CDS Mathematics · Ratio, Proportion and Variation

Equal Ratios and Proportion Algebra

Set every equal ratio to k and substitute; for an equation between two fractions, cross-multiply and factor.

All Ratio, Proportion and Variation notes CDS Mathematics strategy

Why this matters

Fourteen PYQs. Putting each ratio equal to k turns an expression in several letters into one in k, which then cancels. The other move — cross-multiplying and factoring — usually ends in an 'either–or' answer.

Concept 1 of 2: Put each ratio equal to k

\dfrac ab = \dfrac cd = k

, then

a = kb

and

c = kd

. Every letter is now tied to one other, and any expression of the same degree in top and bottom loses

k

Definition

$\dfrac ab = \dfrac cd = k$ gives $a = kb$ , $c = kd$ .
$2a = 3b = 6c = k$ gives $a = \dfrac k2$ , $b = \dfrac k3$ , $c = \dfrac k6$ .
A ratio of homogeneous expressions of the same degree is fixed by the ratios; one that mixes degrees (like $\dfrac{3A^2 + 4B}{3A - 4B^2}$ ) is not.
$\dfrac pq = \dfrac qr = \dfrac rs = k$ gives $\dfrac ps = k^3$ .

Equal ratios

\dfrac ab = \dfrac cd = k \;\Rightarrow\; a = kb,\ c = kd

Worked example

\dfrac a3 = \dfrac b4 = \dfrac c5

, find

\dfrac{a^2 + b^2}{c^2}

and

\dfrac{a + b}{c}

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2023 · CDS (II) 2023 — Elementary Mathematics · Q8Moderate

Example 1 · Ratio, Proportion and Variation · Equal Ratios and Proportion Algebra

\dfrac{2a}{3} = \dfrac{4b}{5} = \dfrac{3c}{4}

, then what is the value of

\dfrac{18}{a}\sqrt{a^2 + c^2 - b^2}

Mixed degrees cannot be determined

An expression like

\dfrac{3A^2 + 4B}{3A - 4B^2}

changes with the actual size of

A

and

B

, not just their ratio. 'Cannot be determined' is then the right option.

Drill 8 more on put each ratio equal to k

Concept 2 of 2: Cross-multiply and factor

An equation between two fractions becomes a polynomial equation when you cross-multiply. Expand, cancel the common terms, and factor what is left — often into two brackets that give two alternatives.

Definition

$\dfrac{p + q}{q + r} = \dfrac{r + s}{s + p}$ leads to $(p - r)(p + q + r + s) = 0$ .
$(4a + 7b)(4c - 7d) = (4a - 7b)(4c + 7d)$ leads to $ad = bc$ , i.e. $\dfrac ab = \dfrac cd$ .
A quadratic in $\dfrac ab$ gives two possible ratios; keep both unless one is ruled out.
A statement like 'each of three expressions equals $t$ ' can force $t = 0$ after multiplying and adding.

Cross-multiplication

\dfrac PQ = \dfrac RS \iff PS = QR

Worked example

\dfrac{3a + 2b}{3a - 2b} = \dfrac{3c + 2d}{3c - 2d}

, show that

a : b = c : d

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (II) 2019 — Elementary Mathematics · Q38Moderate

Example 2 · Ratio, Proportion and Variation · Equal Ratios and Proportion Algebra

(4a + 7b)(4c - 7d) = (4a - 7b)(4c + 7d)

, then which one of the following is correct ?

Keep both signs

A quadratic in

\dfrac ab

gives two ratios, and

\dfrac{a + b}{a - b}

then takes two values of opposite sign. An option listing only the positive one is incomplete.

Drill 4 more on cross-multiply and factor

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Put each ratio equal to k
Equal ratios
$\dfrac ab = \dfrac cd = k \;\Rightarrow\; a = kb,\ c = kd$
Cross-multiply and factor
Cross-multiplication
$\dfrac PQ = \dfrac RS \iff PS = QR$

Watch out for (2)

Mixed degrees cannot be determined
→ Put each ratio equal to k
Keep both signs
→ Cross-multiply and factor

Test yourself on Ratio, Proportion and Variation

20 past CDS questions from this chapter, timed at 24 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Ratio, Proportion and Variation chapter test Drill all 14 questions