IGCSE / O Level Mathematics · Topic 1 Number · Chapter 4 · pp. 30–42

Integers, fractions, decimals and percentages

Fractions, decimals and percentages are three ways of writing the same thing. , 0.5 and 50% are one number in three outfits. This chapter moves between them, calculates with them without a calculator, and finishes with the neat trick that turns a recurring decimal back into a fraction.

Learning objectives

By the end of this chapter you can:

Find a fraction of an amount, and switch between mixed numbers and improper fractions.
Convert between fractions, decimals and percentages.
Do long multiplication, short division and long division by hand.
Add, subtract, multiply and divide fractions and mixed numbers, giving answers in their simplest form.
Change a recurring decimal into a fraction. (Extended)

Book exercises: 4.1 to 4.11 and Student assessment 1 (pp. 30–42).

One number, three forms. Every arrow can be followed both ways.

Next: fractions →

Fractions

🔑 Key ideas

In , the numerator 3 counts the parts taken; the denominator 10 says how many equal parts make a whole.
Proper: top smaller (). Improper or vulgar: top bigger (). Mixed number: whole + proper ().
To find of an amount: divide by the bottom, multiply by the top.

A bar model: split 35 into 5 equal parts of 7, then take 3 of them.

📘 Example 1 · Find of 35

Divide by the denominator to find one part: 35 ÷ 5 = 7.
Multiply by the numerator to take 3 parts: 7 × 3 = 21.

📘 Example 2 · Change to an improper fraction

Each whole is , so 3 wholes are (whole × denominator).
Add the fraction part: + = .

📘 Example 3 · Change to a mixed number

Divide the top by the bottom: 27 ÷ 4 = 6 remainder 3.
The quotient is the whole number and the remainder goes over the denominator: .

✏️ Try it

of 240
Write as an improper fraction.
Write as a mixed number.

Answers

1. 240 ÷ 8 × 7 = 210 · 2. · 3.

Next: decimals and percentages →

Decimals and percentages

🔑 Key ideas

Each decimal place is a fraction: tenths, hundredths, thousandths.
Per cent means “out of 100”: 29% = .
Fraction → decimal: divide the top by the bottom. Decimal → fraction: write it over 10, 100 or 1000, then simplify.

Units	.	tenths	hundredths	thousandths
3	.	2	7
0	.	0	3	8

3.27 = 3 + + = 3 + , and 0.038 = .

📘 Example 4 · Change to a percentage

Find what turns the denominator into 100: 5 × 20 = 100.
Multiply top and bottom by 20: = .
Out of 100 is per cent: 60%.

📘 Example 5 · Change to a decimal

  0. 6 2 5
8)5. 0²0⁴0

Divide 5 by 8. 8 doesn't go into 5, so write 0. and carry 5 into the tenths: 50 ÷ 8 = 6 r 2.
Carry the 2: 20 ÷ 8 = 2 r 4.
Carry the 4: 40 ÷ 8 = 5 exactly. Answer: 0.625.

📘 Example 6 · Change 2.325 to a mixed number

The last digit is in the thousandths, so 0.325 = .
Simplify by the HCF, 25: = .
Answer: .

Fraction
Decimal	0.1	0.2	0.25	0.5	0.75	0.333…
Percent	10%	20%	25%	50%	75%	33⅓%

💡 Learn this row of equivalents by heart. It makes estimating and checking much quicker.

✏️ Try it

as a percentage.
as a decimal.
0.825 as a fraction in its simplest form.

Answers

1. = 68% · 2. 0.4375 · 3. =

Next: the four rules →

The four rules by hand

🔑 Key ideas

Long multiplication: multiply by each digit in turn, remembering its place value, then add the rows.
Division: carry on past the decimal point with zeros instead of leaving a remainder.
Mixed operations follow BIDMAS (Chapter 3).

📘 Example 7 · 184 × 37

    1 8 4
  ×   3 7
  -------
  1 2 8 8   ← 184 × 7
+ 5 5 2 0   ← 184 × 30
  -------
  6 8 0 8

Multiply 184 by the 7 units: 1288.
Multiply 184 by the 3 tens. It is really × 30, so write a 0 first: 5520.
Add the two rows: 1288 + 5520 = 6808.
Estimate to check: 200 × 40 = 8000. Close enough. ✔

📘 Example 8 · 7184 ÷ 23 to 1 d.p.

