1 Ratios & Rates
Ratios
A ratio compares two quantities of the same type. Written as a:b, a/b, or "a to b".
- A class has 12 girls and 18 boys → ratio = 12:18 = 2:3 (simplified)
- To simplify, divide both numbers by their GCF
- Order matters: 2:3 is not the same as 3:2
Rates
A rate compares two quantities of different types (different units).
Rate
quantity A per quantity B (e.g. km/h, $/kg)
Unit rate
rate with denominator = 1 (e.g. 80 km per 1 hour)
✏️A car travels 240 km in 3 hours. Unit rate = 240 ÷ 3 = 80 km/h
Solving Proportions
A proportion says two ratios are equal: a/b = c/d. Cross-multiply to solve.
Cross-multiplication
a/b = c/d → a × d = b × c
✏️
Solve: 3/4 = x/20
3 × 20 = 4 × x
60 = 4x
x = 15
3 × 20 = 4 × x
60 = 4x
x = 15
2 Percentages
The Three Percentage Problems
Find the part
part = (%) / 100 × whole
Find the percent
% = (part / whole) × 100
Find the whole
whole = part / (% / 100)
Percent Increase and Decrease
Percent change
% change = ((new − old) / old) × 100
New value (increase)
new = old × (1 + rate) e.g. +20% → × 1.20
New value (decrease)
new = old × (1 − rate) e.g. −15% → × 0.85
✏️
A shirt costs $40, discounted 25%. New price?
new = 40 × (1 − 0.25) = 40 × 0.75 = $30
Price went from $50 to $65. Percent increase?
% = ((65 − 50) / 50) × 100 = (15/50) × 100 = 30%
new = 40 × (1 − 0.25) = 40 × 0.75 = $30
Price went from $50 to $65. Percent increase?
% = ((65 − 50) / 50) × 100 = (15/50) × 100 = 30%
💡Tax, tip, and discounts are all percent increase/decrease problems. Identify the original value first.
3 Direct Proportion
Two quantities are in direct proportion when one increases, the other increases by the same factor — their ratio stays constant.
General form
y = kx (k is the constant of proportionality)
Finding k
k = y / x (same for every point on the line)
Recognising Direct Proportion
- Graph passes through the origin (0, 0)
- Graph is a straight line
- The ratio y/x is constant for all points
- If x doubles, y doubles; if x halves, y halves
✏️
5 apples cost $3.50. How much do 8 apples cost?
k = 3.50 / 5 = 0.70 (cost per apple)
y = 0.70 × 8 = $5.60
k = 3.50 / 5 = 0.70 (cost per apple)
y = 0.70 × 8 = $5.60
4 Inverse Proportion
Two quantities are in inverse proportion when one increases, the other decreases so that their product stays constant.
General form
y = k / x (or xy = k)
Finding k
k = x × y (same for every point)
Recognising Inverse Proportion
- Graph is a curve (hyperbola), never passes through origin
- As x increases, y decreases; as x decreases, y increases
- The product x × y is constant for all points
- If x doubles, y halves; if x triples, y becomes one-third
✏️
4 workers take 6 days to finish a job. How long for 8 workers?
k = 4 × 6 = 24 (total worker-days)
y = 24 / 8 = 3 days
k = 4 × 6 = 24 (total worker-days)
y = 24 / 8 = 3 days
| Direct Proportion | Inverse Proportion | |
|---|---|---|
| Formula | y = kx | y = k/x |
| Constant | y/x = k | xy = k |
| Graph | Straight line through origin | Hyperbola curve |
| If x doubles | y doubles | y halves |
5 Scale & Similar Figures
Scale Factors
Scale factor
k = image length / original length
Scale on a map
1:n means 1 cm on map = n cm in reality
Similar Figures
Two figures are similar if they have the same shape (angles equal) but possibly different sizes. Corresponding sides are proportional.
✏️
Two similar triangles: sides 3, 4, 5 and 6, ?, ?
Scale factor = 6/3 = 2
Missing sides: 4 × 2 = 8 and 5 × 2 = 10
Scale factor = 6/3 = 2
Missing sides: 4 × 2 = 8 and 5 × 2 = 10
🔑In similar figures: angles are equal (not scaled). Only side lengths scale.
Map Reading
✏️
Map scale 1:50 000. Distance on map = 3.2 cm. Real distance?
Real distance = 3.2 × 50 000 = 160 000 cm = 1.6 km
Real distance = 3.2 × 50 000 = 160 000 cm = 1.6 km
6 Common Mistakes to Avoid
| Mistake | What to do instead |
|---|---|
| Confusing direct and inverse proportion | Ask: "If one goes up, does the other go up (direct) or down (inverse)?" |
| Cross-multiplying incorrectly | In a/b = c/d, multiply diagonals: ad = bc, not ac = bd. |
| Forgetting to convert map scale units | Map scale 1:50 000 → multiply by 50 000 to get real distance in same units. |
| Applying percent increase twice | If price is $100 and increases 20%: new = 100 × 1.20 = $120, not $100 + 20 = $120 only in this case — always use the formula. |
| Angles scale in similar figures | WRONG — angles stay equal. Only side lengths change with the scale factor. |
| Mixing up y/x = k and xy = k | Direct proportion: y/x = k. Inverse proportion: xy = k. Check by graphing. |