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 06 Pathway: N/A

First Principles

Objective: Add, subtract, multiply, and divide decimals confidently, and explain why the decimal point moves.

Concrete scenario: Imagine a 1-litre jerrycan of milk. If you divide it into 10 equal glass cups, each cup is \(\frac{1}{10}\) of a litre, written as \(0.1\) L. Pouring 3 cups gives \(\frac{3}{10} = 0.3\) L. If you take one cup and split it into 10 equal spoonfuls, each spoonful is \(\frac{1}{100}\) of the full litre, or \(0.01\) L.

Geometric insight: Each time you split a piece into ten, you zoom in one place-value level. The decimal point acts as the anchor between wholes (on the left) and fractional parts (on the right). Tenths are first, hundredths second, and thousandths third — each exactly ten times smaller than the spot to its left.

Algebraic rule: Because our number system is base-10, shifting the decimal point one place to the right multiplies the quantity by \(10\). Shifting it one place to the left divides it by \(10\). That is why \(2.5 \times 10 = 25\) and \(25 \div 10 = 2.5\).

Splitting Bar — Tap to Drill In

Level: Whole — Tap the bar to split into tenths

Each place after the decimal point is ten times smaller than the one before it.

Align decimal points when adding or subtracting; count decimal places when multiplying.

Moving the decimal point right multiplies by 10; moving it left divides by 10.

Next lesson: Converting between fractions, decimals, and percentages — and using that fluency to compare quantities in shopping, measurement, and data.

Key Formulas

\(\frac{d}{10^{n}}\) — A digit \(d\) in the \(n\)th place after the decimal point is worth \(d\) divided by \(10^n\). So 0.3 means \(\frac{3}{10}\), and 0.05 means \(\frac{5}{100}\).

\(N \times 10^{k}\) — Move the decimal point \(k\) places to the right. The number gets larger because place values increase by powers of 10.

\(N \div 10^{k}\) — Move the decimal point \(k\) places to the left. The number gets smaller as place values decrease by powers of 10.

\(x + y\) — Align decimal points vertically, pad with trailing zeros if necessary, and add column by column.

\(x - y\) — Align decimal points vertically, pad trailing zeros, and subtract column by column, borrowing across columns as needed.

\(d_1 \times d_2\) — Multiply factors as whole numbers. Sum the total decimal places in both factors, then place the decimal point that many positions from the right.

\(\frac{a}{b}\) — If the divisor is a decimal, multiply both dividend and divisor by \(10^k\) until the divisor is a whole number, then divide normally.

Worked Examples

Example 1 (Easy): Add \(0.4 + 0.35\)

Problem: Find the sum of \(0.4\) and \(0.35\).

Align Place Values: Pad \(0.4\) with a trailing zero to make it \(0.40\) so both numbers have two decimal places.
Add Columns: Line up \(0.40\) and \(0.35\). Hundredths: \(0 + 5 = 5\). Tenths: \(4 + 3 = 7\). Wholes: \(0 + 0 = 0\).
Final Result: \(0.75\).

Answer: \( 0.75 \)

Example 2 (Medium): Multiply \(0.6 \times 0.25\)

Problem: Find the product of \(0.6\) and \(0.25\).

Multiply as Whole Numbers: Ignore the decimal points temporarily: \(6 \times 25 = 150\).
Count Decimal Places: \(0.6\) has 1 decimal place; \(0.25\) has 2 decimal places. Total = \(1 + 2 = 3\) decimal places.
Place the Point: Count 3 positions left from the end of 150: \(0.150\), which simplifies to \(0.15\).

Answer: \( 0.15 \)

Example 3 (Hard): Divide \(4.2 \div 0.07\)

Problem: Compute \(4.2 \div 0.07\).

Make Divisor a Whole Number: The divisor \(0.07\) has 2 decimal places. Multiply both numbers by \(100\) to shift decimal points 2 places right.
Rewrite Equation: \(4.2 \times 100 = 420\) and \(0.07 \times 100 = 7\). The problem becomes \(420 \div 7\).
Divide: \(420 \div 7 = 60\).

Answer: \( 60 \)

Common Mistakes

Mistake \(0.3 + 0.7 = 0.10\)

Correction \(0.3 + 0.7 = 1.0\)

Why it feels right Students often add 3 and 7 to get 10 and tack it onto the end of "0.". However, 3 tenths plus 7 tenths equals 10 tenths, which regroup into 1 full whole unit.

Mistake Thinking \(0.45\) is larger than \(0.8\) because 45 is larger than 8.

Correction \(0.8 > 0.45\)

Why it feels right Students treat decimal digits as separate whole numbers. Line them up by place value: \(0.80\) vs \(0.45\). 8 tenths is larger than 4 tenths.

Mistake \(0.2 \times 0.3 = 0.6\)

Correction \(0.2 \times 0.3 = 0.06\)

Why it feels right Multiplying \(2 \times 3 = 6\) is correct, but forgetting to sum the decimal places (1 + 1 = 2) leads to placing only one decimal digit instead of two.

Real World

A vendor sells passion fruit juice at KSh 45.50 per bottle. If you buy 3 bottles, compute \(45.50 \times 3\): \(45 \times 3 = 135\) and \(0.50 \times 3 = 1.50\), totaling KSh 136.50.

A 5-litre container of cooking oil costs KSh 1,250.50. To find the cost per litre, compute \(1250.50 \div 5 = 250.10\) (KSh 250.10 per litre).

If a tailor needs \(1.4\) metres of kitenge fabric for a shirt and buys a \(5.0\) metre roll, the fabric remaining is \(5.0 - 1.4 = 3.6\) metres.

Practice

Evaluate \(0.65 + 0.28\). (Type only the numerical decimal value, e.g., 0.93)

Review the concepts above.

A Matatu fare costs KSh 80.50. You pay with a KSh 100 note. How much change in KSh should you receive? (Type only the numerical value, e.g., 19.5)

Review the concepts above.

Evaluate \(0.4 \times 0.15\). (Type only the numerical value, e.g., 0.06)

Review the concepts above.

A butcher packages meat into bags weighing \(0.75\) kg each. What is the total mass in kg of 6 such bags? (Type only the numerical value, e.g., 4.5)

Review the concepts above.

Evaluate \(3.6 \div 0.09\). (Type only the numerical integer value, e.g., 40)

Review the concepts above.

A farmer harvested 12.5 kg of passion fruit. She sold 4.8 kg at a market and distributed the remaining fruit equally into 7 small baskets for her neighbours. How many kg of passion fruit were in each basket? (Type only the numerical decimal value, e.g., 1.1)

Review the concepts above.