     3 1 2 . 3 4
23 ) 7 1 8 4 . 0 0
     6 9
       2 8
       2 3
         5 4
         4 6
           8 0
           6 9
           1 1 0
             9 2

23 into 71 goes 3 times (69). Remainder 2; bring down 8 → 28.
23 into 28 goes once. Remainder 5; bring down 4 → 54. 23 into 54 goes twice, remainder 8.
Add a decimal point and zeros: 80 → 3 times, remainder 11; 110 → 4 times.
312.34… to 1 d.p. is 312.3. (Work one place further than you need, to round.)

✏️ Try it, without a calculator

785 × 38
7892 ÷ 15 to 1 d.p.
How many 35-cent stamps can you buy with $10?

Answers

1. 29 830 · 2. 526.1 · 3. 1000 ÷ 35 = 28.57…, so 28 stamps (you can't buy part of one)

Next: calculating with fractions →

Calculating with fractions

🔑 Key ideas

Simplify by dividing top and bottom by their HCF.
Add or subtract: make the denominators the same first.
Multiply: top × top, bottom × bottom. Divide: flip the second fraction and multiply.
Change mixed numbers to improper fractions before multiplying or dividing.

Equivalent fractions cover the same amount. To add and , cut both into sixths: + = .

📘 Example 9 · −

Find a common denominator: the LCM of 5 and 3 is 15.
Rewrite each fraction: = and = .
Subtract the numerators: = .

📘 Example 10 · −

Change to improper fractions: − .
The denominators already match: = .
Simplify and write as a mixed number: = .

📘 Example 11 · ÷

Change to an improper fraction: ÷ .
Flip the second fraction and multiply: × .
Top × top and bottom × bottom: = .

🧪 Try it: fraction calculator

Write fractions as 3/4 and mixed numbers as 2 1/3.

First Second

✏️ Try it

Answers

1. = · 2. · 3. 2

Next: recurring decimals →

Recurring decimals Extended

🔑 Key ideas

A recurring decimal repeats for ever. The book marks the repeating digits with dots; here we write 0.4545…
Every recurring decimal is rational, so it can be written as a fraction.
The trick: multiply by 10, 100 or 1000 so the repeating parts line up, then subtract.

The recurring-decimal method as a flowchart, with 0.4545… worked along the bottom.

📘 Example 12 · Convert 0.444… to a fraction

Let x = 0.444 444…
One digit repeats, so multiply by 10: 10x = 4.444 444…
Subtract: 10x − x = 4.444… − 0.444…, so 9x = 4.
Divide: x = .

📘 Example 13 · Convert 0.0313131… to a fraction

Let x = 0.031 313 1…. Two digits (31) repeat, so multiply by 100: 100x = 3.131 313…
Subtract: 100x − x = 3.131 31… − 0.031 31… gives 99x = 3.1.
Clear the decimal by multiplying both sides by 10: 990x = 31.
Divide: x = .

✏️ Try it

0.777… as a fraction
0.4242… as a fraction in its simplest form
0.0555… as a fraction

Answers

1. · 2. = · 3. 10x = 0.555…, 100x = 5.555…, 90x = 5, x =

Next: summary and quiz →

Chapter summary

A fraction of an amount: ÷ the bottom, × the top. Mixed ↔ improper: whole × bottom + top, or divide with a remainder.
Fraction → decimal: top ÷ bottom. Decimal → fraction: over 10, 100 or 1000, then simplify. Percent means out of 100.
Long multiplication: one row per digit, with place-value zeros. Long division: continue past the point instead of leaving a remainder.
Add/subtract with a common denominator; multiply straight across; divide by flipping and multiplying.
Recurring decimal → fraction: let x = it, multiply by 10ᵏ, subtract, divide.

📄 Practice questions

The book's Exercises 4.1 to 4.11 and Student assessment 1 are in the printable assignment for this chapter.

Mini-quiz

1. What is 3/5 of 35?

2. What is 3 5/8 as an improper fraction?

3. What is 3/5 as a percentage?

4. What is 5/8 as a decimal?

5. What is 4/5 − 1/3?

6. What is 3/8 ÷ 3/4?

7. What is 0.45 as a fraction in its simplest form?

8. What is 0.444… (4 recurring) as a fraction?

Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 4 (Integers, fractions, decimals and percentages). Explanations, the Try it answers, the tools and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.