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Learning Resources

Decimals

Interactive curriculum lessons, worked examples, and geometric problem-solving techniques designed to help Kenyan students master CBC, KPSEA, KCSE, and IGCSE mathematics.

Grade 07 Pathway: N/A

First Principles

Core Concept: Every common fraction \(\frac{p}{q}\) represents the division \(p \div q\). Decimals are simply place-value representations of fractions whose denominators are powers of ten (\(10, 100, 1000, \dots\)).

🔬 Interactive Step-by-Step Long Division Engine

Enter a fraction to see whether it terminates or enters an infinite repeating loop (recurring decimal):

1. Concrete Real-Life Context

Imagine 3 market traders in Nakuru sharing a KSh 100 profit equally. Each trader receives \(100 \div 3\) shillings. In long division:

  • \(100 \div 3 = 33\) remainder \(1\).
  • Multiply remainder by \(10\): \(10 \div 3 = 3\) remainder \(1\).
  • Bring down another zero: \(10 \div 3 = 3\) remainder \(1\) again!

Because the remainder \(1\) repeats endlessly, each trader's exact theoretical share is \(33.3333\dots = 33.\overline{3}\) shillings.

2. The Remainder Principle (Why Decimals Recur)

When dividing by an integer \(q\), the only possible remainders are \(\{0, 1, 2, \dots, q-1\}\). Since there is a finite number of possible remainders:

  • If the remainder becomes 0, the division stops \(\rightarrow\) Terminating Decimal.
  • If the remainder never reaches 0, it must repeat a previous remainder within at most \(q-1\) steps \(\rightarrow\) Recurring Decimal.

Key Formulas

1. Fraction to Decimal Conversion:\[\frac{p}{q} = p \div q\]Carry out standard long division by placing a decimal point and appending trailing zeros to the dividend.
2. Terminating Decimal Criterion:\[\frac{p}{q} \text{ terminates } \iff q = 2^a \times 5^b \quad (a, b \ge 0)\]A fully simplified fraction terminates if and only if the prime factorisation of its denominator contains ONLY 2s and/or 5s.
3. Terminating Decimal to Fraction:\[0.a_1a_2\ldots a_n = \frac{a_1a_2\ldots a_n}{10^n}\]Count the number of decimal places \(n\), use denominator \(10^n\), and simplify to lowest terms.
4. Pure Recurring Decimal to Fraction:\[0.\overline{d_1 d_2 \dots d_k} = \frac{d_1 d_2 \dots d_k}{\underbrace{99\dots 9}_{k \text{ nines}}}\]Example: \(0.\overline{7} = \frac{7}{9}\), \(0.\overline{27} = \frac{27}{99} = \frac{3}{11}\).

Worked Examples

Example 1 (Easy): Converting Terminating Decimal to Fraction

Problem: Convert \(0.625\) into a simplified common fraction in its lowest terms.

Step-by-Step Solution:

  1. Identify place value: There are 3 decimal digits, so the place value denominator is \(10^3 = 1000\):\[0.625 = \frac{625}{1000}\]
  2. Find common factors: Divide numerator and denominator by highest common factor (or step-by-step by 25):\[\frac{625 \div 25}{1000 \div 25} = \frac{25}{40}\]
  3. Simplify completely: Divide numerator and denominator by 5:\[\frac{25 \div 5}{40 \div 5} = \frac{5}{8}\]

Final Answer: \(\mathbf{\frac{5}{8}}\)

Example 2 (Medium): Converting Fraction to Recurring Decimal

Problem: Convert \(\frac{5}{6}\) into decimal notation. State whether it is terminating or recurring.

Step-by-Step Solution:

  1. Test denominator: \(6 = 2 \times 3\). Because there is a prime factor of 3, the decimal must recur.
  2. Perform long division (\(5.000 \div 6\)):
    • \(5 \div 6 = 0\), remainder \(5\). Place decimal point: \(0.\)
    • Bring down 0: \(50 \div 6 = 8\) remainder \(2\) (since \(6 \times 8 = 48\)). Quotient: \(0.8\)
    • Bring down 0: \(20 \div 6 = 3\) remainder \(2\) (since \(6 \times 3 = 18\)). Quotient: \(0.83\)
    • Bring down 0: \(20 \div 6 = 3\) remainder \(2\). Remainder \(2\) repeats indefinitely!
  3. Apply bar notation: Only the digit 3 repeats:\[\frac{5}{6} = 0.8333\dots = 0.8\overline{3}\]

Final Answer: \(\mathbf{0.8\overline{3}}\) (Recurring decimal)

Example 3 (Hard): Converting Recurring Decimal to Fraction via Algebra

Problem: Convert \(0.\overline{45} = 0.454545\dots\) into a simplified common fraction.

