Fractions And Decimals

What Are Fractions?

A fraction represents a part of a whole or a ratio of two integers. It has a numerator (top number) and a denominator (bottom number). For example, 3/4 means 3 parts out of 4 equal parts.

• Proper fractions: numerator < denominator (e.g., 3/5).
• Improper fractions: numerator ≥ denominator (e.g., 9/4).
• Mixed numbers: a whole number plus a fraction (e.g., 2 1/3).
• Equivalent fractions: different-looking fractions that name the same value (e.g., 1/2 = 2/4 = 50/100).
• Simplest form: a fraction whose numerator and denominator share no common factor greater than 1 (e.g., 8/12 simplifies to 2/3).

What Are Decimals?

Decimals use place value to show parts of a whole to the right of the decimal point. Each place is a power of 10: tenths (0.1), hundredths (0.01), thousandths (0.001), and so on.

• Terminating decimals: decimals that end (e.g., 0.75).
• Repeating decimals: decimals with a repeating pattern (e.g., 0.333... for 1/3).
• Rounding: adjust to a specified place for estimation or reporting (e.g., 2.678 rounded to the hundredths place is 2.68).

Converting Between Fractions and Decimals

• Fraction to decimal: divide the numerator by the denominator. Example: 3/4 = 3 ÷ 4 = 0.75. If the division does not end, the decimal repeats (e.g., 1/3 = 0.333...).
• Decimal to fraction (terminating): read the place value and simplify. Example: 0.125 = 125/1000 = 1/8.
• Decimal to fraction (repeating): use an algebraic approach or known patterns. Example: 0.3 repeating = 1/3, 0.16 repeating (16 repeats) = 16/99, then simplify if possible.

Operations With Fractions

• Add/Subtract: use a common denominator, then add or subtract numerators. Example: 1/4 + 3/8 = 2/8 + 3/8 = 5/8.
• Multiply: multiply numerators and denominators; simplify. Example: 2/3 × 9/10 = 18/30 = 3/5.
• Divide: multiply by the reciprocal of the divisor. Example: (5/6) ÷ (2/3) = (5/6) × (3/2) = 15/12 = 5/4.
• Mixed numbers: convert to improper fractions before multiplying or dividing; for addition/subtraction, you can add wholes and fractional parts separately using a common denominator.

Operations With Decimals

• Add/Subtract: line up decimal points by place value, then compute.
• Multiply: ignore decimal points, multiply as whole numbers, then place the decimal so total decimal places match the add-up of factors.
• Divide: move the decimal in the divisor to make it a whole number and move the dividend’s decimal by the same amount; then divide.

Comparing and Ordering

• Using fractions: find a common denominator or compare to benchmarks like 0, 1/2, and 1.
• Using decimals: compare place by place (tenths, hundredths, etc.).
• Convert: switch to either all fractions or all decimals to compare consistently.
• Number line: place values visually to compare and order.

Connections and Applications

• Percent: a percent is a fraction out of 100; 0.75 = 75% = 3/4.
• Ratios and rates: fractions represent part-to-part and part-to-whole comparisons; decimals often express rates (e.g., money, unit price).
• Measurement and data: decimals and fractions express lengths, masses, times, and averages with precision.

Common Pitfalls and Tips

• Do not cancel across addition or subtraction (e.g., you cannot cancel the 3 in (3 + 6)/3 to get 1 + 2).
• Remember that a larger denominator means smaller parts; 1/8 is less than 1/6.
• Always simplify fractions when possible, but keep exact forms when needed.
• Estimate first to check reasonableness (e.g., 0.49 + 0.52 is about 1.01).

Practice Ideas

• Convert 5/8, 2/5, and 7/20 to decimals.
• Add 3/10 + 7/100 and write the answer as a decimal and fraction.
• Multiply 1.25 × 0.4 and explain where the decimal goes.
• Order these from least to greatest: 0.6, 2/3, 0.58, 5/9.

Next Topic