Ordering and Calculating with Decimals

British & IGCSE › Checkpoint 8 › Checkpoint 8 Maths — Complete Course

In Stage 6 and 7 you added, subtracted, multiplied and divided decimals by whole numbers. In Stage 8 you go further: you compare negative decimals and fractions, you multiply a decimal by another decimal, and you divide by a decimal such as 0.6. The written methods are the ones you already know — the new skill is knowing where the decimal point goes and checking with an estimate.

Key words

1. Comparing and ordering decimals

Positive decimals. Compare digit by digit from the left, place by place. It helps to give every number the same number of decimal places by writing zeros on the end — this does not change the value.

Negative decimals. Think of a number line: numbers get smaller as you move left. −0.6 is closer to zero than −0.75, so −0.6 is the bigger number. A quick rule: for two negative numbers, the one that looks 'bigger' without its minus sign is actually smaller. Any positive number is bigger than any negative number.

Fractions and decimals together. Change every fraction to a decimal (divide the numerator by the denominator), then compare the decimals. Learn the common ones by heart:

For harder fractions, divide: 5/6 = 5 ÷ 6 = 0.8333… and 7/9 = 7 ÷ 9 = 0.777… Write enough decimal places to see the difference (at least 3).

2. The six symbols

The symbol always opens towards the bigger number (the 'mouth' eats the bigger one). Remember that ≥ and ≤ are true when the two numbers are equal, but > and < are false when they are equal: 0.75 ≤ 3/4 is true, but 0.75 < 3/4 is false. Also note 0.3 ≠ 1/3, because 1/3 = 0.333…

3. Multiplying decimals

Method — 'ignore the points, then put them back':

1. Estimate by rounding each number (often to 1 significant figure).
2. Remove the decimal points and multiply the whole numbers (use any method: grid, column).
3. Count the decimal places in the question — add the counts for both numbers.
4. Your answer has the same number of decimal places. Count them from the right; write extra zeros at the front if you need them.
5. Only now remove any zeros at the end (4.500 = 4.5). Compare with your estimate.

Why does it work? 0.3 = 3 tenths and 0.4 = 4 tenths. Tenths × tenths = hundredths, so 0.3 × 0.4 = 12 hundredths = 0.12. The diagram shows it as an area.

Big idea: multiplying by a number between 0 and 1 makes the answer smaller (8 × 0.4 = 3.2, which is less than 8). Multiplying by a number bigger than 1 makes it bigger. Use this to check your answer is sensible.

Using a known fact. If you know 23 × 47 = 1081, you can write many related answers without calculating again — just count decimal places: 2.3 × 4.7 = 10.81 (2 d.p.), 0.23 × 47 = 10.81, 2.3 × 0.47 = 1.081 (3 d.p.), 230 × 4.7 = 1081 (the zero and the decimal place cancel out).

4. Dividing by a decimal with one decimal place

Dividing by a decimal like 0.6 is awkward. The trick: multiply both numbers by 10 so the divisor becomes a whole number. The answer does not change, because a division is like a fraction and 7.56/0.6 = 75.6/6 (equivalent fractions).

1. Estimate using friendly numbers, e.g. 7.56 ÷ 0.6 ≈ 7.2 ÷ 0.6 = 12.
2. Multiply both the number and the divisor by 10 (move each decimal point one place right).
3. Do the division by a whole number (short division). Keep the decimal point in the answer above the point in the number.
4. Check: answer × divisor should give the number (12.6 × 0.6 = 7.56 ✓), and the answer should be close to your estimate.

Big idea: dividing by a number between 0 and 1 makes the answer bigger (18 ÷ 0.4 = 45, which is more than 18). That is because you are asking 'how many 0.4s fit into 18?' — and lots of small pieces fit.

Worked examples

Worked example 1 — Write these numbers in order, starting with the smallest: 0.65 −0.6 2/5 −3/4 −0.06 [2]

1. Change the fractions to decimals: 2/5 = 0.4 and −3/4 = −0.75.
2. Negatives first. Their sizes without signs are 0.6, 0.75, 0.06. The biggest of these (0.75) gives the smallest number: −0.75 < −0.6 < −0.06.
3. Positives next: 0.4 < 0.65.
4. Write the answer in the original form: −3/4, −0.6, −0.06, 2/5, 0.65

Common mistake: putting −0.06 first because '6 hundredths looks smallest'. −0.06 is very close to zero, so it is the biggest negative. Another mistake: writing the decimals (−0.75, 0.4) instead of the numbers you were given.

