Decimals and Percentages

Syllabus Identity

Curriculum: Mathematics
Topic ID: topic-csee-basic-mathematics-2005-[[decimal|decimals]]-and-[[percentage|percentages]]
Form: Form I
Hub: Number and Computation
Competence grouping: Number computation and estimation

This is a current Mathematics syllabus topic. It preserves the 2005 Basic Mathematics identity and order for exam-facing mapping. Do not merge it into the 2023 Mathematics transition topic page even when the learning idea overlaps.

Official Scope

Current Mathematics syllabus topic covering decimals; operations on decimals; percentages; word problems involving decimals and percentages.

Subtopics

Decimals
Operations on decimals
Percentages
Word problems involving decimals and percentages

Core Concepts

Decimals

A decimal is a mathematical notation used to represent a fraction whose denominator is a power of $10$ (such as $10$, $100$, $1000$). The decimal point separates the whole number part on the left from the fractional part on the right. The place values to the right of the decimal point represent tenths ($\frac{1}{10}$), hundredths ($\frac{1}{100}$), thousandths ($\frac{1}{1000}$), and so on. For instance, the decimal number $4.56$ consists of $4$ whole units, $5$ tenths, and $6$ hundredths.

Operations on Decimals

Addition and Subtraction: To add or subtract decimals, you must align the numbers vertically by their decimal points. This ensures that you are combining digits with the same exact place value. Fill in empty decimal places with zeros if necessary.
Multiplication: Multiply the numbers as if they were whole numbers (ignoring the decimal point initially). Then, count the total number of decimal places across all the factors being multiplied. The final product must have exactly this combined number of decimal places.
Division: To divide by a decimal, move the decimal point of the divisor to the right until it becomes a whole number. Move the decimal point of the dividend (the number being divided) the same number of places to the right. Then, divide as usual, placing the decimal point in the quotient directly above the new decimal point in the dividend.

Percentages

The term percentage is derived from the Latin per centum, meaning "out of one hundred" or "per hundred". It is simply a specialized fraction with a denominator of $100$, denoted by the symbol $\%$.

Converting a percentage to a fraction or decimal: Divide the percentage value by $100$ and remove the $\%$ sign. For example, $45\% = \frac{45}{100} = 0.45$.
Converting a fraction or decimal to a percentage: Multiply the value by $100$ and attach the $\%$ sign. For example, $\frac{3}{5} = \left(\frac{3}{5} \times 100\right)\% = 60\%$.

Word Problems Involving Decimals and Percentages

Word problems require translating text-based scenarios into mathematical expressions. A common calculation is finding a percentage of a given quantity, which is done using the formula: $\text{Value} = \frac{\text{Percentage}}{100} \times \text{Total Quantity}$.

A critical concept in these problems is dealing with "remaining" amounts. When a portion of a quantity is spent or removed, the remaining amount is the original total minus the portion removed. If a subsequent percentage is given "of the remainder", it must be calculated based on this new balance, not the original starting total.

Worked Examples

Example 1: Operations on Decimals Evaluate $3.45 + 12.6 \times 0.5$, applying the proper order of operations (BODMAS).

Step 1: Perform multiplication first. Multiply $126$ by $5$ to get $630$. There is $1$ decimal place in $12.6$ and $1$ decimal place in $0.5$, giving a total of $2$ decimal places. Thus, $12.6 \times 0.5 = 6.30 = 6.3$.

Step 2: Perform addition. Align the decimal points vertically: $$ \begin{array}{r@{\quad}l} 3.45 \\ + 6.30 \\ \hline 9.75 \end{array} $$ Answer: $9.75$

Example 2: Finding Percentages and Decimals from a Total In a farm with $250$ animals, $150$ are goats, and the rest are sheep. Find the percentage of sheep on the farm, and express the number of goats as a decimal fraction of the total number of animals.

Step 1: Find the number of sheep. $$\text{Number of sheep} = 250 - 150 = 100$$

Step 2: Calculate the percentage of sheep. $$\text{Percentage of sheep} = \left( \frac{\text{Number of sheep}}{\text{Total animals}} \times 100 \right)\%$$ $$\text{Percentage of sheep} = \left( \frac{100}{250} \times 100 \right)\% = \left( \frac{2}{5} \times 100 \right)\% = 40\%$$

Step 3: Express the number of goats as a decimal. $$\text{Fraction of goats} = \frac{150}{250} = \frac{3}{5}$$ $$\text{Decimal} = 3 \div 5 = 0.6$$ Answer: The percentage of sheep is $40\%$, and the goats represent $0.6$ of the total animals.

Example 3: Percentage of a Remaining Quantity John had $40,000$ TZS. He spent $20\%$ of it on food and $25\%$ of the remaining money on transport. How much money did he have left?

Step 1: Calculate the money spent on food. $$\text{Food cost} = 20\% \text{ of } 40,000 = \frac{20}{100} \times 40,000 = 8,000 \text{ TZS}$$

Step 2: Calculate the remaining amount. $$\text{Remaining after food} = 40,000 - 8,000 = 32,000 \text{ TZS}$$

Step 3: Calculate the money spent on transport (from the remaining money). $$\text{Transport cost} = 25\% \text{ of } 32,000 = \frac{25}{100} \times 32,000 = \frac{1}{4} \times 32,000 = 8,000 \text{ TZS}$$

Step 4: Calculate the final money left. $$\text{Final money left} = 32,000 - 8,000 = 24,000 \text{ TZS}$$ Answer: John had $24,000$ TZS left.

NECTA Exam Focus

An analysis of past NECTA CSEE papers reveals several distinct ways this topic is assessed:

Contextual Word Problems: Questions frequently feature real-world scenarios, such as allocating money or grouping individuals (e.g., classifying boys and girls in a classroom). Students are expected to seamlessly convert between basic fractions, decimals, and percentages to find the required values.
"Of the Remaining" Traps: Multi-step expenditure is a highly recurring theme. NECTA will often specify that a second percentage is taken from the remaining amount, not the original total. A common pitfall is adding the percentages together first (e.g., $35\% + 10\% = 45\%$ of the total), which yields an incorrect result. Always calculate the new numerical balance before applying the next percentage.
Rounding to Specified Decimal Places: Decimal concepts are extensively integrated into Geometry and Trigonometry questions. When calculating side lengths of triangles or prisms, NECTA tests your precision by asking you to leave your answer correct to a specific number of decimal places (e.g., "correct to one decimal place"). Failing to follow the standard rules of rounding leads to unnecessary loss of marks.

Practice Problems

[2019 Paper 1] Mary was given $60,000$ shillings by her mother. She spent $35\%$ of the money to buy shoes and $10\%$ of the remaining money to buy books. How much money remained?
[2024 Paper 1] In a class of $40$ students, $17$ students are boys and the rest are girls. Determine:

The percentage of girls and boys.
The number of boys in decimal.

[2025 Paper 1] Three fifths of the pupils in a certain school come from the city centre. What is the percentage of pupils who do not come from the city centre?

Crosswalk Notes

Cross-version relationships are drafted in data/curricula/crosswalks/csee-basic-mathematics-2005-to-mathematics-2023.json. Partial and 2005-only mappings remain reviewable.