Step-by-Step Solution:

  1. Set up algebraic variable: Let \(x = 0.454545\dots\)   (Equation 1)
  2. Multiply by \(10^k\): Since the repeating block has \(k = 2\) digits, multiply by \(10^2 = 100\):\[100x = 45.454545\dots\quad \text{\textit{(Equation 2)}}\]
  3. Subtract Equation 1 from Equation 2:\[\begin{aligned}100x - x &= (45.454545\dots) - (0.454545\dots) \\99x &= 45\end{aligned}\]
  4. Solve for \(x\) and simplify:\[x = \frac{45}{99} = \frac{45 \div 9}{99 \div 9} = \frac{5}{11}\]

Final Answer: \(\mathbf{\frac{5}{11}}\)

Common Mistakes

Mistake 1: Writing \(\frac{1}{3} = 0.33\) as an exact equality.
Why it feels right: Calculators and school worksheets frequently display two decimal places, giving the illusion that division terminates after two steps.
Correction: \(0.33 = \frac{33}{100} \neq \frac{1}{3}\). The exact value is \(0.\overline{3}\) or \(0.333\dots\). Rounding introduces approximation error.
Mistake 2: Putting the recurring bar over non-repeating digits (e.g., writing \(\frac{1}{6} = 0.\overline{16}\)).
Why it feels right: Learners notice both '1' and '6' after the decimal point and bar the whole decimal part.
Correction: In \(\frac{1}{6} = 0.16666\dots\), only the digit 6 repeats. The correct notation is \(0.1\overline{6}\).
Mistake 3: Converting \(0.\overline{7}\) as \(\frac{7}{10}\).
Why it feels right: Confusing terminating decimals with recurring decimals.
Correction: \(0.7 = \frac{7}{10}\), but \(0.\overline{7} = \frac{7}{9}\). Recurring single digits have denominators of 9, not 10.

Real World

M-Pesa and Shared Bills: When 3 university roommates in Nairobi split an electricity bill of KSh 1,000 via mobile money, each share is \(\frac{1000}{3} = 333.333\dots\) KSh. Because payment systems only handle 2 decimal places, two roommates send KSh 333.33 and one sends KSh 333.34 to settle the exact total.
Land Surveying and Construction: In standard civil engineering, plot boundaries measured in 1/8 or 1/16 sections of an acre must be converted into decimal metres. \(\frac{1}{8}\text{ acre} = 0.125\text{ acre}\) terminates cleanly, preventing boundary disputes.
Grain Mill Dispensing: Posho mill operators in Eldoret measure maize by fraction sacks (e.g., \(\frac{3}{4}\) sack = \(0.75\) sack, \(\frac{5}{8}\) sack = \(0.625\) sack) while electronic scales record decimal kilograms.
Textile and Tailoring: Kitenge fabric rolls measured in thirds of a metre require tailors to understand that \(\frac{2}{3}\text{ m} = 0.667\text{ m}\) is rounded, but the actual cut must be exact to prevent running short.

Practice

Convert the common fraction \(\frac{3}{5}\) into a decimal. (Type only the number, e.g., 0.5)
Review the concepts above.
Convert the decimal 0.45 into a fraction in its simplest form \(\frac{a}{b}\). What is the value of the numerator \(a\)? (Type only the number, e.g., 7)
Review the concepts above.
A farmer in Eldoret measures a field section as \(\frac{7}{8}\) of a hectare. Convert \(\frac{7}{8}\) to a decimal. (Type only the number, e.g., 0.125)
Review the concepts above.
Convert the recurring decimal \(0.\overline{6} = 0.666\dots\) into a simplified common fraction \(\frac{a}{b}\). What is the value of the denominator \(b\)? (Type only the number, e.g., 3)
Review the concepts above.
Convert the recurring decimal \(0.\overline{27} = 0.272727\dots\) into a fraction in lowest terms \(\frac{a}{b}\). What is the value of the denominator \(b\)? (Type only the number, e.g., 11)
Review the concepts above.
A shopkeeper at a market packs \(3.375\) kg of green grams into a container. She writes the weight as a mixed number in simplest form \(3\frac{a}{b}\). What is the value of the denominator \(b\)? (Type only the number, e.g., 8)
Review the concepts above.