112−83\frac{11}{2} - \frac{8}{3}

Step 3: Find the LCD of 2 and 3, which is 6. 336−166=176\frac{33}{6} - \frac{16}{6} = \frac{17}{6}

Step 4: Convert to a mixed number. 176=256\frac{17}{6} = 2\frac{5}{6}

Check: We started with about 5.5 and used about 2.7, so about 2.8 remaining. 256≈2.832\frac{5}{6} \approx 2.83. Correct.

Answer: 2562\dfrac{5}{6} gallons remain.

Example 3: Multiplication -- finding a fraction "of" a quantity (medium)​

Problem: A recipe calls for 2142\dfrac{1}{4} cups of flour. Sarah wants to make 23\dfrac{2}{3} of the recipe. How much flour does she need?

Step 1: "Of" means multiply. 23×214\frac{2}{3} \times 2\frac{1}{4}

Step 2: Convert the mixed number. 23×94\frac{2}{3} \times \frac{9}{4}

Step 3: Cross-cancel. 3 and 9 share a factor of 3: 9→39 \to 3, 3→13 \to 1. 21×34\frac{2}{1} \times \frac{3}{4}

2 and 4 share a factor of 2: 2→12 \to 1, 4→24 \to 2. 11×32=32\frac{1}{1} \times \frac{3}{2} = \frac{3}{2}

Step 4: Convert. 32=112\frac{3}{2} = 1\frac{1}{2}

Check: Two-thirds of about 2.25 cups should be about 1.5 cups. Correct.

Answer: Sarah needs 1121\dfrac{1}{2} cups of flour.

Example 4: Division -- equal sharing (medium)​

Problem: A carpenter has a board that is 7127\dfrac{1}{2} feet long. She needs to cut it into pieces that are each 1141\dfrac{1}{4} feet long. How many pieces can she cut?

Step 1: Divide the total length by the length of each piece. 712÷1147\frac{1}{2} \div 1\frac{1}{4}

Step 2: Convert to improper fractions. 152÷54\frac{15}{2} \div \frac{5}{4}

Step 3: Keep, Change, Flip. 152×45\frac{15}{2} \times \frac{4}{5}

Step 4: Cross-cancel. 15 and 5 share a factor of 5: 15→315 \to 3, 5→15 \to 1. Also 4 and 2 share a factor of 2: 4→24 \to 2, 2→12 \to 1. 31×21=61=6\frac{3}{1} \times \frac{2}{1} = \frac{6}{1} = 6

Check: Each piece is about 1.25 feet, and 6×1.25=7.56 \times 1.25 = 7.5. Correct.

Answer: She can cut 6 pieces.

Example 5: Multi-step problem (challenging)​

Problem: A school fundraiser collected \480.Thecommitteedecidedtospend. The committee decided to spend \dfrac14onsuppliesandon supplies and\dfrac13$ on decorations. The remaining money will be split equally among 5 charity projects. How much does each project receive?

Step 1: Find how much was spent on supplies. 14×480=4804=120\frac{1}{4} \times 480 = \frac{480}{4} = 120

Step 2: Find how much was spent on decorations. 13×480=4803=160\frac{1}{3} \times 480 = \frac{480}{3} = 160

Step 3: Find the remaining amount. 480−120−160=200480 - 120 - 160 = 200

Step 4: Divide equally among 5 projects. 200÷5=40200 \div 5 = 40

Check: 120+160+(5×40)=120+160+200=480120 + 160 + (5 \times 40) = 120 + 160 + 200 = 480. All money is accounted for.

Answer: Each charity project receives $40.

Common Mistakes​

Mistake 1: Using the wrong operation because of a misread clue word

❌ "Maria ate 13\frac{1}{3} of the 12 cookies" →\to 13+12=1213\frac{1}{3} + 12 = 12\frac{1}{3}

✅ "Of" means multiply: 13×12=4\frac{1}{3} \times 12 = 4 cookies

Why this matters: The word "of" in fraction contexts almost always signals multiplication, not addition. Misreading clue words is the number one source of errors in word problems.

Mistake 2: Forgetting to convert mixed numbers before operating

❌ 213×3=6132\frac{1}{3} \times 3 = 6\frac{1}{3} (multiplied only the whole number part by 3, then tagged on the fraction)

✅ 213×3=73×3=213=72\frac{1}{3} \times 3 = \frac{7}{3} \times 3 = \frac{21}{3} = 7

Why this matters: The fraction part must be included in the operation. Treating the whole number and fraction separately gives incorrect results. Always convert to an improper fraction first.

Mistake 3: Giving an answer without units or context

❌ Answer: 32\frac{3}{2}

✅ Answer: 1121\frac{1}{2} cups of flour

Why this matters: Word problems ask about real-world quantities. Your answer should include the unit (miles, cups, dollars, pieces) and be expressed in a form that makes sense in context -- usually a mixed number rather than an improper fraction.

Practice Problems​

Try these on your own before checking the answers:

  1. A pizza has 8 slices. Tom ate 38\frac{3}{8} and Lisa ate 14\frac{1}{4}. What fraction of the pizza was eaten?
  2. A rope is 6236\frac{2}{3} meters long. A piece of 2562\frac{5}{6} meters is cut off. How long is the remaining rope?
  3. A garden covers 45\frac{4}{5} of an acre. If 34\frac{3}{4} of the garden is planted with vegetables, how many acres of vegetables are there?
  4. A container holds 4124\frac{1}{2} liters of juice. How many 34\frac{3}{4}-liter glasses can be filled?
  5. In a class of 36 students, 23\frac{2}{3} are girls. Of the girls, 34\frac{3}{4} play a sport. How many girls play a sport?
Click to see answers
  1. 38+14=38+28=58\frac{3}{8} + \frac{1}{4} = \frac{3}{8} + \frac{2}{8} = \frac{5}{8} of the pizza was eaten.
  2. 203−176=406−176=236=356\frac{20}{3} - \frac{17}{6} = \frac{40}{6} - \frac{17}{6} = \frac{23}{6} = 3\frac{5}{6} meters remain.
  3. 34×45=1220=35\frac{3}{4} \times \frac{4}{5} = \frac{12}{20} = \frac{3}{5} of an acre is planted with vegetables.
  4. 92÷34=92×43=366=6\frac{9}{2} \div \frac{3}{4} = \frac{9}{2} \times \frac{4}{3} = \frac{36}{6} = 6 glasses can be filled.
  5. Girls: 23×36=24\frac{2}{3} \times 36 = 24. Girls who play a sport: 34×24=18\frac{3}{4} \times 24 = 18. 18 girls play a sport.

Summary​

  • Always follow the four-step strategy: read, choose the operation, solve, and check.
  • "Of" means multiply. "How many groups" or "shared equally" means divide.
  • Convert mixed numbers to improper fractions before performing any operation.
  • Include units in your answer and express results as mixed numbers when it makes sense in context.
  • Estimate before and after solving to catch errors -- if your answer does not make sense in the real-world scenario, re-check your work.

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