3.2 Decimals & Recurring Decimals

Decimal questions run through the whole ICAS range: Paper D (Year 6) tests decimal place value and operations directly, money problems appear at every level, and by Papers G-H (Years 9-10) you must convert terminating and recurring decimals to fractions. Since personal calculators are banned, the written and mental methods below are exactly what the exam rewards.

Decimal Place Value

Each place is ten times the one to its right. In 3.472 there are 3 ones, 4 tenths, 7 hundredths and 2 thousandths. Adding zeros after the last decimal digit changes nothing (0.5 = 0.50 = 0.500), but a zero between the point and a digit matters enormously: 0.5 is ten times 0.05.

Ordering Decimals

Line up the decimal points and compare digit by digit from the left, padding with zeros so every number has the same length. To order 0.6, 0.58 and 0.605, write 0.600, 0.580, 0.605: clearly 0.580 < 0.600 < 0.605. The classic trap is claiming 0.58 > 0.6 because 58 > 6 — pad the numbers and the mistake disappears.

Adding and Subtracting

Align the decimal points and pad with zeros: 3.6 + 0.75 becomes 3.60 + 0.75 = 4.35, and 5 − 2.37 becomes 5.00 − 2.37 = 2.63. Most errors in ICAS decimal questions come from misaligned columns, not from the arithmetic itself.

Multiplying and Dividing

Multiplication: ignore the points and multiply the whole numbers, then place the point so the answer has as many decimal places as the two factors have combined. For 0.4 × 0.3: 4 × 3 = 12, with two decimal places in total, so the answer is 0.12. Estimation check: multiplying two numbers both below 1 must give something below 1, so an answer like 1.2 is instantly wrong.

Division: make the divisor a whole number by shifting both decimal points the same number of places. 4.5 ÷ 0.3 = 45 ÷ 3 = 15. Check by multiplying back: 15 × 0.3 = 4.5, which matches.

Multiplying and Dividing by 10, 100, 1000

The digits move; the decimal point stays put. Multiplying by 10 shifts digits one place left: 3.47 × 10 = 34.7 and 3.47 × 100 = 347. Dividing by 1000 shifts three places right: 52 ÷ 1000 = 0.052. Count the zeros in the multiplier to know how many places to shift — this is a frequent one-step ICAS question.

Money Problems

Money is decimals in disguise, and the cents column is where marks are lost: $4.05 is four dollars and five cents, not $4.50. Worked example: three items cost $2.75, $4.20 and $13.05. Add the cents (75 + 20 + 5 = 100 cents = $1) and the dollars (2 + 4 + 13 = 19), giving $20.00. Change from $50 is $30.00.

Converting Between Fractions and Decimals

Fraction to decimal: divide the numerator by the denominator, so 3/8 = 3 ÷ 8 = 0.375. Decimal to fraction: write the digits over the place value and simplify, so 0.075 = 75/1000 = 3/40. Worth memorising: 1/2 = 0.5, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2, 1/8 = 0.125, 3/8 = 0.375, 5/8 = 0.625, 7/8 = 0.875 and 1/3 = 0.333…

Terminating vs Recurring Decimals (Papers G-H)

A terminating decimal ends (0.375); a recurring decimal repeats a block forever (0.333…, written with a dot over the 3; or 0.3636…). Here is the test ICAS expects you to know: a fraction in lowest terms terminates only when its denominator's prime factors are nothing but 2s and 5s. So 3/8 terminates (8 = 2³) but 5/6 recurs (6 = 2 × 3, and the 3 spoils it).

To convert a recurring decimal to a fraction, shift and subtract:

• One repeating digit: x = 0.777…, so 10x = 7.777… and 10x − x = 7, giving 9x = 7 and x = 7/9.
• Two repeating digits: x = 0.3636…, so 100x = 36.3636… and 99x = 36, giving x = 36/99 = 4/11.
• A non-repeating part first: x = 0.1666…, so 10x = 1.666… and 100x = 16.666…, giving 90x = 15 and x = 15/90 = 1/6.

Always simplify the final fraction — ICAS options appear in lowest terms.

Rounding Decimals

Look at the first digit you are dropping: if it is 5 or more, round up. So 3.746 to two decimal places is 3.75, to one decimal place is 3.7, and to the nearest whole number is 4. In multi-step problems keep full precision until the final step, and watch for the phrase 'correct to', which tells you exactly the precision ICAS wants.

Estimation: Your Built-In Check

Before accepting any decimal answer, sanity-check it: 4.9 × 6.1 should be near 5 × 6 = 30, and 23.7 ÷ 4 should be near 6. With no personal calculator, this habit catches the place-value slips that ICAS distractor options are designed around.