LESSON 09 / 100 | 9% COMPLETE | STAGE I — FOUNDATIONS
Stage I — Foundations
Decimals & Percentages
Three ways to say the same thing — and when to use each.
Section 01
Understanding Decimals
A decimal is a way of writing fractions whose denominators are powers of ten (10, 100, 1000, …). The decimal point separates the integer part (left) from the fractional part (right).
Place Value Chart — the number 4.7382
4
Ones
7
Tenths (1/10)
3
Hundredths (1/100)
8
Thousandths (1/1000)
2
Ten-thousandths
Reading a Decimal
Read the integer part normally, say “and” for the decimal point, then read the digits after as a whole number followed by the place-value name of the last digit.
4.7382 → “four and seven thousand three hundred eighty-two ten-thousandths”
Rounding Decimals
Look at the digit immediately to the right of your target place. If it is 5 or more, round up. If it is 4 or less, round down (keep as is).
Worked Example — Round 3.7461 to 2 decimal places
1
Target: hundredths place → the digit is 4
2
Look right: next digit is 6 ≥ 5 → round up
3
Answer: 3.75
Section 02
Converting Between Fractions and Decimals
Fraction → Decimal
Divide the numerator by the denominator.
1
3/8
2
3 ÷ 8 = 0.375
3
Answer: 0.375
Decimal → Fraction
Place digits over the matching power of 10, then simplify.
1
0.36 → 2 decimal places → over 100
2
36/100
3
GCF = 4 → 9/25
Recurring (Repeating) Decimals
Some fractions produce decimals that repeat forever. A dot (or bar) above the digit marks the repeating block.
Common Recurring Decimals
| Fraction | Decimal | Notation |
|---|---|---|
| 1/3 | 0.333… | 0.3̄ |
| 2/3 | 0.666… | 0.6̄ |
| 1/7 | 0.142857142857… | 0.1̄4̄2̄8̄5̄7̄ |
| 1/6 | 0.1666… | 0.16̄ |
Section 03
Understanding Percentages
Per cent means per hundred. A percentage is a fraction with a denominator of 100. The symbol % replaces “/100”.
n% = n/100 = n ÷ 100
Percentages are the universal language of comparison. A pay rise, a test score, a sale discount, and a tax rate are all statements about parts of 100 — algebra simply reveals the number hiding behind the symbol.
The Conversion Triangle
FRACTION
e.g. 3/4
e.g. 3/4
⇄
DECIMAL
e.g. 0.75
e.g. 0.75
⇄
PERCENT
e.g. 75%
e.g. 75%
| Conversion | Method | Example |
|---|---|---|
| Fraction → Percent | Divide, then multiply by 100 | 3/4 → 0.75 → 75% |
| Percent → Fraction | Write over 100, simplify | 45% → 45/100 → 9/20 |
| Decimal → Percent | Multiply by 100 (shift point right 2) | 0.032 → 3.2% |
| Percent → Decimal | Divide by 100 (shift point left 2) | 7.5% → 0.075 |
Common Mistake — Shifting the Point
Students often shift in the wrong direction. Remember: percent means “out of 100” — so going to a decimal means dividing by 100, which moves the point left. Going to a percent means multiplying by 100, which moves the point right.
Section 04
Percentage Calculations
Type 1 — Finding a Percentage of a Number
x% of y = (x/100) × y
Worked Example — Find 35% of 140
1
Convert: 35% = 0.35
2
Multiply: 0.35 × 140 = 49
3
Answer: 49
Type 2 — Expressing One Number as a Percentage of Another
Percentage = (part / whole) × 100
Worked Example — 18 out of 60 as a percentage
1
18 / 60 = 0.3
2
0.3 × 100 = 30
3
Answer: 30%
Type 3 — Percentage Increase & Decrease
Increase
New = Original × (1 + r)
r = rate as decimal
e.g. 20% increase → × 1.20
Decrease
New = Original × (1 − r)
e.g. 15% decrease → × 0.85
Worked Example — A jacket costs R800, reduced by 30%. Find the sale price.
1
Decrease multiplier: 1 − 0.30 = 0.70
2
800 × 0.70 = 560
3
Sale price: R560
Percentage Change Formula
% change = ((new − original) / original) × 100
Note — Positive vs Negative Result
A positive result is a percentage increase. A negative result is a percentage decrease. The sign tells the direction automatically.
Section 05
Benchmark Values to Memorise
| Fraction | Decimal | Percentage |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/4 | 0.25 | 25% |
| 3/4 | 0.75 | 75% |
| 1/3 | 0.333… | 33.3̄% |
| 2/3 | 0.666… | 66.6̄% |
| 1/5 | 0.2 | 20% |
| 1/8 | 0.125 | 12.5% |
| 1/10 | 0.1 | 10% |
Memorise the benchmark values the way a musician memorises scales. You will reach for them instinctively when algebra problems arrive as word problems dressed in “percent” clothing.
Exercises
Practice Set
1. Convert 7/8 to a decimal and to a percentage.
7 ÷ 8 = 0.875. 0.875 × 100 = 87.5%
2. Convert 0.048 to a fraction in simplest form.
0.048 = 48/1000. GCF(48, 1000) = 8. 48 ÷ 8 = 6, 1000 ÷ 8 = 125. Answer: 6/125
3. Find 23% of 350.
0.23 × 350 = 80.5
4. A student scored 54 out of 72. Express this as a percentage (to 1 d.p.).
(54 / 72) × 100 = 0.75 × 100 = 75%
5. A salary of R24 000 is increased by 8.5%. What is the new salary?
Multiplier: 1 + 0.085 = 1.085. 24 000 × 1.085 = R26 040
6. A laptop dropped in price from R9 800 to R7 840. Calculate the percentage decrease.
Change = 7 840 − 9 800 = −1 960. % change = (−1 960 / 9 800) × 100 = −20% (a 20% decrease)
7. Round 0.08461 to (a) 3 decimal places and (b) 2 significant figures.
(a) Look at 4th decimal: 6 ≥ 5 → round up. Answer: 0.085. (b) First 2 significant figures are 8 and 4; next digit is 6 → round up. Answer: 0.085 (same here — 8.5 × 10⁻²)
8. Challenge: A price is increased by 20%, then the new price is decreased by 20%. Is the final price the same as the original? Justify algebraically.
Let original = P. After 20% increase: 1.2P. After 20% decrease: 1.2P × 0.8 = 0.96P. 0.96P ≠ P. The final price is 4% less than the original. The operations do not cancel — this is a classic percentage trap.
Summary
Lesson Checklist
You Can Now
- Read and interpret decimal place values
- Round decimals to any specified number of places
- Convert fluently between fractions, decimals, and percentages
- Recognise and work with recurring (repeating) decimals
- Calculate a percentage of a number and express one quantity as a percentage of another
- Apply percentage increase and decrease using multipliers
- Calculate percentage change and interpret the sign of the result
Discover why the order in which you perform operations matters — and the single rule that makes every calculation unambiguous.