Decimals and Percentages

Decimals and percentages are ways of writing parts of a whole — just like fractions. In this lesson you will learn how they work, how to calculate with them, and how fractions, decimals, and percentages are all connected.

---

Understanding Decimals

A decimal number uses a decimal point to separate the whole number part from the fractional part.

Ones	.	Tenths	Hundredths	Thousandths
3	.	4	5	6

This number is 3.456:

• 3 is the ones part
• 4 tenths = 4/10
• 5 hundredths = 5/100
• 6 thousandths = 6/1000

So 3.456 = 3 + 4/10 + 5/100 + 6/1000.

Comparing decimals: Always compare digit by digit from left to right.

• 0.42 vs 0.4: 0.42 has 4 tenths and 2 hundredths; 0.4 has only 4 tenths. So 0.42 > 0.4.
• Do not be fooled by length: 0.9 > 0.35 (9 tenths is more than 3 tenths).

---

Multiplying and Dividing by 10, 100, 1000

When you multiply by 10, each digit moves one place to the left (the number gets 10 times bigger).

• 3.45 x 10 = 34.5
• 3.45 x 100 = 345
• 3.45 x 1000 = 3,450

When you divide by 10, each digit moves one place to the right (the number gets 10 times smaller).

• 670 / 10 = 67
• 670 / 100 = 6.7
• 670 / 1000 = 0.67

Important: The decimal point does NOT move — the digits do. But it is often easier to think of the decimal point "moving".

---

Adding and Subtracting Decimals

Line up the decimal points, then add or subtract just like whole numbers.

Example: 4.72 + 3.8

4.72

• 3.80

---

8.52

Pad with a zero so both numbers have the same number of decimal places, then add column by column.

Example: 6.4 - 2.75

6.40

• 2.75

---

3.65

---

Rounding Decimals

To round a decimal, look at the digit one place past where you are rounding to:

• If it is 5 or more, round up.
• If it is 4 or less, round down.

Rounding to 1 decimal place (1 d.p.):

• 3.47 → look at the hundredths digit (7). 7 ≥ 5, so round up → 3.5
• 6.82 → look at the hundredths digit (2). 2 < 5, so round down → 6.8
• 9.95 → look at the hundredths digit (5). 5 ≥ 5, so round up → 10.0 (write the trailing zero)

Rounding to 2 decimal places (2 d.p.):

• 4.567 → look at the thousandths digit (7). 7 ≥ 5, so round up → 4.57
• 2.341 → look at the thousandths digit (1). 1 < 5, so round down → 2.34

---

Ordering and Comparing Decimals

To compare decimals, write them with the same number of decimal places (pad with zeros), then compare digit by digit from the left.

Example: Put in order from smallest to largest: 0.6, 0.35, 0.4, 0.09

Rewrite with 2 decimal places: 0.60, 0.35, 0.40, 0.09. Compare tenths first, then hundredths:

• 0.09 (0 tenths) → 0.35 (3 tenths) → 0.40 (4 tenths) → 0.60 (6 tenths)

Answer: 0.09, 0.35, 0.4, 0.6

Example: Which is greater, 0.8 or 0.75? 0.8 = 0.80. Compare: 0.80 vs 0.75 → 80 hundredths > 75 hundredths → 0.8 is greater.

---

Multiplying and Dividing Decimals

Multiplying a Decimal by a Whole Number

Remove the decimal, multiply, then put the decimal point back.

Example: 1.4 × 3

• Ignore the decimal: 14 × 3 = 42.
• 1.4 has 1 decimal place, so the answer has 1 decimal place: 4.2

Example: 0.36 × 5

• Ignore the decimal: 36 × 5 = 180.
• 0.36 has 2 decimal places, so the answer has 2 decimal places: 1.80 = 1.8

Dividing a Decimal by a Whole Number

Use short division, carrying the decimal point straight up.

