Solving Fraction Word Problems: Addition and Subtraction

Word problems involving fractions can often seem daunting, but with a step-by-step approach, you can tackle them with confidence. In this article, we’ll dive into how to effectively solve word problems that require the addition and subtraction of fractions. We'll look at helpful strategies, break down examples, and provide some practice problems for you to hone your skills.

Step-by-Step Approach to Solving Word Problems

When faced with a word problem, it’s essential to follow a systematic approach to simplify the process:

  1. Read the Problem Carefully: Understand what the problem is asking. Highlight or underline key information.
  2. Identify the Fractions: Look for the fractions mentioned in the problem. Pay attention to their denominators.
  3. Determine the Operation: Decide if you need to add or subtract the fractions based on the wording of the problem (words like "together," "in total," or "combined" suggest addition, while "remaining," "left," or "after" suggest subtraction).
  4. Find a Common Denominator: Before you can add or subtract fractions, they must have the same denominator.
  5. Perform the Operation: Add or subtract the numerators while keeping the common denominator. Simplify your answer if possible.
  6. Check Your Work: Make sure your final answer makes sense in the context of the problem.

Example 1: Addition of Fractions

Problem: Jane has \(\frac{1}{3}\) of a chocolate bar, and her friend gives her \(\frac{1}{6}\) of another bar. How much chocolate does Jane have now?

Solution:

  1. Identify the fractions: Jane has \(\frac{1}{3}\), and she receives \(\frac{1}{6}\).
  2. Determine the operation: Since we need to find out how much chocolate she has now, we will add the two fractions.
  3. Find a common denominator: The least common denominator (LCD) for 3 and 6 is 6.
  4. Convert fractions:
    • \(\frac{1}{3} = \frac{2}{6}\)
    • \(\frac{1}{6} = \frac{1}{6}\)
  5. Add the fractions: \[ \frac{2}{6} + \frac{1}{6} = \frac{3}{6} \]
  6. Simplify if necessary: \[ \frac{3}{6} = \frac{1}{2} \] Final Answer: Jane has \(\frac{1}{2}\) of a chocolate bar now.

Example 2: Subtraction of Fractions

Problem: A recipe calls for \(\frac{3}{4}\) cup of flour. If you have already used \(\frac{1}{4}\) cup, how much flour is left?

Solution:

  1. Identify the fractions: The total requested is \(\frac{3}{4}\), and the amount used is \(\frac{1}{4}\).
  2. Determine the operation: Since we need to find how much flour is left, we will subtract.
  3. We already have a common denominator of 4.
  4. Subtract the fractions: \[ \frac{3}{4} - \frac{1}{4} = \frac{2}{4} \]
  5. Simplify if necessary: \[ \frac{2}{4} = \frac{1}{2} \] Final Answer: There is \(\frac{1}{2}\) cup of flour left.

Example 3: Complex Addition of Fractions

Problem: A gardener is planting flowers. She has \(\frac{2}{5}\) of a bag of fertilizer for roses and \(\frac{3}{10}\) for tulips. How much fertilizer does she have in total?

Solution:

  1. Identify the fractions: Fertilizer for roses is \(\frac{2}{5}\), and for tulips is \(\frac{3}{10}\).
  2. Determine the operation: We will add the two fractions.
  3. Find a common denominator: The LCD for 5 and 10 is 10.
  4. Convert fractions:
    • \(\frac{2}{5} = \frac{4}{10}\)
    • \(\frac{3}{10} = \frac{3}{10}\)
  5. Add the fractions: \[ \frac{4}{10} + \frac{3}{10} = \frac{7}{10} \] Final Answer: The gardener has \(\frac{7}{10}\) of a bag of fertilizer in total.

Example 4: Mixed Addition and Subtraction

Problem: Emma had \(\frac{5}{6}\) of a pizza. She ate \(\frac{1}{3}\) of it and then ordered another pizza, from which she ate \(\frac{1}{2}\). How much pizza does she have left?

Solution:

  1. Identify the fractions: Initial pizza: \(\frac{5}{6}\), amount eaten from the first pizza: \(\frac{1}{3}\), and amount eaten from the second pizza: \(\frac{1}{2}\).
  2. Determine the operations: First, we will subtract what Emma ate from the first pizza and then subtract what she ate from the second pizza.
  3. Finding a common denominator: The LCD for 3 and 6 is 6 (to subtract from the first pizza), and for 2 and 6 is 6 (to subtract from the second).
  4. Convert fractions:
    • \(\frac{1}{3} = \frac{2}{6}\)
    • \(\frac{1}{2} = \frac{3}{6}\)
  5. Subtract from the first pizza: \[ \frac{5}{6} - \frac{2}{6} = \frac{3}{6} \] This simplifies to \(\frac{1}{2}\).
  6. Now subtract from the second pizza:
    • Emma now has \(\frac{1}{2}\) pizza remaining.
    • From the second ordered pizza, she ate \(\frac{3}{6}\), which can be written as \(\frac{1}{2}\). \[ \frac{1}{2} - \frac{1}{2} = 0 \] Final Answer: Emma has 0 pizza left.

Tips for Success with Fraction Word Problems

  1. Practice: The more you practice, the easier these types of problems will become.
  2. Visual Aids: Sometimes drawing a picture or using fraction bars can help visualize the problem.
  3. Check Your Work: Always double-check your calculations and your final answer to prevent simple mistakes.
  4. Explain Your Reasoning: If you can explain how you solved the problem, it will help reinforce your understanding of the concepts.

Practice Problems

To solidify your understanding, try solving these practice problems on your own:

  1. Sam drank \(\frac{3}{8}\) of a smoothie, and then he drank another \(\frac{1}{4}\). How much smoothie did he drink in total?

  2. A recipe requires \(\frac{2}{3}\) of a cup of sugar. If you only have \(\frac{1}{6}\) of a cup left, how much more sugar do you need to buy?

  3. Lisa had \(\frac{7}{10}\) of a gallon of paint. She used \(\frac{1}{5}\) of a gallon for a project. How much paint does she have left?

  4. A baker made \(\frac{5}{12}\) of a cake and later added \(\frac{1}{3}\) of another cake. What fraction of the cakes does he have in total?

Conclusion

Solving word problems involving the addition and subtraction of fractions might initially seem complicated, but with practice and the right approach, you can manage them easily. Stick to the steps outlined above, and remember to practice regularly. Soon, you'll be solving these problems with ease, ensuring you ace your math tasks!