3.2 Decimals: Operations, Terminating and Recurring Decimals, and Equivalence

Decimals extend the base-$10$ place-value system to parts of a whole using tenths ($\frac{1}{10} = 0.1$), hundredths ($\frac{1}{100} = 0.01$), thousandths ($\frac{1}{1000} = 0.001$), and beyond. On the LANTITE Numeracy test, decimal items frequently assess whether you can perform non-calculator arithmetic accurately and avoid common place-value misconceptions.

Ordering Decimals of Different Lengths

A widespread misconception among school students—and a classic trap under exam time pressure—is the whole-number length misconception: assuming that a decimal with more digits after the decimal point is larger (falsely thinking $0.408 > 0.45 > 0.5$ because $408 > 45 > 5$).

In reality, place value decreases by a factor of $10$ with each step to the right. To compare decimals reliably, pad each number with trailing zeros so all values have the same number of decimal places, then compare from left to right:

• $0.408 = \frac{408}{1000}$
• $0.45 = 0.450 = \frac{450}{1000}$
• $0.5 = 0.500 = \frac{500}{1000}$

Comparing thousandths makes the true ascending order unmistakable: $0.408 < 0.45 < 0.5$. Be careful in athletics track events (such as a $100\text{ m}$ sprint), where the smallest time in seconds represents the fastest runner, versus field events (such as long jump), where the largest distance in metres wins.

Operations with Decimals Without a Calculator

Adding and Subtracting Decimals

Line up the decimal points vertically so that tenths sit directly beneath tenths, hundredths beneath hundredths, and thousandths beneath thousandths. Always insert trailing zeros when adding or subtracting decimals of different lengths so that every column has a visible digit.

• Addition with ragged decimals: To add $12.8 + 4.075 + 0.96$, rewrite the sum with three decimal places as $12.800 + 4.075 + 0.960 = 17.835$.
• Subtraction from a shorter decimal or whole number: If a relay swimmer finishes a leg in $48.76\text{ seconds}$ against a qualifying target of $54.2\text{ seconds}$, pad $54.2$ to $54.20$ before subtracting: $54.20 - 48.76 = 5.44\text{ seconds}$ under the target. Similarly, change from a $20 note for a $13.45 book order is $20.00 - 13.45 = 6.55$, or $6.55.

Multiplying Decimals: Counting Total Decimal Places

When multiplying decimals in Section 2, do not line up the decimal points. Instead:

1. Ignore the decimal points and multiply the numbers as whole integers.
2. Count the total number of digits after the decimal point across both factors.
3. Place the decimal point in the product so that it has that exact total number of decimal places, inserting leading zeros if needed.

Consider the classic exam trap $0.06 \times 0.4$:

• Multiply the non-zero integers: $6 \times 4 = 24$.
• Count decimal places: $0.06$ has $2$ decimal places and $0.4$ has $1$ decimal place, giving a total of $2 + 1 = 3$ decimal places.
• Place the decimal point three places from the right: $0.024$ (never $0.24$).

Dividing by a Decimal

Long division directly by a decimal divisor is error-prone. Instead, write the division as a fraction and multiply both the dividend and the divisor by the same power of $10$ ($10$, $100$, or $1000$) so that the divisor becomes a whole number. Because you scale numerator and denominator equally, the quotient does not change.

For example, to evaluate $14.4 \div 0.12$, multiply both numbers by $100$ (shifting both decimal points two places to the right):

$$14.4 \div 0.12 = \frac{14.4 \times 100}{0.12 \times 100} = \frac{1440}{12} = 120$$

Notice that $14.4$ becomes $1440$, not $144$, because shifting two places to the right requires appending a zero.

Trap: Why Dividing by a Decimal Between $0$ and $1$ Gives a Larger Result

Candidates often expect multiplication to make numbers larger and division to make numbers smaller. That intuition only holds when the multiplier or divisor is greater than $1$. When operating on a positive number $N$ with a decimal $d$ between $0$ and $1$ ($0 < d < 1$):

• Multiplying ($N \times d$) produces a smaller result because you are taking only a proper fraction of $N$ (for example, $15 \times 0.2 = 3$).
• Dividing ($N \div d$) produces a larger result because you are finding how many small fractional parts of size $d$ fit into $N$ (for example, $15 \div 0.2 = 150 \div 2 = 75$).

Converting Between Fractions and Decimals

Terminating vs. Recurring Decimals

To convert any fraction $\frac{a}{b}$ to a decimal without a calculator, either scale the denominator to $10$, $100$, or $1000$, or divide the numerator by the denominator ($a \div b$) using short division with trailing zeros ($3 \div 8 = 3.000 \div 8 = 0.375$).

A simplified fraction $\frac{a}{b}$ produces a terminating decimal if and only if the prime factorisation of its denominator $b$ contains only $2$s and $5$s (since $10 = 2 \times 5$). If the simplified denominator contains any other prime factor (such as $3$, $7$, or $11$), the division never ends and produces a recurring (repeating) decimal, denoted by a dot above the repeating digit(s).

Converting Decimals Back to Simplified Fractions

To convert a terminating decimal to a fraction, read the place value of the final digit, write the digits over that power of $10$ ($10$ for tenths, $100$ for hundredths, $1000$ for thousandths), and divide by the HCF:

$$0.625 = \frac{625}{1000} = \frac{625 \div 125}{1000 \div 125} = \frac{5}{8}, \qquad 0.045 = \frac{45}{1000} = \frac{45 \div 5}{1000 \div 5} = \frac{9}{200}$$

In Section 1 (Calculator), avoid rounding recurring decimals prematurely during multi-step calculations: if an intermediate step involves $\frac{1}{3}$ or $\frac{2}{3}$ of a budget, multiply by $1$ and divide by $3$ (or multiply by $2$ and divide by $3$) rather than typing $0.33$ or $0.67$, which introduces rounding error.