Ratio compares quantities of the same unit —a:bmeansaxandbxfor somex. Proportion says two ratios are equal (a/b = c/d). The art is findingxand chaining ratios without solving twice.
Comparison of two same-unit quantities — how many times one contains the other.
a : b or a/ba, Consequent = b in
a:b
x: ratio 3:4 does NOT mean numbers are 3 and
4 — they are 3x and 4x. Finding x is usually the
solve.
Equation that two ratios are equal: a : b = c : d or
a/b = c/d
Cross-multiplication (extreme = means):
a × d = b × c
Given separate ratios A:B and B:C, get A:B:C or
A:C.
If A:B = x:y and B:C = m:n, make the common term
B equal:
A : B : C = (x×m) : (y×m) : (y×n)
Alternatively for A/C only: multiply fractions (A/B)×(B/C)=A/C →
(x/y)×(m/n)=xm/yn.
A:B:C = (2×4):(3×4):(3×5)=8:12:15. Check: B is 12 in both now. If A:B=3:2 and
B:C=3:4, make B equal via LCM 6 → A:B=9:6, B:C=6:8 → A:B:C=9:6:8.
| Type | Relation | Formula | Example |
|---|---|---|---|
| Direct | Both ↑ together | y = kx or x₁/y₁ = x₂/y₂ |
Distance ∝ Time (fixed speed); circumference ∝ radius |
| Inverse | One ↑, other ↓ | xy = k or x₁y₁ = x₂y₂ |
Time ∝ 1/workers; speed ∝ 1/time (fixed distance) |
Most common advanced pattern — ratio changes after adding/subtracting a constant.
Scenario: ratio is a:b, after +N to numerator and
−M to denominator, new ratio is c:d.
ax and bx.(ax + N)/(bx − M) = c/dd(ax + N) = c(bx − M), solve for x.ax, bx.(3x+5)/(4x−6)=5/3 → 3(3x+5)=5(4x−6) →
9x+15=20x−30 → 11x=45 → x=45/11 → originals
135/11 and 180/11. (Numbers fractional here indicate the constants
were large relative to ratio — check problem statement for integer expectation.)
| Dimension | What scales | Ratio |
|---|---|---|
| 1D | Length, perimeter, radius | a : b |
| 2D | Area | a² : b² — square it |
| 3D | Volume | a³ : b³ — cube it |
Example: Two squares sides 2:3 → perimeters 2:3, areas 4:9, if they were cubes volumes 8:27. Doubling a side quadruples area and octuples volume.
Mixture X A:B = p:q, Mixture Y A:B = r:s, mix in equal quantities:
A=p/(p+q), B=q/(p+q), Y:
A=r/(r+s), B=s/(r+s)
2/5=0.4 and
3/7≈0.4286 → combined A = 0.8286, B =
0.6+0.5714=1.1714 → ratio ≈ 0.8286:1.1714 = 29:41 after scaling. For
equal mixing, you can also do (p/(p+q)+r/(r+s)) : (q/(p+q)+s/(r+s)).
Focus on the component not being added (e.g., milk when adding water):
Final = Initial × (1 − Replacement/Total)ⁿ — same as Module 10.
Profit Ratio = (Investment A × Time A) : (Investment B × Time B)
— full details in Module 09. If time same → C₁:C₂; if investment same →
T₁:T₂.
| Symbol | Meaning |
|---|---|
x, k |
Common multiplier / constant of proportionality |
a, b, c, d |
Terms of ratio |
N, M |
Quantities added/subtracted |
Ratio 3:4 means numbers are?
a:b = c:d → ?
A:B=2:3, B:C=4:5 → A:B:C?
Direct proportion: y = ?
Sides ratio a:b → area ratio?
Before-after: ratio a:b → (ax+N)/(bx−M)=c/d → next step?
Mix (2:3) and (3:4) equally — method?
a:b ⇒ ax, bx (x is multiplier) ; a:b=c:d ⇒ ad=bc
A:B=x:y, B:C=m:n ⇒ A:B:C = xm : ym : yn (N / zig-zag)
A/C ⇒ (x/y)×(m/n)
Direct ⇒ y=kx ; Inverse ⇒ xy=k
a:b → (ax+N)/(bx−M)=c/d → cross → solve x
1D a:b ; 2D a²:b² ; 3D a³:b³
Mix ⇒ p/(p+q) fractions → add ; Replace ⇒ Initial×(1−x/V)ⁿ
Primary source: none pinned yet — drop your preferred video/book resource into RESOURCES.md and it will be linked here. Ask me anything that's unclear.
Questions? Ask your agent — you can follow up on any concept, quiz answer, or get extra practice problems tuned to this module.