Worked example 2 — Write one of the symbols =, ≠, <, > in each box. (a) −0.45 ☐ −0.5 (b) 3/8 ☐ 0.375 (c) 7/9 ☐ 0.78 [3]

• (a) −0.45 is closer to zero than −0.50, so −0.45 > −0.5.
• (b) 3/8 = 3 ÷ 8 = 0.375, so 3/8 = 0.375.
• (c) 7/9 = 0.7777…, and 0.7777… < 0.7800, so 7/9 < 0.78 (≠ would also be true, but < gives more information; read what the question asks).

Worked example 3 — (a) Estimate 3.6 × 1.25. (b) Work out 3.6 × 1.25 exactly. Show your working. [3]

• (a) 3.6 ≈ 4 and 1.25 ≈ 1, so the estimate is 4 × 1 = 4.
• (b) Ignore the points: 36 × 125. Split it: 36 × 100 = 3600 and 36 × 25 = 900, so 36 × 125 = 4500.
• Decimal places: 3.6 has 1, 1.25 has 2 → 3 in total. 4500 → 4.500.
• Answer: 4.5. Check: close to the estimate 4 ✓.

Common mistake: removing the zeros first (4500 → 45) and then counting 3 places to get 0.045. Count the decimal places before you remove the end zeros.

Worked example 4 — You are told that 23 × 47 = 1081. Use this to write down the answers to (a) 2.3 × 4.7 (b) 0.23 × 470 (c) 10.81 ÷ 2.3 [3]

• (a) 2.3 × 4.7: two decimal places in total → 10.81.
• (b) 0.23 × 470 = 0.23 × 47 × 10. 0.23 × 47 = 10.81, then × 10 = 108.1. (Check: 0.2 × 500 = 100 ✓)
• (c) Division is the inverse: since 2.3 × 4.7 = 10.81, then 10.81 ÷ 2.3 = 4.7.

Worked example 5 — Work out 7.56 ÷ 0.6. Show your working. [2]

• Estimate: 7.2 ÷ 0.6 = 72 ÷ 6 = 12, so the answer is about 12.
• × 10 both: 7.56 ÷ 0.6 = 75.6 ÷ 6.
• Short division: 6 into 7 = 1 r 1; 6 into 15 = 2 r 3; decimal point; 6 into 36 = 6. So 75.6 ÷ 6 = 12.6.
• Answer: 12.6. Check: 12.6 × 0.6 = 7.56 ✓

Common mistake: multiplying only the divisor by 10 (7.56 ÷ 6 = 1.26). You must change both numbers in the same way.

Worked example 6 — A shop has 9.3 kg of rice. It packs the rice into bags of 0.6 kg. (a) How many full bags can it make? (b) How much rice is left over? [3]

• (a) 9.3 ÷ 0.6 = 93 ÷ 6 = 15.5. Only full bags count, so the answer is 15 bags (do not round up to 16 — there is not enough rice for a 16th bag).
• (b) 15 bags use 15 × 0.6 = 9 kg. Left over: 9.3 − 9 = 0.3 kg.
• Exam technique: in a real-life question, always ask 'does my answer make sense?' — a decimal number of bags is not a final answer.

Practice

All pages in this course

Start here

• Stage 8 Maths: The Tests and How to Prepare

Number

• Order of Operations and Calculating with Integers
• Factors, Multiples, Primes, HCF and LCM
• Types of Number and Index Laws
• Squares, Cubes and Roots (Positive and Negative)
• Multiplying and Dividing by 0.1 and 0.01, and Significant Figures
• Fractions: Recurring Decimals, Mixed Numbers and Order of Operations
• Percentage Increase, Decrease and Absolute Change
• Ordering and Calculating with Decimals
• Ratio and Proportion

Algebra

• Algebraic Expressions: Simplifying, Expanding and Factorising
• Writing Expressions and Formulae, and Changing the Subject
• Equations and Inequalities
• Sequences and the nth Term
• Functions and Straight-line Graphs (y = mx + c)
• Reading and Interpreting Real-life Graphs

Geometry and measure

• Quadrilaterals and Regular Polygons
• Circumference of a Circle, and Miles and Kilometres
• Area of Parallelograms and Trapezia
• 3D Shapes: Euler's Formula and Plans and Elevations
• Volume of Prisms and Surface Area
• Angles in Triangles and on Parallel Lines
• Constructions with Ruler and Compasses
• Bearings and Midpoints
• Translation, Reflection, Rotation and Enlargement

Statistics and probability

• Data Collection and Sampling
• Representing Data
• Averages, Range and Comparing Distributions
• Probability: Complementary and Combined Events

Practice tests

• Stage 8 Maths — Practice Test 1
• Stage 8 Maths — Practice Test 2
• Stage 8 Maths — Practice Test 3