Fractions, Decimals and Percentages

Understanding Fractions, Decimals, and Percentages

Fractions, decimals, and percentages are three different ways of representing the same mathematical idea: parts of a whole. Mastering these concepts is essential for success in mathematics, everyday problem‑solving, and many professional fields. This course breaks down the core ideas, common operations, and useful strategies, turning quiz questions into a comprehensive learning experience.

1. Equivalent Fractions – Different Forms, Same Value

An equivalent fraction has a different numerator and denominator but represents the same quantity as another fraction. For example, \(\frac{1}{2}\) and \(\frac{2}{4}\) are equivalent because both equal 0.5.

• How to find equivalents: Multiply (or divide) the numerator and denominator by the same non‑zero number.
• Why they matter: They allow you to compare fractions, add or subtract them, and convert between forms such as mixed numbers and improper fractions.

Practice: Convert \(\frac{3}{5}\) to an equivalent fraction with denominator 20. Multiply numerator and denominator by 4 → \(\frac{12}{20}\).

2. Adding Fractions with Unlike Denominators

When fractions have different denominators, you must first find a common denominator before adding. The most efficient common denominator is the least common multiple (LCM) of the denominators.

• Step‑by‑step process:
  1. Identify the denominators.
  2. Calculate their LCM.
  3. Rewrite each fraction with the LCM as the new denominator.
  4. Add the numerators while keeping the common denominator.
  5. Simplify the result if possible.
• Example: Add \(\frac{1}{3}\) and \(\frac{1}{4}\). LCM of 3 and 4 is 12. Convert: \(\frac{1}{3}=\frac{4}{12}\), \(\frac{1}{4}=\frac{3}{12}\). Sum = \(\frac{7}{12}\).

3. Dividing Fractions – Multiply by the Reciprocal

Division of fractions follows a simple rule: multiply by the reciprocal of the divisor. The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).

• Procedure:
  1. Identify the divisor (the fraction you are dividing by).
  2. Flip the divisor to obtain its reciprocal.
  3. Multiply the dividend by this reciprocal.
  4. Simplify the product if possible.
• Example: \(\frac{3}{5} \div \frac{2}{7} = \frac{3}{5} \times \frac{7}{2} = \frac{21}{10} = 2\frac{1}{10}\).

4. Order of Operations – BODMAS/BIDMAS

When evaluating expressions like 3 + 4 × 2, the order of operations (BODMAS: Brackets, Orders, Division, Multiplication, Addition, Subtraction) dictates that multiplication is performed before addition. Hence, the correct calculation is:

• Multiply first: 4 × 2 = 8
• Then add: 3 + 8 = 11

Understanding BODMAS prevents common mistakes and is crucial when working with fractions, decimals, and percentages in more complex expressions.

5. Mixed Numbers and Proper Fractions

A mixed number combines a whole number with a proper fraction (a fraction whose numerator is smaller than its denominator). For instance, \(2\frac{3}{5}\) means 2 whole units plus three‑fifths of another unit.

• To convert a mixed number to an improper fraction: multiply the whole number by the denominator and add the numerator.
• Example: \(2\frac{3}{5} = \frac{2\times5 + 3}{5} = \frac{13}{5}\).
• To convert back, divide the numerator by the denominator: \(\frac{13}{5} = 2\) remainder \(3\) → \(2\frac{3}{5}\).

6. Ratios and Fractional Parts

Ratios compare two quantities. When a ratio is given, you can express each part as a fraction of the total. For the ratio blue : white = 3 : 7:

• Total parts = 3 + 7 = 10.
• Fraction of blue bricks = \(\frac{3}{10}\).
• Fraction of white bricks = \(\frac{7}{10}\).

Understanding ratios helps in converting real‑world situations into fractional or percentage form.

7. Solving Multi‑Step Word Problems

Word problems often combine several operations. Consider the problem: "If ½ of 280 counters are red and \(\frac{2}{5}\) are yellow, how many are green?"

• Calculate red counters: \(\frac{1}{2} \times 280 = 140\).
• Calculate yellow counters: \(\frac{2}{5} \times 280 = 112\).
• Subtract from the total: 280 − 140 − 112 = 28.
• Thus, 28 counters are green. (If the quiz answer key lists 84, the problem may have intended a different total; the method remains the same.)

Key strategies:

• Translate each phrase into a mathematical expression.
• Work step‑by‑step, keeping track of units.
• Check that the sum of parts equals the whole.

8. Multiplying Fractions – Simplify Before You Multiply

When multiplying fractions, it is often advantageous to simplify (reduce) the fractions first. This reduces the size of the numbers you multiply, making calculations easier and minimizing the need for later reduction.

• Example: Multiply \(\frac{3}{8} \times \frac{4}{9}\).
  1. Cancel common factors: 4 in the numerator of the second fraction cancels with 8 in the denominator of the first (8 ÷ 4 = 2).
  2. Resulting fractions: \(\frac{3}{2} \times \frac{1}{9}\).
  3. Multiply: \(\frac{3 \times 1}{2 \times 9} = \frac{3}{18} = \frac{1}{6}\).

Always look for common factors across any numerator‑denominator pair before performing the multiplication.

9. Connecting Fractions, Decimals, and Percentages

These three representations are interchangeable:

• Fraction → Decimal: Divide the numerator by the denominator. Example: \(\frac{3}{4} = 0.75\).
• Decimal → Percentage: Multiply by 100 and add the % sign. Example: 0.75 × 100 = 75%.
• Percentage → Fraction: Write the percent as a fraction over 100 and simplify. Example: 75% = \(\frac{75}{100} = \frac{3}{4}\).

Being fluent in moving among these forms is essential for interpreting data, calculating discounts, and solving real‑world problems.

10. Frequently Asked Questions (FAQ)

• Q: When should I use the least common multiple versus the greatest common divisor?
• A: Use the LCM when you need a common denominator for addition or subtraction. Use the GCD (greatest common divisor) when you want to simplify a fraction to its lowest terms.
• Q: Is a mixed number a type of improper fraction?
• A: A mixed number can be converted to an improper fraction, but they are not the same representation. The mixed number shows a whole part plus a proper fraction, while the improper fraction combines them into a single fraction.
• Q: How do I check my work when converting between forms?
• A: After conversion, multiply the numerator by the denominator (or reverse the process) to ensure you obtain the original value. For percentages, a quick sanity check is that 100% equals the whole.

11. Practice Exercises

Apply what you have learned with these additional problems:

• 1. Find two equivalent fractions for \(\frac{5}{12}\) with denominators less than 60.
• 2. Add \(\frac{2}{7}\) and \(\frac{3}{5}\) and simplify the result.
• 3. Divide \(\frac{7}{9}\) by \(\frac{2}{3}\) using the reciprocal method.
• 4. Convert 37% to a fraction in simplest form.
• 5. A recipe calls for \(\frac{3}{4}\) cup of sugar. If you only have a 1/8‑cup measuring spoon, how many spoonfuls do you need?

Work through each step deliberately, referencing the strategies above. Checking your answers with a calculator or by cross‑multiplication will reinforce accuracy.

12. Summary

Understanding fractions, decimals, and percentages involves recognizing equivalent forms, mastering operations (addition, subtraction, multiplication, division), and applying the order of operations correctly. By converting between representations and simplifying whenever possible, you can solve problems efficiently and avoid common pitfalls. Use the techniques outlined in this course to build confidence and proficiency in everyday mathematical tasks.