Example: 6.4 ÷ 4

   1 . 6
4 | 6 . 4
6 ÷ 4 = 1 remainder 2; bring down 4 → 24; 24 ÷ 4 = 6
Answer: **1.6**

Example: 8.25 ÷ 5

   1 . 6 5
5 | 8 . 2 5
8 ÷ 5 = 1 r3; 32 ÷ 5 = 6 r2; 25 ÷ 5 = 5
Answer: **1.65**

---

Understanding Percentages

Percent means "out of 100". The symbol is %.

• 50% means 50 out of 100 = 50/100 = 1/2.
• 25% means 25 out of 100 = 25/100 = 1/4.
• 10% means 10 out of 100 = 10/100 = 1/10.

Finding a percentage of an amount:

The easiest way is to find 10% first (divide by 10), then build up.

• 10% of 80 = 80 / 10 = 8
• 20% of 80 = 2 x 8 = 16
• 5% of 80 = half of 10% = 4
• 15% of 80 = 10% + 5% = 8 + 4 = 12
• 1% of 80 = 80 / 100 = 0.8
• 35% of 80 = 30% + 5% = 24 + 4 = 28

---

Conversion Triangle: Fractions, Decimals, Percentages

Every value can be written three ways. Use this triangle to move between forms:

flowchart TD
    F["Fraction<br/>e.g. 3/4"]
    D["Decimal<br/>e.g. 0.75"]
    P["Percentage<br/>e.g. 75%"]
    F -->|divide top by bottom| D
    D -->|x 100, add %| P
    P -->|/ 100, drop %| D
    D -->|write over 10/100/1000<br/>then simplify| F
    F -->|x 100/denominator| P
    P -->|over 100<br/>then simplify| F

---

Fraction, Decimal and Percentage Equivalences

These are the key equivalences you must know:

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
2/5	0.4	40%
1/10	0.1	10%
1/100	0.01	1%

Converting a fraction to a decimal: divide the numerator by the denominator.

• 3/8 = 3 / 8 = 0.375

Converting a decimal to a percentage: multiply by 100.

• 0.65 = 0.65 x 100 = 65%

Converting a percentage to a fraction: write as a fraction over 100 and simplify.

• 35% = 35/100 = 7/20

---

Percentage Increase and Decrease

Percentage increase: find the percentage and add it on.

• Increase 80 by 25%: 25% of 80 = 20. New value: 80 + 20 = 100.

Percentage decrease: find the percentage and take it off.

• Decrease 60 by 30%: 30% of 60 = 18. New value: 60 - 18 = 42.

---

Practice

1. What is the value of the digit 7 in 3.074? 7 hundredths (7/100)
2. 4.56 x 100 = ? 456
3. 3.7 + 1.85 = ? 5.55
4. What is 35% of 200? 70
5. Write 0.6 as a fraction in its simplest form. 3/5
6. Write 3/8 as a decimal. 0.375
7. Increase 120 by 15%. 138
8. Which is greater: 0.7, 3/4, or 68%? 3/4 (= 0.75 = 75%)

---

Worked Example: Sale Price at the Bookshop

A bookshop has a "20% off" sale. A pupil wants to buy a book originally priced at £18.50 and a workbook priced at £6.00. They have £20.00 saved. Can they afford both books in the sale?

Step 1 — Find 20% of each price. The easiest way is to find 10% first, then double it.

10% of £18.50 = £1.85    (divide by 10 — move the decimal point one place left)
20% of £18.50 = £3.70    (double 10%)

10% of £6.00 = £0.60
20% of £6.00 = £1.20

Step 2 — Subtract the discount from each original price.

Book sale price:     £18.50 - £3.70 = **£14.80**
Workbook sale price: £6.00  - £1.20 = **£4.80**

Use column subtraction with decimal points lined up.

£18.50

• £ 3.70

---

£14.80

Step 3 — Add the two sale prices.

£14.80 + £4.80 = **£19.60**

Step 4 — Compare with available money. £19.60 < £20.00, so YES, the pupil can afford both books. They will have £0.40 (40p) change.

Step 5 — A different shop offers 25% off. Would that be cheaper than 20% off?

25% of £18.50 = £18.50 / 4 = **£4.625** ≈ **£4.63** (rounded to 2 d.p.)
Sale price = £18.50 - £4.63 = **£13.87**