Proportion and Percentages

By the end of this lesson you will be able to:

1. Convert between fractions, decimals, and percentages
2. Find a percentage of a quantity
3. Express one quantity as a percentage of another
4. Solve direct proportion problems using the unitary method
5. Solve inverse proportion problems and tell them apart from direct proportion

Your Five Learning Objectives Today

By the end of this lesson you will:

1. Convert between fractions, decimals, and percentages
2. Find a percentage of a quantity
3. Express one quantity as a percentage of another
4. Solve direct proportion problems using the unitary method
5. Solve inverse proportion problems and distinguish them from direct

Naledi's Stall: 15 of What?

Naledi sells cloth at the market.

A sign reads: "15% discount on all fabric today."

Her customer asks: "15% — 15 of what?"

• Per cent means "out of 100"
• 15% means
• We know , so:

Four Ways to Convert Between Forms

Worked: All Four Conversion Directions

To percentage (×100):

From percentage (÷100):

Sanity check: 60% > 50%, so the decimal must be > 0.5. ✓

Check-In: Try Two Conversions Yourself

Work these in your exercise book.

1. Write as a percentage.

2. Write as a decimal.

(Hint: what operation takes you from fraction/% to %/decimal?)

Answers:

Common Equivalents — Know These

Memorise these — sanity-check anchors.

Chunk 2: Percentage of a Quantity

Naledi's customer: cloth marked 2,000 sh, discount 15%.

• 15% means
• "of" means multiply (fractions-01)

Guided: Find 25% of 3,200 sh

1. Write as fraction:
2. "Of" means multiply:

3. Multiply numerator × quantity, then ÷ 100:

Shortcut: 25% = ¼, so 3,200 ÷ 4 = 800 ✓

Reversing the Question: Part as Percentage

12 out of 40 pupils passed. What percentage?

Rule: part on top, whole on bottom — then × 100.

18 out of 24 books sold:

Check-In: Express as a Percentage

15 out of 60 pupils are absent.

What percentage is absent?

Answer:

Misconception Patrol: Three Common Errors

M1 drop /100: 15×2000 = 30,000 ✗ → 15/100×2000 = 300 ✓

M2 one place: 0.75 = 7.5% ✗ → 0.75×100 = 75% ✓

M3 flip: 40/12×100 = 333% ✗ → 12/40×100 = 30% ✓

Direct Proportion: What Stays the Same?

5 books cost 15,000 sh. How much for 8 books?

• More books → more money: this is direct proportion
• The cost per book (the rate) stays constant

In direct proportion: as one quantity increases, the other increases at the same rate.

The Unitary Method: Find One First

Step 1: Find the value of ONE unit.

Step 2: Scale to the number wanted.

The Unitary Method: Bar Model

The middle bar — the unit — is the pivot. Divide to it, then multiply away from it.

Worked Example: Fuel and Distance

6 litres → 90 km. How far on 8 litres?

• More fuel → more distance: direct proportion

Same two steps — different context.

Check-In: Solve a Direct Proportion

12 pens cost 6,000 sh. How much for 9 pens?

1. Direct or inverse? How do you know?
2. Find the cost of 1 pen.
3. Find the cost of 9 pens.

Answer: Direct (more pens, more cost). 1 pen = 500 sh. 9 pens = 4,500 sh.

Chunk 4: The Deciding Question

4 workers dig a trench in 6 days.

8 workers dig instead — more or less time?

• More workers → finish faster → less time
• "More → less" is the signature of inverse proportion

Inverse Proportion: The Product Stays Constant

Total work is fixed: worker-days.

Changing workers? Days must adjust to keep the product at 24.

Inverse: product stays constant. Direct: rate stays constant.

Worked: Workers and the Trench

4 workers, 6 days. How long for 8 workers?

Step 1: worker-days (constant product)

Step 2: days

Sense check: more workers → less time. 3 < 6 ✓

Worked: Taps Filling a Tank

3 taps, 8 hours. How long for 4 taps?

More taps → less time: inverse proportion

Step 1: tap-hours

Step 2: hours

Sense check: 4 taps > 3 — time must be less than 8. 6 < 8 ✓

Check-In: Solve an Inverse Proportion

2 painters take 12 days to paint a building.

How long for 4 painters?

1. More painters — more or less time?
2. Find the constant product.
3. Divide to find the answer.

Answer: Less time. painter-days. days.

Direct vs Inverse: Side by Side

Unscaffolded Practice: Three Integration Tasks

1. Write as a percentage.

2. Naledi earns 4,500 sh. She saves 30% of it. How much?

3. 6 workers build a road section in 8 days. How long for 9 workers?

No scaffolding — apply everything from this lesson.

Key Takeaways: Five Rules to Remember

✓ % = out of 100
✓ ×100 to %; ÷100 from % — two places
✓ "of" means ×
✓ Part ÷ whole × 100
✓ Direct: rate fixed. Inverse: product fixed.

M1: drop /100 | M2: one place | M3: flip | M4: wrong direction